EDUCAIDE SOFTWARE
Algebra I
Database Module
an add-on to Acces
“The Teacher’s Database”
Copyright (c) 1989, 1994 by EAS EducAide Software, Inc. All rights reserved.
About this Module
Title:
Algebra I
Publisher : EducAide Software
Database code :
ALG
Last updated : 22 Jan. 2001
Catalog version:
2.6
License, Copyright Notice
This database module, or item bank, is an add-on to Acces, “The Teacher’s Database.” Both
are produced and copyrighted by EducAide Software [Acces (c) 1992–2001, items (c) 1989–1994].
All rights are reserved. Users of Acces and this database module are subject to the terms and
conditions set forth by EducAide Software.
For your convenience, the contents of this module are shown in a problem catalog. The catalog may
be available to you in printed or electronic form, as a PDF or Adobe Acrobattm file. Regardless of
its form, you must treat the catalog as a part of Acces, not as a separate product. For licensing
purposes, it is considered software documentation, and its use is governed by the same license as
Acces. Here are the main points:
You are not allowed to transfer the PDF file by any electronic or other means, not even to other
licensed end-users of Acces. The only legitimate way to put the file on a computer is through
Acces’ Setup program.
You are not allowed to put the PDF file on a Web site or otherwise make it available for
download through a computer network. The only exception to this rule is if the file is placed on
a network filer server in accordance with your software license (which means that only licensed
end-users have access to it).
You may view or print the PDF file using Adobe Acrobattm or Acrobat Readertm , in accordance
with Adobe’s software license. Very important: If you produce a printed catalog, or hard copy,
it must be for your own personal use.
You must treat a printed catalog like a book. This is true even if you print it from the PDF
file. The catalog is copyrighted by EducAide Software and it may not be reproduced in printed
or electronic form without permission.
You may use the items in the catalog only in accordance with your software license, regardless
of whether you use Acces to reproduce the items. In particular, you may not convert, transfer, or
enter the items into any other data base or text retrieval system, nor use them for any commercial
purpose.
Features and Specifications
18,628 problems
171 categories (topics)
105 pictures
Requires 3.1 MB of disk space
Covers real numbers, simplifying and evaluating expressions, ratio and proportion, polynomials,
factoring, first and second degree equations, rational and radical equations, inequalities, coordinate
geometry and graphing, and systems of equations.
All problems are free response; they are intended for regular classroom instruction and testing in
grades 8–11; they may also be useful for review or remediation at other levels.
Credits
Project manager :
Dan Levin
Contributors : Stuart Kumaishi, Gale Bach, Michelle Openshaw, Arne Lim, Natalie Docktor,
Kathi Dominguez, Steve Apfelberg
Hints for Use
Selecting Problems in Acces
The code for this database module is ALG. To select problems from the catalog in Acces:
1. Type the three-character database
code in the first column.
2. Type the two-letter category code
(shown at the top of each catalog
page) in the second column.
ALG
AB
7
3. Type the desired problem number in
the third column.
4. You do not need to fill in the other
columns. They are for customizing
your document. To learn more about
them, consult your Reference Manual.
5. To see how the problem will appear
in your document, choose Utilities >
Preview.
Directions
Every problem in the database module has been assigned a default direction, which is shown at
the beginning of the category in the printed problem catalog. These can be overridden by typing
a new number in Acces’ Directions column. You can learn the meaning of each number by placing
the cursor in the Directions column and choosing Utilities > List Choices.
If more than one direction is shown at the beginning of a category, the first is the default, and the
others are suggested alternatives. See the Reference Manual for information about how to write
your own directions.
Database Organization
Problems in this module are organized in groups of two or four “clones”. In other words, every
two or four problems in a row are similar—they cover the same topic and are of the same level of
difficulty. This makes it very easy to produce multiple versions of a test or quiz, review worksheets,
benchmarks, etc.
Generally speaking, level of difficulty increases slowly from the beginning of a category to the end.
Difficulty is subjective, however, so you should always review the problems you select and make
sure they meet your own criteria.
One way to take advantage of both the grouping and gradual increase in difficulty is to use Acces’
“Auto Advance” and “Select by” features. These allow you to choose a regular series of problems,
such as “every other odd”. Refer to your Reference Manual for more information about these
features.
Other Database Modules
Here is a list of modules that are available from EducAide Software, as of April, 2002.
Newer modules and updates may be available at the time you are reading this. Please visit
www.educaide.com for a current list. Also, note that textbook-aligned modules are available from
CORD Communications (CCI) and Pearson Education Canada. For information about them,
please contact the publisher.
Code
Title
PRE
ALG
GEO
TRI
MMA
APC
SAT
CA1
CM1
CM2
CM3
NC1
NC2
NC3
NC4
NC5
NY1
NY2
NY5
NY6
NY7
NYM
OH1
TX2
TX3
TX4
TX5
TX6
MMF
T2S
T3S
TX6
MCC
MCH
NSM
BCC
NCC
WCC
UNC
IEC
ISC
Pre-Algebra
Algebra I
Geometry
Algebra II/Trigonometry
Mid-level Math Assessment
AP Calculus
SAT Math Prep.
Calif. Math Stds. 6–7, Alg
Canadian Math 11–12
Canadian Math 8–10
Canadian Math 4–7
NC Math Stds. 6–12, calculus
NC Math Testlets 3–5
NC Algebra I
NC Reading 3–8
NC Math Testlets 6–8, Alg
NY Regents Math (Course I–III)
NY Regents English
NY Regents Biology
NY Regents Chemistry
NY Regents Physics
NY Math 7–8, Math A
Ohio Math Proficiency
Texas Elem. Math (TAAS)
Texas Sec. Math/Algebra EOC
Texas Elem. Reading
Texas Sec. Reading
Texas Elem. Math (TEKS)
French Translation of MMA
Spanish Translation of TX2
Spanish Translation of TX3
Spanish Translation of TX6
MATHCOUNTS Competitions
MATHCOUNTS Handbooks
North Suburban Math League
British Columbia Math Contests
NC State Math Contests
West. Carolina Math Contests
UNC Charlotte Math Contests
Illinois Math Contests 3–8
Illinois Math Contests 9–12
Problems Categories Pictures
15018
18628
5104
15955
5000
1767
2144
3154
6382
5824
3150
1749
1316
2126
929
1713
5797
3589
4086
3257
2054
3242
2184
1945
4012
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1513
3004
5000
1945
4012
3004
6654
5139
2401
1571
2910
3047
1156
1928
4759
164
171
139
161
112
68
60
51
107
66
36
103
24
52
104
33
56
22
56
130
25
79
19
52
53
54
34
52
112
52
53
52
51
51
42
94
50
38
50
26
84
334
105
860
325
545
118
388
585
547
820
873
260
925
600
85
635
968
—
1112
291
1218
610
180
860
610
32
15
1273
545
860
610
1273
705
548
413
357
481
544
139
438
576
Type
free-response
free-response
free-response
free-response
multiple-choice
multiple-choice
multiple-choice
multiple-choice
various
various
various
various
various
multiple-choice
various
various
various
various
multiple-choice
multiple-choice
multiple-choice
various
multiple-choice
multiple-choice
multiple-choice
multiple-choice
multiple-choice
multiple-choice
multiple-choice
multiple-choice
multiple-choice
multiple-choice
free-response
free-response
free-response
multiple-choice
multiple-choice
multiple-choice
multiple-choice
free-response
free-response
How to Contact EducAide
For information about Acces and its various database modules, please call or write:
EducAide Software
PO Box 1048
Vallejo, CA 94590
800-669-9405, 707-554-9600 fax
email: [email protected]
Internet: www.educaide.com
Note: If you discover an error in any database module (typo, wrong answer, etc.), you may be
entitled to a “finder’s fee”. Normally, EducAide pays $3–5 to the person who first reports an
error. For more details, please visit the “error report” section of EducAide’s web site.
Algebra I
Table of Contents
A.
Rational Numbers
Adding integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AA
Subtracting integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AB
Expressions which emphasize grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AC
Combined methods, including parantheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AD
Multiplying integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AE
Dividing integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AF
Adding and subtracting fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AG
Adding and subtracting decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AH
Multiplying and dividing fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A I
Multiplying and dividing decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A J
Converting fractions to decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AK
Understanding rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AL
B.
Exponents
Using exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BA
Laws of exponents: multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BB
Laws of exponents: power to a power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BC
Laws of exponents: division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BD
Laws of exponents: combined methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BE
Variable exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BF
Negative and zero exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BG
Using scientific notation I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BH
Using scientific notation II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B I
Simplifing expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B J
Mixed practice and review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BK
Word problems with scientific notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BL
C.
Basic Concepts
Order of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CA
Absolute value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CB
Order on number line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CC
Properties of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CD
Using distributive property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CE
Translating algebraic expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CF
Statements about real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CG
Understanding the roots of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CH
D.
Evaluating and Simplifying Expressions
Evaluate for given value(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DA
Evaluate for given replacement set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DB
Evaluate for w = 6, x = 3, y = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DC
Evaluate for a = 5, b = 7, c = −2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DD
Evaluate for n = −3, p = 4, r = −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DE
Evaluate for x = −2, y = −3, z = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DF
Evaluate for a = −4, b = 8, c = 2, d = −3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DG
Evaluate for negative and zero exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DH
Adding variable terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D I
Combining like terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D J
Multiplying monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DK
Dividing monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DL
Combined methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DM
Applications of monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DN
E.
Polynomials
Adding polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EA
Subtracting polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EB
Multiplying monomials and polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EC
Multiplying binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ED
Binomial Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EE
Difference of squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EF
Multiplying polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EG
Dividing monomials and polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EH
Dividing polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E I
Combined methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E J
Applications of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EK
Mixed practice with polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EL
3 Dimensional figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EM
F.
Averages, Percents, Simple Formulas
Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FA
Percents I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FB
Percents II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FC
Percents III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FD
Applications of averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FE
Applications of percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FF
Explicit formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FG
Area, perimeter, volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FH
Time, distance, rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F I
Interest and investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F J
G.
First Degree Equations
Addition and subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GA
Multiplication and division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GB
Equations of the form: ax + b = c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GC
Mixed practice and review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GD
Combining terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GE
Variables on both sides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GF
Advanced equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GG
Solving for other variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GH
H.
Word Problems For First Degree Equations
Number problems (simple) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HA
Number problems (advanced) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HB
Consecutive integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HC
Coins and stamps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HD
Age problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HE
Time, distance, rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HF
Mixture problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HG
Interest and investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HH
Area, perimeter, volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H I
Triangles, supplements, complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H J
I.
Factoring
Prime factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finding the missing factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Factoring out monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Factoring trinomials of the form: x2 + bx + c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Factoring trinomials of the form: x2 − bx + c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Factoring trinomials of the form: x2 + bx − c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Factoring trinomials of the form: x2 − bx − c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Factoring trinomials of the form: ax2 + bx + c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mixed practice and review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Factoring difference of squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Factoring perfect square trinomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Combined methods of factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Factoring by grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Factoring sums and differences of cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems with variable exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
J.
IA
IB
IC
ID
IE
IF
IG
IH
II
IJ
IK
IL
IM
IN
IO
Solving Equations By Factoring
Using zero-product rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quadratic equations (simple) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quadratic equations (advanced) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Higher order equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solving for other variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Word problems involving factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
JA
JB
JC
JD
JE
JF
K.
Ratio and Proportion
Solving proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KA
Writing ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KB
Ratios of x to y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KC
Word problems involving ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KD
Word problems involving proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KE
L.
Rational Expressions and Equations
Reciprocals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LA
Least common multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LB
Greatest common factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LC
Values for which an expression is undefined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LD
Simplifying rational expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LE
Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L F
Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LG
Addition and subtraction (common demoninators) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LH
Addition and subtraction (different denominators) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L I
Simplifying complex fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L J
Rational equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LK
Advanced rational equations (quadratic solutions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L L
Solving for other variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LM
Word problems involving rational equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LN
M. Square Roots
Rational roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MA
Simplifying square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MB
Multiplying square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MC
Dividing square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MD
Adding and subtracting square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ME
Combined methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MF
Multiplying binomials with square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MG
Simplifying complex fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MH
Solving equations with square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M I
Using Pythagorean theorem I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MJ
Using Pythagorean theorem II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MK
Word problems involving square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ML
N.
Completing The Square and Quadratic Formula
Perfect square trinomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NA
Solving equations with perfect squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NB
Solving equations by completing square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NC
Solving equations using quadratic formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ND
Solving for other variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NE
Finding a quadratic equation from roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NF
Understanding the roots of an equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NG
Word problems involving the quadratic formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NH
O.
Inequalities and Absolute Value
Simple inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Advanced inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Compound Inequalties (“and”, “or”) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solving equations with absolute value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inequalities and absolute value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
P.
OA
OB
OC
OD
OE
Graphs, Equations of Lines
Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PA
Midpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PB
Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PC
Points on a line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PD
Graphing lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PE
Writing equations of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PF
Mixed practice and review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PG
Parallel and perpendicular lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PH
Graphing inequalities in two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P I
Graphing the intersection of inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P J
Other equations and graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PK
Ordered pairs and equations of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PL
Writing systems of inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PM
Word problems involving graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PN
Q.
Systems of Equations
Graphing and substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QA
Addition, elimination, determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QB
Advanced methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QC
Systems of 3 variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QD
Word problems involving two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QE
AA
Topic:
Adding integers.
Directions:
6—Simplify.
1.
7 + (−2)
5.
−4 + 8
9.
4 + (−9)
5
4
34. −25 + 8
−13
42. −48 + 0
−75
0
−30
57. (−13) + (−28)
61. −45 + 17
−41
65. (−45) + (−24)
69. −36 + 144
−69
73. 57 + (−96)
77. (−63) + (−42)
81. −42 + 19
89. −33 + 93
−110
93. 66 + (−87)
97. (−105) + 59
101. 57 + (−57)
105. 321 + (−105)
109. −286 + 145
216
−141
117. (−147) + (−83)
−230
9
20. (−5) + (−6)
−9
24. −11 + 1
−14
−30
2
−11
−10
28. (−7) + (−7)
−14
32. 15 + (−9)
6
17
39. −16 + 42
26
40. −15 + 51
36
−48
43. 0 + (−36)
−8
48. 18 + (−48)
−30
0
52. 26 + (−26)
0
51. 15 + (−15)
54. −67 + 17
−50
55. −37 + 30
−52
66. (−61) + (−34)
−95
78. (−72) + (−24)
−96
86. 34 + (−106)
91. −45 + 86
30
−24
−58
106. 476 + (−97)
379
110. −197 + 104
−93
92. −57 + 99
118. (−325) + (−85)
−108
−18
−20
42
−15
96. 48 + (−71)
−23
99. (−92) + 58
−34
100. (−80) + 33
−47
115. 0 + (−125)
108. −372 + 504
−126
−566
0
132
112. 238 + (−454)
116. 0 + (−94)
−125
119. (−445) + (−121)
ALG catalog ver. 2.6 – page 1 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
104. 165 + (−165)
0
142
111. 329 + (−455)
−410
−45
95. 57 + (−72)
107. −287 + 429
−43
88
88. 88 + (−108)
41
−74
80. (−92) + (−16)
84. −34 + 16
−114
103. (−228) + 228
0
−96
−19
87. 51 + (−165)
−72
68. (−53) + (−21)
76. 28 + (−73)
−35
−56
−59
72. −70 + 158
55
79. (−54) + (−42)
83. −71 + 52
−15
−59
−5
60. (−42) + (−14)
64. −74 + 15
67. (−26) + (−33)
75. 49 + (−84)
−28
−37
−43
71. −55 + 110
59
56. −20 + 15
−7
59. (−25) + (−12)
63. −55 + 12
−14
−20
−20
0
58. (−16) + (−36)
44. 0 + (−20)
−36
47. 17 + (−37)
50. −21 + 21
114. −43 + 0
−72
−10
−17
102. 84 + (−84)
0
16. −6 + 15
16
31. 19 + (−17)
98. (−121) + 63
−46
1
36. −34 + 17
94. 72 + (−96)
−21
−6 + 7
−6
90. −42 + 72
60
8.
6
12. 3 + (−13)
−4
27. (−15) + (−15)
−10
9 + (−3)
35. −32 + 26
82. −64 + 49
−23
85. 32 + (−142)
113. −72 + 0
−105
7
4.
−17
74. 37 + (−65)
−39
−3 + 10
23. −15 + 1
9
70. −38 + 97
108
7.
1
19. (−4) + (−5)
−10
−8
62. −32 + 18
−28
6 + (−5)
15. −3 + 19
8
46. 27 + (−35)
−4
3.
11. 7 + (−11)
−2
38. −19 + 36
29
53. −54 + 24
4
30. 17 + (−8)
4
45. 29 + (−33)
49. −8 + 8
−5 + 9
26. (−5) + (−5)
−16
29. 15 + (−11)
41. −75 + 0
6.
2
22. −10 + 2
25. (−8) + (−8)
37. −15 + 44
8 + (−6)
18. (−8) + (−2)
−13
−6
33. −22 + 9
2.
14. −8 + 16
7
17. (−7) + (−6)
21. −8 + 2
98—Perform the indicated operation(s).
10. 8 + (−10)
−5
13. −7 + 14
1—Add.
−216
−94
120. (−176) + (−322)
−498
AA
More than two terms
121. −4 + (−8) + 15
3
125. 6 + (−2) + (−14)
129. (−32) + 67 + 25
−10
122. −8 + 24 + (−7)
9
126. 14 + (−26) + 9
−3
130. 36 + 12 + (−14)
60
133. (−27) + 52 + (−13)
138. −33 + 51 + 16
4
146. (−5) + 19 + (−13)
140. 32 + 28 + (−43)
9
−60
148. (−18) + 20 + (−3)
1
149. (−10) + (−8) + 12 + 22
16
150. 23 + 14 + (−16) + (−18)
151. (−9) + 25 + (−17) + 14
13
152. −21 + 6 + (−8) + 35
153. −8 + 15 + (−24) + 17
155. −21 + 15 + 27 + (−21)
3
12
154. 13 + (−11) + (−16) + 14
0
156. (−36) + 25 + (−8) + 19
0
0
0
157. (−26) + (−44) + 14 + 36
−20
158. 47 + 15 + (−27) + (−45)
−10
159. 41 + (−39) + 18 + (−34)
−14
160. (−56) + 24 + 43 + (−17)
−6
ALG catalog ver. 2.6 – page 2 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
17
144. (−26) + (−56) + 22
−26
147. −17 + 6 + 12
45
136. 18 + (−26) + (−16)
143. (−38) + 40 + (−28)
1
−7
−24
139. 25 + 46 + (−62)
−87
−1
132. (−22) + 43 + 24
38
−14
34
6
128. 16 + (−8) + (−15)
−8
135. 15 + (−12) + (−17)
142. 15 + (−67) + (−35)
−44
124. −5 + (−12) + 23
2
131. 13 + (−33) + 58
34
11
141. (−15) + (−49) + 22
145. −15 + 3 + 11
127. 8 + (−19) + 3
134. (−33) + 61 + (−17)
12
137. −56 + 42 + 18
123. −3 + 12 + (−7)
−1
AB
Topic:
Subtracting integers.
Directions:
6—Simplify.
1.
20 − 13
5.
103 − 87
9.
3−8
7
16
21. −4 − 3
37 − 14
6.
238 − 159
23
79
22. −2 − 7
−7
45 − 27
7.
141 − 25
18
116
−14
15. 25 − 58
−9
18. 92 − 185
−52
3.
11. 5 − 19
−3
14. 27 − 36
−12
17. 112 − 164
98—Perform the indicated operation(s).
2.
10. 7 − 10
−5
13. 19 − 31
2—Subtract.
−33
19. 147 − 202
−93
23. −5 − 8
−9
4.
62 − 35
8.
157 − 94
27
63
12. 12 − 18
−6
16. 48 − 75
−27
20. 45 − 162
−55
24. −4 − 6
−13
−117
−10
25. −13 − 18
−31
26. −12 − 17
−29
27. −18 − 7
29. −17 − 19
−36
30. −18 − 16
−34
31. −46 − 21
−67
32. −21 − 58
−79
33. −66 − 96
−162
34. −35 − 78
−113
35. −43 − 77
−120
36. −84 − 86
−170
37. −124 − 99
38. −76 − 154
−223
41. −12 − 0
−12
42. −15 − 0
−15
45. 0 − (−7)
7
46. 0 − (−9)
9
49. 0 − 32
50. 0 − 27
−32
39. −217 − 48
−230
43. −24 − 0
47. 0 − 12
−27
28. −5 − 18
−25
−23
40. −139 − 102
−265
44. −19 − 0
−24
48. 0 − 16
−12
−241
−19
−16
51. 0 − (−48)
48
52. 0 − (−35)
23
56. 14 − (−17)
31
60. 78 − (−33)
111
35
53. 21 − (−11)
32
54. 14 − (−10)
24
55. 17 − (−6)
57. 34 − (−34)
68
58. 25 − (−25)
50
59. 46 − (−58)
61. −3 − (−2)
−1
62. −6 − (−3)
−3
63. (−5) − (−3)
−2
64. (−8) − (−7)
−1
67. −22 − (−8)
−14
68. −29 − (−11)
−18
3
72. (−2) − (−9)
7
76. −3 − (−15)
12
65. (−19) − (−4)
69. −4 − (−9)
−15
66. (−13) − (−7)
70. −3 − (−6)
5
73. (−12) − (−18)
6
−6
104
71. (−7) − (−10)
3
74. (−8) − (−15)
7
75. −2 − (−16)
14
0
77. 31 − 31
0
78. 17 − 17
0
79. (−5) − (−5)
81. 24 − 38
−14
82. 33 − 45
−12
83. 21 − 65
85. (−55) − 46
89. −32 − 33
−101
90. −42 − 56
−65
93. 54 − (−36)
−11
101. (−37) − (−81)
105. −76 − (−76)
109. 732 − 448
113. −142 − 226
44
0
−368
117. −92 − (−128)
36
−21
88. −47 − 39
−62
0
−86
−98
91. (−24) − 72
−96
92. (−31) − 24
−55
80
95. 82 − (−24)
106
96. 74 − (−31)
105
98. −82 − (−73)
99. (−51) − (−33)
−9
−18
100. (−57) − (−43)
20
103. −29 − (−54)
106. (−42) − (−42)
0
107. 496 − 496
0
108. 127 − 127
0
111. 275 − 346
−71
112. 385 − 627
−242
114. −251 − 132
62
−383
118. −175 − (−305)
130
115. 65 − (−335)
25
31
116. 392 − (−108)
400
119. −814 − (−492)
ALG catalog ver. 2.6 – page 3 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
104. −31 − (−62)
−14
102. (−42) − (−62)
110. 519 − 457
284
84. 56 − 77
−44
87. −36 − 26
−121
94. 64 − (−16)
90
97. −88 − (−77)
86. (−69) − 52
80. −8 − (−8)
−322
120. −663 − (−571)
500
−92
AC
Topic:
Expressions which emphasize grouping (commutative and associative properties of addition).
Directions:
6—Simplify.
98—Perform the indicated operation(s).
1.
98 + 27 + 2
127
2.
5 + 49 + 95
149
3.
190 + 86 + 10
5.
4 + 28 + 56
88
6.
78 + 15 + 2
95
7.
26 + 37 + 24
9.
2 + 135 + 148 + 15
10. 4 + 199 + 66 + 1
270
286
87
11. 298 + 74 + 2 + 6
380
4.
25 + 37 + 75
8.
19 + 37 + 1
137
57
12. 395 + 8 + 5 + 82
490
300
14. 950 + 57 + 50 + 3
13. 990 + 25 + 10 + 75
1100
15. 47 + 15 + 13 + 785
1060
17. (−50) + (−28) + (−50)
18. −75 − 48 − 25
22. 6 + (−5) + 4
25
29. −23 + 6 + 23
6
33. 16 − 25 + 24
15
4
30. −38 + 14 + 38
34. 12 − 14 + 38
10
38. −21 + 45 − 9
41. −17 + 14 + 3
0
42. −40 + 25 + 15
45. (−23) + (−6) + (−17)
−46
12
14
31. 14 − 7 − 14
39. −38 + 45 − 2
15
−267
10
46. (−5) + (−12) + (−25)
−51
50. (−16) + 29 + (−4) + 11
−20
57. 4 + (−5) + (−1) + 2
0
−40
58. 3 + (−7) + 6 + (−2)
51. 8 + (−13) + 12 + (−17)
237
47
69. (26 + 1) + (99 + 24)
150
73. 10 + (67 + 190) + 3
270
77. (15 − 32) + (25 − 18)
−10
81. −4 + [−6 + (−8)]
30
59. (−5) + 3 + (−8) + 10
0
85
63. 8 + (12 + 19)
66. (32 + 19) + 8
59
67. 150 + (18 + 50)
39
218
71. (7 + 24) + (13 + 6)
50
70
74. 990 + (150 + 10) + 50
75. 50 + (86 + 950) + 4
1090
79. (12 − 29) + (48 − 11)
0
82. −11 + [−19 + (−15)]
0
48. (−18) + (−8) + (−12)
52. 13 + (−24) + 7 + (−26)
56. −5 + 22 − 5 + 28
40
60. 7 + (−11) + 10 + (−6)
64. 7 + (13 + 48)
68
68. 1 + (36 + 99)
136
72. (23 + 18) + (7 + 12)
60
1200
78. (−36 + 17) + (−4 + 23)
20
0
62. (45 + 36) + 4
70. (15 + 18) + (2 + 35)
40. −13 + 60 − 27
−30
55. −7 + 31 − 13 + 19
0
45
−38
−10
54. 8 − 56 + 12 − 4
21
−5
36. −17 + 35 + 27
44. −31 + 38 − 7
0
47. (−34) + (−11) + (−6)
20
53. 14 − 21 + 16 − 29
5
4
28. (−6) + 31 + (−4)
32. 19 − 5 − 19
67
43. −13 + 35 − 22
0
3
−7
35. −25 + 17 + 75
36
24. 3 + (−6) + 7
12
27. (−1) + 23 + (−19)
−42
49. (−7) + 5 + (−13) + 25
61. (37 + 199) + 1
23. −8 + 3 + 17
5
26. (−17) + 32 + (−3)
37. −14 + 30 − 6
−18
20. −150 − 67 − 50
−177
25. (−3) + 14 + (−7)
65. (16 + 27) + 4
1250
19. (−99) + (−77) + (−1)
−148
−128
21. −5 + 11 + 9
16. 43 + 60 + 7 + 1140
860
20
83. [−29 + (−18)] − 12
−45
ALG catalog ver. 2.6 – page 4 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−59
76. 97 + (16 + 3) + 24
140
80. (−14 + 28) + (−16 + 12)
10
84. [−16 + (−35)] − 5
−56
AD
Topic:
Combined methods (adding and subtracting integers, parantheses).
Directions:
6—Simplify.
1.
−(−4)
5.
− [−(−12)]
9.
−(11 − 4)
4
−12
21. −(65 − 28)
−(−0)
6.
− [−(−7)]
0
−7
14. −(−8 + 17)
17
17. −(−26 − 12)
2.
10. −(15 − 28)
−7
13. −(−20 + 3)
98—Perform the indicated operation(s).
22. −(43 − 29)
−37
−(−16)
7.
− [−(−0)]
16
0
11. −(45 − 25)
13
−9
18. −(−14 − 29)
38
3.
43
−(−8)
8.
− [−(−25)]
8
12. −(6 − 21)
−20
−25
15
15. −(−32 + 16)
16
16. −(−19 + 33)
−14
19. −(−18 − 12)
30
20. −(−21 − 14)
35
23. −(72 − 54)
−14
4.
24. −(61 − 32)
−18
−29
25. − [−(−8 + 5)]
−3
26. − [−(−9 + 16)]
7
27. − [−(−7 + 12)]
5
28. − [−(−17 + 6)]
29. − [−(86 − 46)]
40
30. − [−(33 − 58)]
−25
31. − [−(40 − 90)]
−50
32. − [−(76 − 21)]
33. − [(−7) − (−11)]
37. − [12 − (−37)]
−4
−49
41. − [(−105) + (−65)]
170
34. − [(−33) − (−18)]
38. − [24 − (−26)]
15
−50
42. − [(−48) + (−56)]
35. − [(−17) − (−9)]
39. − [28 − (−34)]
−36
−62
43. − [(−35) + (−85)]
104
45. − [17 − (−19)]
8
−11
55
36. − [(−9) − (−12)]
40. − [25 − (−42)]
−3
−67
44. − [(−14) + (−87)]
120
101
46. − [15 − (−17)]
−32
47. − [(−9) − (−12)]
50. −12 + 15 − 16
−13
51. −7 + 24 − 5
−3
48. − [(−11) − (−12)]
−1
More than two terms
49. 17 − 23 + 4
−2
53. −19 + 12 − (−7)
0
57. 8 + (−10) − (−7)
5
54. 5 − (−8) − 6
55. 11 − 18 − (−6)
7
58. −32 + (−5) − (−28)
1
65. 4 − (−8) − 19 + (−6)
−13
62. 9 − 15 + 8 − 7
0
−5
4
56. −23 − (−7) + 5
−11
60. 24 + (−19) − (−16)
21
63. 13 − 18 + 10 − 9
−4
64. −4 + 23 − 8 + 5
16
66. −16 + (−9) + 23 − (−4) 67. −5 + 27 − (−14) + (−8) 68. 13 + (−38) − (−42) − 17
2
28
69. −32 + (−7) − (−40) + 6 70. 11 − 26 − (−19) + (−4)
7
−1
59. −11 − (−23) + (−12)
−9
61. −8 + 17 − 12 + 4
52. 7 − 21 + 18
12
0
71. 4 + (−20) − 18 − (−13)
−21
0
72. 10 − 14 + (−24) − (−26)
−2
Grouping symbols
73. 12 − (25 − 21)
74. 25 − (−28 + 30)
8
77. −11 − (−13 + 35)
78. −7 − (43 − 38)
23
75. 8 − (36 − 18)
−10
76. 20 − (46 − 9)
−17
−12
79. −13 − (14 + 9)
−36
80. −24 − (7 + 22)
36
83. 6 − (−43 + 35)
14
84. 43 − (9 − 29)
87. −15 − (7 − 22)
0
88. −22 − (−4 − 18)
−53
−33
81. 24 − (33 − 54)
45
85. 14 − (−9 + 23)
0
89. −11 − (18 − 25)
82. 19 − (−31 + 14)
86. 29 − (39 − 10)
−4
0
90. −34 − (−45 + 31)
91. −5 − (7 − 21)
9
63
92. −19 − (−36 + 15)
−20
93. −57 + [(−61) − 39]
−157
94. 61 + [(−58) − 41]
95. [(−72) − 31] + 33
−38
ALG catalog ver. 2.6 – page 5 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−70
96. [(−49) − 29] + 21
−57
0
2
AD
97. (27 − 42) + (46 − 34)
98. (47 − 54) + (19 − 17)
−3
99. (52 − 42) + (36 − 47)
−5
101. −36 − (58 − 22) + 72
102. −38 − (23 + 45) − 42
105. 43 − (−31 + 42) − 21
106. 25 − (−51 + 36) + 17
11
−2
103. 18 − (54 + 15) + 24
−148
0
100. (21 − 15) + (23 − 31)
−1
104. 45 − (21 − 62) − 22
−27
64
107. −7 − (−36 + 14) − 19
108. −42 − (−79 + 24) + 13
−4
57
26
109. −50 − (−6 − 24) − (−15)
−5
110. −34 − (−42) − (24 − 38)
111. −113 − (32 − 77) − (−68)
0
112. 57 − (26 − 31) + (−27)
25
114. −18 − (−12 − 21 + 57)
−42
116. −21 − (−12 + 16 − 25)
0
113. 26 − (125 − 74 − 81)
115. 15 − (13 − 21 + 17)
56
6
22
117. 15 − [12 − (−18 + 7)]
−8
118. 25 − [−8 − (13 − 46)]
119. −23 − [14 − (9 − 16)]
−44
120. −50 − [−29 − (−13 + 35)]
121. − [−5 − (−8)] − [−9 + (−2)]
123. [17 + (−9)] − [6 − (−11)]
−9
8
0
1
122. − [8 − (−3)] − [−4 − (−15)]
124. [−12 + (−4)] − [3 − (−8)]
ALG catalog ver. 2.6 – page 6 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−22
−27
AE
Topic:
Multiplying integers.
Directions:
6—Simplify.
1.
13(5)
5.
(23)(7)
9.
6 · 55
65
161
13. (−5)(10)
6.
(8)(26)
−84
25. 12(−15)
−180
45. −9(−18)
162
7.
(11)(15)
64
165
19. −7(13)
−126
4(18)
8.
(24)(11)
72
264
360
16. (−7)(14)
−108
20. −15(5)
−91
23. (12)(−9)
−132
4.
12. 8 · 45
300
15. (−18)(6)
−80
−98
−75
24. (16)(−8)
−108
−128
−144
27. 14(−20)
−280
28. 11(−17)
−187
30. −9 · 21
−189
31. −10 · 24
−240
32. −15 · 20
−300
35. −16(21)
−336
36. 42(−12)
−504
39. (−1)(24)
−24
−405
−18
42. (−5)(−11)
84
16(4)
26. 18(−8)
38. −1 · 18
−26
3.
11. 75 · 4
297
34. (27)(−15)
−432
41. (−14)(−6)
208
22. (11)(−12)
−198
33. (−36)(12)
80
18. −9(14)
21. (6)(−14)
37. 26(−1)
5(16)
98—Perform the indicated operation(s).
14. (−10)(8)
−50
−84
29. −33 · 6
2.
10. 33 · 9
330
17. −12(7)
3—Multiply.
43. (−12)(−6)
55
46. −24(−9)
216
47. −9(−21)
40. −1(35)
−35
44. (−4)(−17)
72
189
68
48. −15(−9)
135
49. −13(−16)
208
50. −14(−8)
112
51. −15(−11)
165
52. −8(−27)
216
53. (−1)(−32)
32
54. −15(−1)
15
55. (−17)(−1)
17
56. −1(−48)
48
57. (−13)(0)
61. 8(−8)
58. −15 · 0
0
−64
65. (−11)(−11)
121
59. (0)(−25)
0
62. (−15)(15)
−225
63. −12(12)
66. (−9)(−9)
81
67. −14(−14)
60. 0(−35)
0
0
64. 16(−16)
−144
−256
196
68. −13(−13)
169
−2214
69. −31(48)
−1488
70. 57(−23)
−1311
71. −39 · 61
−2379
72. (−41)(54)
73. 321(−5)
−1605
74. −2(721)
−1442
75. −472(4)
−1888
76. 3(−375)
−1125
More than two factors
77. (3)(−5)(2)
81. (−10)(−8)(2)
85. −9 · 8 · 4
78. (−6)(4)(3)
−30
160
89. (−6)(−3)(−4)
82. (−2)(4)(−8)
86. −5 · 7 · 8
−288
−72
93. (2)(−9)(−5)(−3)
79. (5)(−4)(8)
−72
83. (5)(−3)(−8)
64
87. −3 · 6 · 7
−280
90. (−11)(−5)(−3)
94. (7)(2)(−7)(5)
−165
−490
80. (4)(9)(−3)
−160
84. (−11)(−2)(5)
120
88. −3 · 4 · 11
−126
91. (−4)(−7)(−3)
95. (−4)(6)(3)(−5)
−108
−84
360
110
−132
92. (−9)(−4)(−5)
96. (3)(−2)(6)(−4)
−180
144
−270
97. (−4)(6)(−5)(−6)
98. (11)(−1)(7)(4)
−308
99. (−8)(−1)(−12)(−9)
−720
ALG catalog ver. 2.6 – page 7 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
864
100. (−7)(−2)(−5)(−3)
210
AF
Topic:
Dividing integers.
Directions:
6—Simplify.
1.
294 ÷ 7
5.
9.
4—Divide.
98—Perform the indicated operation(s).
42
2.
315 ÷ 7
108 ÷ 12
9
6.
128 ÷ 16
567 ÷ 27
21
10. 483 ÷ 21
13. 63 ÷ (−9)
17. −84 ÷ 3
45
3.
222 ÷ 6
8
7.
192 ÷ 16
23
11. 396 ÷ 18
14. 54 ÷ (−6)
−7
18. −91 ÷ 7
−28
4.
432 ÷ 8
12
8.
182 ÷ 13
14
22
12. 512 ÷ 16
32
37
15. 72 ÷ (−4)
−9
−18
19. −144 ÷ 9
−13
−16
21. 51 ÷ (−17)
−3
22. 78 ÷ (−13)
−6
23. 153 ÷ (−9)
25. −352 ÷ 16
−22
26. −546 ÷ 21
−26
27. −288 ÷ 12
29. −48 ÷ (−6)
30. −42 ÷ (−6)
8
33. −165 ÷ (−5)
34. −144 ÷ (−8)
33
37. −81 ÷ (−9)
38. −169 ÷ (−13)
9
39. −64 ÷ 8
13
41.
−36
4
−9
42.
−96
16
−6
43.
−84
6
−14
44.
−108
6
47.
434
−14
−31
48.
416
−16
−26
49.
−60
−15
4
50.
−90
−18
53.
−216
−9
24
54.
−256
−8
32
55.
−324
−36
9
56.
−336
−42
59.
−100
−10
10
60.
−144
−12
12
61.
−725
−25
29
62.
930
−30
65.
−16
0
67.
0
22
69. 0 ÷ 5
66.
0
70. 0 ÷ (−17)
73. −15 ÷ (−15)
77.
26
−1
27
0
undef.
1
−12
−1
79.
12
−39
−39
41
−14
−19
−23
6
36. −306 ÷ (−9)
34
40. 121 ÷ (−11)
−11
45.
247
−13
−19
46.
204
−12
−17
51.
−56
−4
14
52.
−75
−5
15
8
57.
49
−7
58.
−210
14
−31
63.
−1175
−25
64.
−1080
40
−18
5
−7
47
68.
0
0
−15
72. 8 ÷ 0
undef.
75. 25 ÷ (−1)
−1
20. −168 ÷ 12
32. −48 ÷ (−8)
5
−8
71. −18 ÷ 0
0
74. 27 ÷ (−27)
78.
−26
undef.
−12
28. −345 ÷ 15
−24
35. −287 ÷ (−7)
18
16. 96 ÷ (−8)
24. 152 ÷ (−8)
−17
31. −40 ÷ (−8)
7
54
0
undef.
76. −32 ÷ (−1)
−25
80.
1
18
−18
32
−1
Rational answers
81. 4 ÷ 12
82. 4 ÷ 16
1
3
85. −24 ÷ 64
89. 18 ÷ (−42)
− 37
93. −40 ÷ (−75)
97.
24
−120
101.
−63
−81
86. −40 ÷ 72
− 38
8
15
− 15
7
9
83. 15 ÷ 20
1
4
18
−108
102.
−40
−88
6
17
− 16
5
11
105. 45 ÷ (−20)
− 94
106. 64 ÷ (−12)
109. −96 ÷ −14
48
7
110. −105 ÷ (−56)
− 17
91. 40 ÷ (−96)
− 34
94. −18 ÷ (−51)
98.
87. −15 ÷ 105
− 59
90. 27 ÷ (−36)
3
4
15
8
88. −5 ÷ 70
1
− 14
96. −12 ÷ (−64)
2
9
99.
−33
121
3
− 11
100.
−42
140
6
− 20
103.
−34
−85
2
5
104.
−39
−78
1
2
107. −54 ÷ 15
− 16
3
2
3
92. 42 ÷ (−54)
5
− 12
95. −18 ÷ (−81)
84. 18 ÷ 27
111. −102 ÷ (−68)
ALG catalog ver. 2.6 – page 8 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
108. −78 ÷ 4
− 18
5
3
2
− 79
3
16
− 39
2
112. −60 ÷ (−54)
10
9
−15
−27
AG
Topic:
Adding and subtracting fractions.
Directions:
6—Simplify.
1.
2
− +3
5
5.
2−
9.
2.
13
5
10
− 43
3
1
2
+ −
3
3
13. −
6.
3
9
+
10 10
1 11
25. − +
6 14
37.
1 2
−
3 3
41.
2
7
−
10 5
45.
9
1
−
10 5
49.
1 7
−
5 6
2 7
53. − −
9 9
2
15
− 11
8
1
+ (−4) − 318
8
1
4
+ −
− 35
5
5
3
7
+ −
2
6
14 2
−
15 3
7
10
46.
− 29
30
50.
5
3
69. − − −
8
8
−1
11
3
− 13
6
8 13
−
7 12
5
84
47.
7
17
−
12 18
5 15
−
3
8
5
− 24
51.
1
3
−
10 6
11 7
−
6
6
55. −
−3
5
2
− −
18
9
67.
1
2
− 35
− 13
36
2
15
3
5
− −
8
12
17
8
11
1
75. − − −
10
6
ALG catalog ver. 2.6 – page 9 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
2
− 27
1
4
−4
4
3
28.
3
1
+ −
4
6
7
12
32.
1
5
+ −
3
8
7
− 24
8
7
36. − + −
5
10
40.
9 1
−
2 2
44.
11 5
−
24 6
48.
5 7
−
6 8
52.
19 2
−
20 3
68.
19
24
2
7
− −
10
10
7 4 1
+ −
5 5 5
− 52
− 23
10
4
− 38
1
− 24
17
60
12
5
−
13 13
− 17
13
8 23
60. − −
− 11
4
5 20
5
1
64.
− −
1
6
6
2
71. −
79.
5
2
5
7
3
5
+ −
8
8
56. −
− 13
11
2
− 15
− 12
− 17
4
1
7 7
59. − −
− 72
6 3
10
4
63.
− −
7
7
13
11
− −
15
15
7
9
3
−
−
10 10 10
1
− 21
4
9
−
11 11
−5 +
7 5
24. − +
6 2
7
3
− 16
3
7
35. − + −
4
2
1 17
−
4 20
8.
5
3
20. − + −
2
2
−2
11
5
+ −
4
12
43.
4.
2
1+ −
7
16.
2
3
3
7
19. − + −
5
5
3
2
+
10 15
3 1
12. − +
7 7
5
1
+ −
6
6
4
15
− 27
14
5
− 39
1 2
31. − +
3 7
1
3
17
17
74. − − −
8
4
78.
− 14
9
7 2
−
5 5
70. −
− 14
11
11
73. − − −
6
2
4 2 1
77. − + −
3 3 3
66.
3
4
4
−2
9
27. −
5
− 24
1
7
58. − −
− 56
4 12
7
5
4
62.
− −
3
9
9
3
9
− −
10
20
7.
39.
54. −
−1
23.
4
5
34. − + −
3
6
42.
− 32
− 19
30.
3
10
6
4+ −
5
15.
5
5
+ −
12
8
3 5
−
7 7
3.
8 5
11. − +
9 9
1
2
26.
38.
− 13
98—Perform the indicated operation(s).
1
3
7 2
22. − +
9 3
− 13
5
5
1
57. − −
− 78
6 24
3
7
5
61.
− −
4
8
8
65.
5
− +2
3
11
7
18. − + −
12
12
− 53
13
21
5
3
33. − + −
8
4
2—Subtract.
1 3
14. − +
4 4
3
5
9
7
+ −
10
2
2 4
29. − +
3 5
10.
− 13
5
5
17. − + −
6
6
21.
1—Add.
1
2
− 14
15
11
1
− −
15
6
72. −
9
10
9
12
− −
13
13
3
13
5
3
76. − − −
6
4
1
− 12
1 11 5
80. − +
+
6
6
6
5
2
AG
81.
1 1 1
+ +
2 4 8
82.
7
8
3
7
2
85. − −
+
5 10 15
Mixed numbers
1
1
89. 2 + −7
2
2
97. −3
3
3
+6
10
5
−5
2
1
105. 5 − 7
3
3
109. 6
3
2
−5
10
5
−9 12
4
3
95. 4 + −3
5
5
5
1
−4
6
6
2
2
121. −7 − −5
7
3
3
5
107. 8 − 2
8
8
−3 13
2
3
125. −11 − −9
5
5
10 56
−1 45
3 14
15
1
2
126. −13 − −7
3
3
−5 23
−4 23
3
3 20
9
7 10
3
3
130. −8 − −13
4
8
2
3
96. −4 + −2
5
10
7
−6 10
5
3
100. 7 + −8
7
14
1
5
104. −5 + 11
2
9
− 12
1
6 18
2 37
−8 18
3
2
5
115. −5 − 7
−13 12
4
3
11
1
119.
− −3
4 16
12
4
5
3
7
116. −4 − 8
−13 12
6
4
3
11
120. 7 − −
8 13
5
15
1
2
123. 5 − −2
2
3
3
5
124. 1 − −3
4
6
127. −9
8 16
8
3
− −17
11
11
131. −11
4 58
− 17
20
−20
3
3
112. −4 − 3
4
8
−11 12
6
7 11
2
3
129. −7 − −15
5
10
1
4
5
1
92. −7 + −12
6
6
1 5
108. 3 −
7 7
5 34
2
5
111. −6 − 4
3
6
−2 16
2
3
122. − − −4
3
5
−1 13
21
2
1
103. 9 + −6
5
4
3
−2 14
−17
1 15
3
5
99. 7 + −12
4
12
−3 85
7
1 2
+ +
12 6 3
3 4
9
88. − + −
4 5 10
7
9
2 13
2
1
114. 5 − 8
−2 56
3
2
1
2
118. 10 − −
6
3
2
1
1
113. 9 − 11
−2 10
5
2
3
1
117. − −5
5 78
4
8
3 1 5
− −
2 6 9
1
2
94. −1 + 3
3
3
3
11
110. 7 − 9
4
12
9
10
87.
84. −
7
5
2
5
91. −11 + −5
7
7
106.
−1 23
4 3
9
+ −
5 2 10
−1
5
1
102. −7 + 5
7
2
2
−3 15
1
6
83.
1
1
90. −4 + 3
6
6
1
7
98. −11 + 7
2
8
3
3 10
2
4
101. 3 + −6
3
5
0
1 2
5
86. − + −
7 3 14
− 76
2
3
93. −5 + −4
7
14
1 5 1
− +
3 6 2
128. −13
7
5 12
7
5
− −18
12
12
4 56
5
3
− −9
16
4
5
2
132. −13 − −7
9
3
−5 89
9
−1 16
1
3
2
133. 5 − 4 − −3
5
5
5
4
5
2
7
134. 7 − −2
−1
9
9
9
8
1
3
1
135. 11 − −5
− 12
2
4
4
5
1
4
137. −4 + −8
5
5
−5 25
+7
3
5
3
5
1
138. −13 + 4 + −5
4
8
4
0
2
1
139. 2 + −5
9
3
−14 38
ALG catalog ver. 2.6 – page 10 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
4 16
5
3
1
136. 4 − 11 − −7
8
4
8
+7
5
18
4
2
7
140. 1 + −6
+8
5
3
15
3 35
AH
Topic:
Adding and subtracting decimals.
Directions:
6—Simplify.
1.
0.8 + 0.7
5.
4.95 + 1.08
9.
25.18 + 36.04
13. 4 − 1.6
1.5
6.03
2.
0.5 + 0.9
6.
2.76 + 5.39
0.7
2.05
25. 34.2 − 28.4
1.4 + 0.8
7.
0.42 + 6.74
2.2
7.16
11. 0.327 + 0.064
39.2
15. 3 − 0.2
2.2
4.
0.7 + 2.6
8.
9.18 + 2.02
3.3
11.2
12. 0.608 + 0.512
0.391
16. 7 − 5.1
2.8
1.12
1.9
0.3
19. −1.1 + 2.6
1.5
20. 1.7 − 0.6
22. 6.7 − 0.82
5.88
23. 3.12 − 0.45
2.67
24. −5.6 + 10.3
4.7
28. 4.35 − 3.57
0.78
27. −0.622 + 0.819
12.06
31. 1.9 − 5
−3.5
34. 0.22 − 0.9
−0.14
3.
18. −0.5 + 0.8
30. 0.5 − 4
−5.8
33. −0.74 + 0.6
8.15
26. −8.04 + 20.1
5.8
98—Perform the indicated operation(s).
1.4
14. 10 − 7.8
21. −2.05 + 4.1
29. 6.2 − 12
2—Subtract.
10. 18.52 + 20.68
61.22
2.4
17. 0.9 − 0.2
1—Add.
32. 3.7 − 15
−3.1
35. 2.1 − 4.5
−0.68
0.197
−2.4
1.1
−11.3
36. −8.2 + 3.6
−4.6
37. 1.6 + (−2)
−0.4
38. 0.25 + (−3)
−2.75
39. 0.03 + (−1)
−0.97
40. 2.4 + (−4)
−1.6
41. 1.28 − 6.08
−4.8
42. −7.16 + 2.9
−4.26
43. −4.2 + 0.05
−4.15
44. 4.82 − 8.3
−3.48
46. 17.02 − 20.7
−3.68
47. 0.27 − 3.06
50. 0.2 − (−1.8)
2
51. 3.5 − (−2.5)
6
52. −1.6 − 0.4
54. −4.13 − 4.87
−9
55. −2.02 − 0.74
−2.76
56. 7.1 − (−0.46)
7.56
60. −0.29 − 1.26
−1.55
45. −42.9 + 25.4
49. −0.3 − 0.9
−17.5
−1.2
53. 1.49 − (−2.51)
57. −18.03 − 6.04
61. −4 + 3.2
4
−24.07
62. −1 + 0.4
−0.8
65. (−1.4) − (−0.7)
58. 7.25 − (−12.43)
−0.7
69. (−2.75) + (−3.25)
−6
19.68
66. (−0.6) + (−2.8)
59. 1.91 − (−3.08)
63. −3 + 0.85
−0.6
−3.4
70. (−2.01) − (−8.01)
6
48. −0.228 + 0.028
−2.79
4.99
64. −2 + 0.01
−2.15
67. (−2.5) + (−3.5)
−2
−1.99
68. (−5.1) − (−1.1)
−6
71. (−10.4) − (−18.6)
−0.2
8.2
−4
72. (−17.9) + (−3.9)
−21.8
73. (−50.2) − (−26.7)
−23.5
74. (−21.55) + (−10.65)
75. (−2.94) + (−0.07)
−32.2
ALG catalog ver. 2.6 – page 11 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−3.01
76. (−0.624) − (−0.283)
−0.341
AI
Topic:
Multiplying and dividing fractions.
Directions:
6—Simplify.
3—Multiply.
4—Divide.
1.
3
· 32
4
24
2.
7
· 30
6
5.
5 1
·
7 2
5
14
6.
1 3
·
10 2
9.
3 1
·
10 9
10.
4 7
·
5 4
14.
20 3
·
27 16
1
30
5 9
3
·
21 25 35
5
17. 36 −
−20
9
13.
21.
7
· 28
4
26.
− 25
3
3
3
29.
−
4
4
33.
2 5
·
5 2
34.
1
1 2 2
·
7 3 21
4
2
41.
−
−
9
5
45.
7
3
−
9
5
49.
3
1
7
3
8
45
7
− 15
1
7
8 11 22
·
9 12 27
10
7
57.
21
16
11
16
65.
7
−
11
−
6 5
·
7 12
5
36
15.
15 4
·
32 9
−25
19.
9
(−80)
10
4
7
3
−
4
8
27
−
7
4
5
23. 56 ·
8
18
31.
− 25
36
−
4
7
1 7
7
·
2 10 20
5
1
42.
−
9
2
46.
−
5
6
50.
−
3
4
4
9
−
7
10
− 11
28
21
44
62.
−
4
5
7
12
5
9
66.
−
8
11
4 4 4
· ·
5 5 5
4.
24 ·
14
15
8.
3 3
·
8 4
5
14
12.
4 3
·
9 16
5
24
16.
7 12
·
10 35
64
125
15
2
4
25
−1
4 1 4
·
5 5 25
3
3
43.
7
4
5
− 18
7
12
− 13
9
28
−
2
7
5
8
51.
−
5
8
9
13
8
9
− 45
88
4
5
67.
7
9
ALG catalog ver. 2.6 – page 12 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
5 3 16
· ·
6 4 15
1 1
·
3 3
36.
3
10
1
9
−
10
3
11
15
3
4
5
9
52.
10
11
−1
− 38
20
63
2
3
−
9
− 20
11
20
9
10
9
− 11
9 4 9
·
16 7 28
4
3
60.
−
15
16
56.
6
13
11 9 33
·
12 10 40
9
7
75.
−
10
9
79.
32.
48
− 45
4
71.
11
17
5
(−18)
8
48.
2
3
63.
28.
5
− 28
3 4 3
·
8 5 10
9
5
59.
−
20
6
7
− 15
−36
1 1 1
·
8 5 40
3
3
44.
−
4
5
55.
2
15
6
25
40.
47.
−
1
12
4
24. −36 −
3
2
−
5
11 4
·
4 11
9
32
35
33
4
20. − · 63
7
−72
39.
7 8 14
·
12 9 27
11
12
74.
−
−
12
17
78.
2
−
5
35. −
1
70.
− 11
15
11
8
18
3
27. −25 −
10
21
2
1 5 1
·
10 9 18
8
5
58.
−
−
25
12
5
24
9 5 15
·
11 12 44
13
11
73.
−
15
13
2 2 2
· ·
3 3 3
11.
54.
69.
77.
2 7
·
5 3
38.
53.
3
· 14
4
5 5
30. − ·
6 6
9
− 16
37.
61.
7.
2
22. − (−81)
9
49
5
25. − · 10
6
45 ·
3
20
5
8
2
5
3.
35
7
5
18. −40 ·
98—Perform the indicated operation(s).
64.
−
68.
2
−
9
13
25
−
5
9
7
6
13 15 39
·
20 16 64
7
15
76.
15
16
1
− 20
7
− 27
72.
7
− 10
80.
3 15 2
·
·
8 18 5
1
8
7
16
13
45
AI
4 21 3 1
·
·
9 16 7 4
3
3
5
85.
−
4
10
6
81.
30 12 2
·
·
6
8 5 3
4
1
14
86.
−
7
8
15
82.
3
− 16
89. 14 ÷
1
− 15
4
7
2 3 8
÷
5 4 15
2
97. 10 ÷ −
5
3
4
94.
5
105. −15 ÷ −
6
15 2 45
÷
17 3 34
4
1
129. ÷ −
5
20
1 1
130. − ÷
3 6
Mixed numbers
1
3
1
133. −3
− 28
5
2
5
1
3
44
137. −5
−1
5
2
5
4
1
141. 1
−2
− 21
5
5
3
3
1
4
145. − ÷ −2
15
5
4
7 3
149. −2 ÷
8 4
−42
107. 24 ÷
−20
146.
61
30
−2
2
7
÷4
8
3
3
16
3
3
÷ −
10
5
3
8
45
4
÷ (−10) − 252
5
2
1
115. − ÷ −
11
4
11
2
2
1
154. −6 ÷ 1
− 16
3
3
4
7
2
158. 5 ÷ −4
− 26
21
9
3
119. −
8
1
÷
15 6
3 2
÷
4 5
123.
8
11
1 78
4
5
−25
5
÷ 15
12
1
36
16 4
÷
27 9
− 39
32
4
3
3
5
21
÷ −
50
10
7
7
7
128.
÷ −
− 23
12
8
124. −
14 4
÷
− 76
15 5
3
3
131. − ÷ −
8
16
127. −
2
132.
3
6
25
135. −4
−2
2
8
7
1
5
139. 9
−4
−44
3
7
2
1
143. 8
4
39
3
2
140.
5
7
147. − ÷ 1
6
8
148.
− 49
1
4 39
÷
12 9 16
5
1
155. 8 ÷ −4
8
2
151. 1
ALG catalog ver. 2.6 – page 13 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
5 1
÷
8 3
3
8
116. − ÷
4 13
120.
− 16
5
27
1
100. −11 ÷ −
22
2
5
104. 15 ÷ −
−18
6
112.
15
8
6
2
159. −3 ÷ 4
7
3
8
9
108. −20 ÷
64
111.
2
27
3
134.
−
− 15
4
5
3
4
138. 1
2
4
7
5
1
4
16
142. −
−4
21
6
7
1
6
4
150. −3
− 23
6
1
3 7
153. 5 ÷ 3
4
4 5
5
3
157. −7 ÷ −3
8
4
2
7
6 1 36
÷
7 6 7
5
2
5
÷ −
118.
− 12
18
3
2
5
122. ÷ −
− 16
15
3
8
11
3
44
126. − ÷ −
39
13
4
−16
96.
1
2 10
3
−15
5
1
103. −15 ÷ −
3
114.
125.
92. 24 ÷
30
99. −9 ÷
9
8
110. − ÷ (−12)
9
− 14
27
3
5
3 2
÷
5 7
95.
9
16
3
106. 30 ÷ −
2
18
2
7
91. 18 ÷
25
3
2
109. − ÷ 14 − 491
7
4
5
113. ÷ −
− 28
25
5
7
25
5
5
117. − ÷ −
7
42
6
2 3
121. − ÷
9 7
2
3
102. −12 ÷
16
6
5
3 2
÷
8 3
98. 6 ÷
−25
3 5 7
7
· ·
20 9 12 144
16
1
15
88.
−
−
5
6
28
84.
3
5
90. 10 ÷
49
2
93.
101. 12 ÷
7 9 35 49
·
·
12 10 6 16
8
27
5
87.
−
−
15
20
6
83.
− 81
98
136.
144.
152.
− 23
12
156.
9 1
÷
2 4
18
5
1
2
5
14
8
3
7
1
1
−
− 78
10
4
7
2
4
−2
− 65
6
8
9
1
3
5
÷ −2
− 14
4
10
2
5
10 ÷ −
− 64
5
3
6
1
2
30
−7 ÷ −1
7
7
3
4
1
160. 2 ÷ 5
5
2
28
55
AJ
Topic:
Multiplying and dividing decimals.
Directions:
6—Simplify.
1.
14(0.2)
2.8
5.
(0.8)(0.5)
9.
(3.4)(0.8)
7(0.9)
0.4
6.
(0.5)(0.6)
2.72
10. (2.6)(0.7)
−0.32
21. −33(−0.3)
9.9
25. (−0.6)(0.4)
0.3
7.
(0.4)(0.9)
1.82
11. (0.5)(4.7)
4(0.3)
0.36
8.
(0.7)(0.8)
0.56
2.35
12. (0.9)(3.6)
3.24
15. (0.3)(−84)
−1.88
19. −14(0.03)
−0.42
22. −28(−0.2)
5.6
23. −16(−0.6)
9.6
27. (0.9)(−0.8)
−2.8
31. (−3.2)(8)
−13.2
20. −9(0.06)
−43.4
−0.56
24. −22(−0.4)
8.8
28. (0.5)(−0.9)
−7.2
32. (−5.7)(3)
−25.6
35. (−0.3)(−7.3)
1.68
1.2
16. (0.7)(−62)
−2.52
−0.6
34. (−0.4)(−4.2)
1.89
4.
4.8
18. 12(−0.05)
30. (3.3)(−4)
−7.2
33. (−0.9)(−2.1)
8(0.6)
26. (−0.4)(0.7)
−2.4
98—Perform the indicated operation(s).
3.
6.3
14. (−0.4)(47)
−1.92
29. (2.4)(−3)
4—Divide.
2.
13. (−0.6)(32)
17. 8(−0.04)
3—Multiply.
−4.5
−17.1
36. (−0.8)(−5.1)
2.19
4.08
37. (−3.7)(0.8)
−2.96
38. (−6.9)(0.7)
−4.83
39. (7.3)(−0.5)
−3.65
40. (8.2)(−0.6)
−4.92
41. (9.1)(−4.6)
−41.86
42. (8.3)(−6.3)
−52.29
43. (−5.4)(3.2)
−17.28
44. (−6.9)(4.1)
−28.29
45. (−1.4)(−3.8)
46. (−1.7)(−4.3)
5.32
49. (0.453)(−10000)
47. (−2.6)(−3.2)
7.31
50. (−100)(−0.204)
20.4
48. (−3.4)(−2.8)
8.32
51. (0.005)(−100)
−0.5
9.52
52. (−1000)(−0.27)
270
−4530
53. (−0.08)(−0.01)
57. (0.0002)(−8)
61. 24 ÷ 0.6
65. 5.6 ÷ 8
0.0008
−0.0016
40
0.7
69. −3.28 ÷ (−100)
0.0328
54. (−0.1)(96)
55. (−0.001)(−2.5)
−9.6
58. (0.0007)(−7)
−0.0049
59. (−6)(0.0006)
62. 36 ÷ 0.6
60
63. 45 ÷ 0.9
66. 7.2 ÷ 12
0.6
67. 4.8 ÷ 3
70. 450 ÷ (−10000)
0.0025
−0.0036
56. (−33)(0.001)
−0.033
60. (−9)(0.0003)
−0.0027
64. 42 ÷ 0.7
50
68. 7.2 ÷ 4
1.6
71. −0.1 ÷ (−1000)
0.0001
60
1.8
72. 650 ÷ (−100)
−6.5
−0.045
73. −5.1 ÷ (0.01)
77. 32 ÷ (−0.8)
−510
85. −1.2 ÷ 0.4
78. 27 ÷ (−0.3)
−40
81. −25 ÷ (−0.25)
100
93. (−21) ÷ (−0.07)
97. −0.56 ÷ (7)
300
−0.08
101. 0.12 ÷ (−0.4)
−0.3
105. −0.32 ÷ (−0.8)
0.4
200
94. (−28) ÷ (−0.04)
700
−0.05
102. 0.16 ÷ (−0.8)
83. −33 ÷ (−0.3)
91. −7.8 ÷ 13
−0.4
−0.2
106. −0.24 ÷ (−0.8)
109. −0.18 ÷ 0.3
−0.6
110. −0.16 ÷ 0.4
−0.4
113. 4.2 ÷ (−0.6)
−7
114. 3.5 ÷ (−0.7)
−5
0.3
−63000
76. −0.42 ÷ (−0.01)
80. (−32) ÷ 0.4
−60
87. 5.4 ÷ (−0.6)
−7
98. −0.45 ÷ (9)
75. 63 ÷ (−0.001)
79. (−42) ÷ 0.7
−90
90. 7.2 ÷ (−18)
−0.2
0.8
82. −50 ÷ (−0.25)
86. −6.3 ÷ 0.9
−3
89. 4.8 ÷ (−24)
74. −0.08 ÷ (−0.1)
−80
84. −44 ÷ (−0.2)
110
88. 2.5 ÷ (−0.5)
−9
92. −4.2 ÷ 14
−0.6
95. (−72) ÷ (−0.09)
800
42
220
−5
−0.3
96. (−81) ÷ (−0.09)
900
99. 0.36 ÷ (−4)
−0.09
100. 0.64 ÷ (−8)
−0.08
103. −0.15 ÷ 0.3
−0.5
104. −0.24 ÷ 0.4
−0.6
107. −0.36 ÷ (−0.9)
111. 0.21 ÷ (−0.7)
115. −4.8 ÷ (0.6)
ALG catalog ver. 2.6 – page 14 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
0.4
−0.3
−8
108. −0.40 ÷ (−0.8)
112. 0.28 ÷ (−0.4)
116. −8.1 ÷ (0.9)
0.5
−0.7
−9
AJ
117. −0.48 ÷ 0.12
121. (−0.09) ÷ (−0.3)
125. 0.04 ) 2.66
66.5
129. −0.3 ) −0.57
133. −0.8 ) 33.2
137. 0.05 ) −2.91
118. −0.64 ÷ 0.16
−4
1.9
−41.5
−58.2
0.3
119. 0.55 ÷ (−0.11)
−4
122. (−0.16) ÷ (−0.4)
126. 0.8 ) 40.4
50.5
130. 0.6 ) −2.4
−4
134. −0.05 ) 4.17
0.4
123. (−1.44) ÷ (−1.2)
127. 0.005 ) 0.037
131. −0.5 ) 37
−83.4
138. −0.006 ) −0.255
42.5
120. 0.44 ÷ (−0.11)
−5
7.4
−74
1.2
−4
124. (−1.21) ÷ (−1.1)
128. 0.02 ) 5.29
264.5
132. 0.7 ) −0.84
−1.2
1.1
135. −0.2 ) −0.63
3.15
136. −0.04 ) 0.162
−4.05
139. 0.07 ) −0.266
−3.8
140. −0.3 ) −4.05
13.5
ALG catalog ver. 2.6 – page 15 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
AK
Topic:
Converting fractions to decimals.
Directions:
42—Write as a decimal.
1.
−
5.
3
4
9.
−
13. −
17.
3
10
−0.3
0.75
1
8
−0.125
49
50
1
25
−0.98
0.04
21. −
7
1000
25. −
5
2
29. −
27
20
33. 1
44
25
45.
1
3
1.3125
1.76
5
6
11
15
57. −
2
9
61. −
1
11
10
11
69. −
73.
−0.93
0.3
49. −2
65.
−1.35
93
100
41.
53.
−2.5
5
16
37. −
−0.007
5
7
55
27
−2.83
0.73
−0.2
−0.09
0.90
−0.714285
2.037
1
2
2.
−
6.
4
5
10.
1
16
−0.5
0.8
0.0625
14. −
1
100
18. −
11
20
22.
30.
7
4
34. −2
38.
101
25
46. −
2
3
50. 1
1.16
29
30
58.
5
9
62.
5
11
66. −
−0.96
6
11
14
27
23. −
27.
35. 3
43.
0.45
59
10000
6
1000
1
6
55. −
8
9
63. −
−0.006
−0.16
5.6
2
15
−0.13
8.
−
12.
5
8
16.
1
50
−0.63
0.1
1
5
−0.2
0.625
0.02
7
25
−0.28
24.
27
1000
0.027
28.
21
2
10.5
32.
13
5
2.6
40.
44. −
48.
3
20
64.
0.85
203
50
5
6
−4.06
0.83
52. −10
56.
−5.15
850
1000
7
30
60. −
0.8
7
11
1
10
36. −5
1.44
2
3
51. 5
−1.1
3.875
36
50
47. −
−0.0059
3.25
11
10
4.
20. −
0.05
7
8
39. −
−0.4375
0.31
13
4
31. −
59.
0.5
22
7
74. −
−0.32
−0.6
1
6
54. −
70.
1
20
0.0133
−0.25
7
16
19.
12.13
42. −
11. −
1
4
−0.55
−2.34
1213
100
−
31
100
1.375
17
50
7.
0.4
15.
−1.75
11
8
2
5
0.01
133
10000
26. −
3.
4
9
8
11
1
3
−10.3
0.23
−0.4
0.72
−0.54
67.
4
11
0.36
68. −
3
11
−0.27
3.142857
71.
4
13
0.307692
72. −
25
13
−1.923076
−0.518
75. −
ALG catalog ver. 2.6 – page 16 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
79
37
−2.135
76.
9
37
0.243
AL
Topic:
Understanding rational numbers.
Directions:
50—Write as a rational number, if possible.
126—Which are rational numbers?
44—Write as a fraction.
1.
0.02
1
50
2.
−0.05
1
− 20
3.
0.15
5.
0.66
33
50
6.
−3.32
− 83
25
7.
−0.54
9.
−2.08
13. −0.0023
17. 0.875
29.
26. 4 − π
irrational
√
2
√
√
√
30. − 5
√
34.
9 31
irrational
− 21
20
13
200
irrational
− 13
50. 0.6
53. 0.27
3
11
54. −0.45
2
9
61. −0.016
1
− 60
65. 0.36363636 . . .
4
11
− 411
200
√
48
56. −0.09
60. 0.8
62. 0.083
1
12
63. 0.06
1
15
64. −0.03
67. −0.54545454 . . .
68. 0.72727272 . . .
1
12
75. −4.93
− 74
15
76. −3.56
− 107
30
80. 0.227
5
22
79. −0.681
82. 0.162
6
37
83. 1.185
25
99
1
− 30
72. 0.083
− 41
33
− 10
13
8
9
5
12
78. −1.24
86. −0.769230
6
− 11
1
− 11
71. 0.416
86
33
89. 0.25
1
6
9
11
77. 2.60
3
7
52. 0.16
55. 0.81
74. 0.13
85. 0.428571
irrational
− 19
2
15
irrational
48. −0.010110111 . . .
59. −0.1
− 13
6
11
1
irrational
− 59
− 10
11
irrational
44. 0.515515551 . . .
58. −0.5
66. −0.90909090 . . .
irrational
√
40. − 50
− 56
11
30
− 35
37
√
32. − 3
√
36.
121
irrational
73. 0.36
81. −0.945
28. π + 5
irrational
51. −0.83
70. −2.16
− 11
6
101
10000
24. −2.055
47. 2.303303330 . . .
5
− 11
16. 0.0101
17
16
irrational
2
3
− 41
50
23. 1.0625
irrational
49. −0.3
17
4
4
− 125
43. −0.090090009 . . .
46. 7.505005000 . . .
4.25
20. −0.032
39.
irrational
8.
6
− 25
1
− 250
irrational
45. −0.292292229 . . .
69. −1.83
irrational
−0.24
19. −0.004
π
irrational
3
√
31.
10 irrational
√
35. − 169 − 131
42. −0.121221222 . . .
irrational
49
1000
27.
irrational
√
38. − 24
irrational
41. 0.101001000 . . .
57. 0.2
− 67
8
4.
12. −0.82
39
20
15. 0.049
317
− 1000
22. −8.375
157
50
33. − 4
37.
18. 0.065
− 27
50
11. 1.95
9
20
14. −0.317
23
− 10000
7
8
21. 3.14
25. π
10. 0.45
− 52
25
3
20
− 15
22
84. −3.259
32
27
87. −0.142857
− 17
− 88
27
88. 0.461538
90. 0.47
47
99
91. 3.31
328
99
92. 1.76
80
33
95. 0.57
19
33
96. 0.723
241
333
100. 4.713
4709
999
93. 0.402
134
333
94. 2.42
97. 3.027
333
110
98. 0.054
6
110
99. 1.049
ALG catalog ver. 2.6 – page 17 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1048
999
175
99
6
13
8
11
BA
Topic:
Using exponents. See also category IA (prime factorization).
Directions:
6—Simplify.
1.
x(x)
5.
4·x·x·x
53—Rewrite using exponents.
x2
4x3
54—Rewrite without any exponents.
2.
3a(−a)
−3a2
3.
−2r(−r)
6.
y·y·y
y3
7.
a·a·a·a
2r2
4.
(p)(p)(p)
a4
8.
−7 · y · y · y · y · y
p3
−7y 5
9.
2·y·y·3·y
6y 3
10. 5 · m · m · 2 · m · m
11. −3 · k · k · k · 4 · k
10m4
13. a · a · a · b · b · a · a · b · a · b
21. 122
27
22. 152
144
16. c · d · k · c · c · d · k · k · k · c
84m3 p5
18. 33
125
30r4
14. 2 · 3 · w · w · 2 · x · x · 3 · w · w · x
a6 b4
15. 6 · m · p · p · p · 7 · p · 2 · m · m · p
17. 53
12. 2 · r · 5 · r · r · 3 · r
−12k 4
225
36w4 x3
c4 d2 k 4
19. 92
81
20. 132
23. 27
128
24. 35
169
243
25. (−8)2
64
26. (−2)6
64
27. (−3)4
81
28. (−9)2
81
29. (−2)5
−32
30. (−5)3
−125
31. (−4)3
−64
32. (−2)7
−128
33. (−1)7
−1
34. (−1)10
35. (−1)19
−1
36. (−1)14
37. −(−11)2
−121
41. −(−3)3
27
1
38. −(−5)4
−625
39. −(−9)2
42. −(−2)5
32
43. −(−10)3
45. −162
−256
46. −142
−196
49. (−a)4
a4
50. (−x)6
x6
53. −(−c)9
54. −(−m)15
c9
47. −26
1000
−64
51. (−y)7
m15
−81
−y 7
55. −(−p)10
−p10
1
40. −(−10)4
−10000
44. −(−2)7
128
48. −202
−400
52. (−k)5
−k 5
56. −(−w)8
−w8
57.
1 5
2
1
32
58.
1 4
3
1
81
59.
1 3
4
1
64
60.
1 3
5
1
125
61.
7 2
8
49
64
62.
3 3
5
27
125
63.
2 4
3
16
81
64.
6 2
7
36
49
65.
1
− 12
1
144
66.
− 10
11
100
121
67.
− 17
2
1
49
68.
− 95
69.
− 25
8
− 125
70.
− 32
− 27
8
71.
− 43
3
− 64
27
72.
1
− 10
2
3
2
3
73. − − 37
2
9
− 49
5
74. − − 12
77. − − 25
3
8
125
78. − − 14
81.
−1 12
2
85. (0.2)3
2 41
0.008
82.
−1 14
2
3
3
86. (0.1)6
25
− 144
1
64
−1 61
64
9
75. − − 11
79. − − 12
83.
2 21
3
2
81
− 121
5
1
32
2
81
25
3
1
− 1000
76. − − 23
4
− 16
81
80. − − 43
3
64
27
15 58
84.
3 31
2
0.000001
87. (0.5)3
0.125
88. (0.3)3
11 19
0.027
89. (0.03)2
0.0009
90. (0.08)2
0.0064
91. (0.1)5
0.00001
92. (0.02)4
93. (0.13)2
0.0169
94. (0.15)2
0.0225
95. (1.1)2
1.21
96. (1.4)2
97. (−1.7)2
2.89
98. (−0.2)6
0.000064
99. (−0.07)2
0.0049
100. (−0.3)4
0.0081
101. (−0.2)5
−0.00032
102. (−0.5)3
−0.125
103. (−0.04)3
−0.000064
104. (−0.1)7
−0.0000001
ALG catalog ver. 2.6 – page 18 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
0.00000016
1.96
BB
Topic:
Laws of exponents: multiplication. See also categories DK and DM (multiplying monomials).
Directions:
6—Simplify.
3—Multiply.
1.
(a · a · a · a)(a · a)
a6
2.
(r · r · r)(r · r · r · r)
3.
(y · y)(y · y · y)(y)
y6
4.
(x · x · x)(x · x)(x · x · x · x)
5.
(3 · y · y)(5 · y · y · y)
6.
(−8 · m · m · m · m)(2 · m · m · m)
7.
(4 · p · p · p)(−6 · p · p · p)
−24p6
8.
(7 · w · w)(−2 · w · w · w)(−3 · w · w)
9.
x2 x
x3
10. a · a2
13. r3 r5
r8
14. y 4 y 5
15y 5
11. c2 c2
a3
y9
m10
20. k 3 k 9
z8
19. m5 m5
21. w10 w5
w15
22. x8 x8
x16
23. y 25 y 5
29. 87 (86 )
30. 75 (75 )
813
33. (−10)8 (−10)3
−1011
31. 98 · 97
37. y 3 y 2 y
y6
38. m2 m2 m3
41. r4 · r2
r6
42. c3 · c7
1012
915
35. −74 (−7)5
−59
39. p · p5 · p3
m7
46. (d4 )(d2 )(d6 )
d12
47. z 7 · z · z 5 · z
49. x · x2 · y 2
x3 y 2
50. a2 · a3 · b · b2
a5 b3
51. m4 · p3 · p2 · p
57. −x(x3 )
−x4
61. (−c)(−c)3
65. −x2 · x5
54. g 5 · h · h6 · g 3
c4
−x7
g 8 h7
58. (−y)3 (y 2 )
−y 5
62. (m)(−m)4
m5
66. −y 4 (−y)6
−y 10
67. (−x)3 (−x)3
59. (−a)2 (−a)
m4 p6
w6 x5
28. 10 · 105
106
32. 610 · 610
620
811
70. −x2 (−x)2
−x4
71. (−w)5 (−w)3
73. (−y 3 )(−y 2 )
y5
74. (x2 )(−x5 )
−x7
75. (−a)(−a3 )(a4 )
w8
a8
r6 w6
a4 b8
−r3
−k 8
68. (y)(−y)5 (−y)
x6
−a5
b15
56. a · a · b3 · a2 · b5
64. −k 4 (−k)4
r4
w11
m20
52. r · r5 · w3 · w3
60. −r(−r)2
−a3
69. (−a)2 (−a)3
ALG catalog ver. 2.6 – page 19 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
a31
48. b · b4 · b9 · b
z 14
55. w2 · x2 · w4 · x3
63. −r2 (−r2 )
k 12
44. m8 · m3 · m9
k 13
p12
c3 d3
h7
40. w4 · w2 · w2 · w3
p9
45. p(p8 )(p3 )
53. c · d3 · c2
p5
36. −(−8)7 (−8)4
79
43. k 2 · k 6 · k 5
c10
42w7
24. a17 a14
y 30
27. 104 · 108
1010
710
34. −56 · 53
12. p3 p2
c4
16. h4 · h3
18. z 7 · z
26. 102 (108 )
−16m7
d6
q8
107
x9
15. d2 · d4
17. q 4 · q 4
25. 103 (104 )
r7
y7
72. (−p)4 (−p)3 (−p)2
−p9
76. (−c3 )(−c3 )(−c4 )
−c10
BC
Topic:
Laws of exponents: power to a power. See also categories DK and DM (multiplying monomials).
Directions:
6—Simplify.
3—Multiply.
1.
(x · x)(x · x)(x · x)
3.
(a · a · a)(a · a · a)(a · a · a)
5.
(−4 · c · c · c)(−4 · c · c · c)
7.
(7 · y · y · y · y · y)(7 · y · y · y · y)
9.
(23 )2
64
10. (22 )3
64
11. (32 )3
729
12. (33 )2
729
13. (x2 )4
x8
14. (y 3 )2
y6
15. (a2 )5
a10
16. (c4 )3
c12
17. (x5 )7
x35
18. (r8 )4
r32
19. (y 7 )10
21. (103 )3
(y · y · y · y)(y · y · y · y)
a9
4.
(m · m)(m · m)(m · m)(m · m)(m · m)
16c6
6.
(3 · p · p)(3 · p · p)(3 · p · p)
8.
(−5 · x · x · x)(−5 · x · x · x)(−5 · x · x · x)
49y 9
22. (104 )5
109
25. (y 3 )(y 2 )3
2.
x6
y9
29. (a3 )2 (a4 )2
a14
m10
27p6
20. (r12 )5
y 70
23. (86 )4
1020
y8
r60
24. (75 )6
824
26. (k 2 )3 (k 4 )
k 10
27. p7 (p4 )2
30. (c5 )2 (c2 )3
c16
31. x3 (x4 )2 (x5 )
−125x9
730
28. (r)(r3 )2
p15
r7
32. (y 2 )4 (y 3 )4 (y 6 )
x16
33. (−k 2 )2
k4
34. (−a3 )4
a12
35. −(−x5 )2
−x10
36. −(−p6 )4
−p24
37. (−c3 )5
−c15
38. (−y 2 )3
−y 6
39. −(−r4 )7
r28
40. −(−w3 )5
w15
41. (−a)3 (−a2 )3
a9
42. (−c)2 (−c3 )2
43. −(x2 )4 (−x3 )2
c8
−x14
y 26
44. (−y 4 )(−y 3 )(−y 2 )5
−y 17
45.
−(−m)3
6
m18
46.
−(−r)4
49. (2a3 )3
8a9
50. (5x3 )2
53. (7p8 )2
49p16
54. (12r7 )2
58. (4a5 b)2
57. (6w2 y 4 )2
36w4 y 8
3
−r12
47.
−(−p)2
6
48.
p12
−(−k)5
51. (9y 4 )2
81y 8
52. (4m5 )3
144r14
55. (5k 5 )3
125k15
56. (3x2 )4
16a10 b2
59. (5m3 p6 )4
625m12 p24
60. (6cd3 )3
25x6
4
k 20
64m15
81x8
216c3 d9
61. (p4 r2 q)3
p12 r6 q 3
62. (km3 n5 )2
k 2 m6 n10
63. (w2 x2 y)4
w8 x8 y 4
64. (a2 bc3 )5
a10 b5 c15
65. (−3r4 )3
−27r12
66. (−10p2 )3
−1000p6
67. (−m2 n)5
−m10 n5
68. (−rq 2 )7
−r7 q 14
71. (−r3 s2 )4
r12 s8
72. (−xy 3 )6
x6 y 18
69. (−10m3 )2
100m6
70. (−8y 9 )2
73. (−a2 b4 c)3
−a6 b12 c3
74. (−w6 xy 4 )5
77.
2 3 2
3x
81.
1 3 2 4
2a b
85. (0.4x)2
89. (0.1cd2 )3
4 6
9x
1 12 8
16 a b
78.
1 4 3
3y
82.
4
5 2
5 wx
64y 18
1 12
27 y
86. (0.1y 3 )2
0.16x2
0.001c3 d6
−w30 x5 y 20
90. (0.2r5 p4 )3
16 2 10
25 w x
75. (−m2 p2 r3 )6
3
79.
− 15 m2
83.
− 57 m2 b
0.008r15 p12
1
− 125
m6
2
87. (0.7w7 p)2
0.01y 6
m12 p12 r18
25 4 2
49 m b
0.49w14 p2
91. (−1.1k 4 m)2
1.21k 8 m2
76. (−cdk 2 )4
c4 d4 k 8
2
80.
− 34 p5
84.
1 3 5
− 10
c d
9 10
16 p
3
88. (0.5a8 b4 )2
1
− 1000
c9 d15
0.25a16 b8
92. (−0.3x4 yz 7 )3
−0.027x12 y 3 z 21
93.
(x3 )2
6
94.
x36
5
97. − −(−p)3
−p15
(y 5 )5
5
95.
y 125
3
98. − −(−m)4
m12
(n4 )3
2
2
99. − −(−r2 )3
ALG catalog ver. 2.6 – page 20 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
96.
n24
−r12
(a3 )3
3
a27
3
100. − −(−a4 )2
a24
BD
Topic:
Laws of exponents: division. See also categories DL and DM (dividing monomials).
Directions:
6—Simplify.
4—Divide.
1.
x·x
x
5.
c·c·c
c·c·c·c·c
9.
26
2
32
10.
53
5
25
11.
34
33
15.
6
63
1
36
16.
3
34
1
27
17.
107
102
21.
x4
x
x3
22.
y8
y
y7
23.
27.
d12
−d3
28.
−y 15
y 11
33.
y4
y4
34.
k9
k9
39.
−b16
−b7
b9
40.
−w21
−w19
45.
−q 14
q8
−q 6
46.
m17
−m9
51.
c
c6
52.
a
a8
57.
x6 y
x2 y 3
58.
k8 m
k 5 m2
63.
w2 x4
−w2 x5
64.
g 2 h6
−g 8 h6
x
1
c2
−d9
1
1
c5
x4
y2
−
69. 107 ÷ 1010
73. a6 ÷ a2
1
1000
y2
3.
a·a·a·a
a·a·a
6.
m·m
m·m·m
1
m
7.
k
k·k·k·k·k
1
k4
12.
25
23
13.
52
54
105
18.
104
103
19.
106
10
m7
m
m6
24.
p5
p
25.
−r9
r5
29.
w2
w6
1
w4
30.
m4
m5
31.
a3
a8
35.
−r6
r6
36.
c13
−c13
37.
x14
−x3
w2
41.
d11
d13
42.
x16
x19
43.
p5
−p12
−
−m8
47.
−y 25
y9
48.
x20
−x12
−x8
49.
b5
−b
53.
w3
w3 y
1
y
54.
k 5 m2
k
m2
55.
a2 b6
b6
59.
p4 r 3
p3 r 5
p
r2
60.
a10 b2
a 4 b9
a6
b7
61.
−c7 d5
cd5
65.
mp2 r4
mp5 r
66.
a6 bc2
a7 bc5
1
ac3
67.
xy 8 z 2
x2 y 9 z
−y 4
1
1
a7
k3
m
−
1
g6
74. y 12 ÷ y 3
82. c7 ÷ c
89. cd3 h3 ÷ c4 dh2
d2 h
c3
93. −a4 x3 y 2 ÷ a6 x3 y
−1
1
d2
y
a2
1
y 16
r3
p3
10
p4
1
m
−1
1
x3
k6
75. x5 ÷ x11
83. p ÷ p5
c6
1
x
94. −kr9 q 2 ÷ k 3 rq 5
r8
k2 q3
w·w·w·w
w·w·w·w
20.
103
105
1
100
26.
c5
−c2
−c3
32.
b5
b9
−x11
38.
a10
−a2
−a8
1
p7
44.
−r11
r17
−
−b4
50.
−d11
d
−d10
a2
56.
wz 9
z8
−c6
62.
−x3 y 2
x3 y
−y
z
xy
68.
k 7 n3 r 3
k 2 n2 r 3
k5 n
105
−r4
1
a5
1
m5
1
p4
80. −k 9 ÷ (−k 9 )
84. a ÷ a6
−
1
w
88. −c7 ÷ c6
wy
x5
1
22
26
76. m4 ÷ m9
1
p3
14.
1
25
1
x6
95. c2 d3 f ÷ (−c2 d3 f 7 )
ALG catalog ver. 2.6 – page 21 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
8.
10
91. w3 xy 2 ÷ w2 x6 y
1
− 6
f
p·p·p·p·p
p·p
72. 109 ÷ 108
87. −w14 ÷ w15
1
a2 c7
4.
1
79. −w14 ÷ (−w14 )
−1
−
a
4
71. 108 ÷ 108
90. a2 b7 c ÷ a4 b7 c8
−
−
y9
86. x6 ÷ (−x7 )
−y
3
1
10
78. −q 16 ÷ q 16
−1
x8
85. y 5 ÷ (−y 4 )
y·y·y·y
y·y
70. 107 ÷ 108
a4
77. −r7 ÷ r7
81. x9 ÷ x
1
x
2.
1
16
1
b4
1
r6
wz
1
1
a5
−c
92. k 5 r2 p9 ÷ k 5 r6 p3
p6
r4
96. m5 pr9 ÷ (−m6 p2 r10 )
−
1
mpr
BE
Topic:
Combined methods (laws of exponents). See also categories DK–DM.
Directions:
6—Simplify.
6
2.
−
6.
5
4d
3
10.
xy 2
14.
mr4
− 2
p
18.
7d3
12a4
22.
y8
(y 2 )(y 3 )
26.
(k 2 )(k 8 )
k 11
1.
a 3
3b
5.
−
3m
2
4
81m4
16
9.
−
1
km
5
−
13.
w2
xy 3
17.
−
21.
(a5 )(a6 )
a9
25.
m7
(m4 )(m3 )
29.
(x2 y 6 )(x2 y)
x5 y 6
33.
p5 w
(−pw2 )(p3 w3 )
37.
(km)2
km
km
38.
(xy)3
xy
41.
(p2 )6
p2 · p 6
p4
42.
r3 · r4
(r3 )4
45.
(−x4 )2
−x4
46.
−a3
(−a3 )2
−
49.
r 3 p6
(rp2 )4
50.
c4 d5
(c3 d)3
d2
c5
53.
(a2 b5 )2
(−a2 b)5
−
54.
(−g 3 h)3
(g 2 h2 )4
−
57.
10x3 z
(5xz 2 )2
2x
5z 3
58.
(6cd2 )2
24c5 d4
3
2c3
61.
−r3 p(−rp)5
(r4 p2 )2
62.
(−km2 )4
(km)3 (km5 )
a3
27b3
6
11a5
6c3
1
k 5 m5
w12
x6 y 18
2
121a10
36c6
a2
1
y
x
30.
−
−x4
1
rp2
b5
a6
p2
p
w4
34.
2c
d
64c6
d6
3x
y
8.
−
1
2d
12.
−
7
rw
2
49
r2 w2
16.
c5
k2 m
3
c15
k 6 m3
20.
2r2
3p
24.
x3
(x6 )(x)
1
x4
28.
(c9 )
(c3 )(c5 )
c
32.
(ab3 c2 )(abc3 )
(a6 b2 c)
b2 c4
a4
36.
(a2 x5 )(−ax4 )
(a3 x7 )
−x2
40.
(rw)4
(rw)6
44.
(c3 )2
(c3 )(c2 )
c
1
48.
(−k 2 )3
(−k 3 )2
−1
−y
52.
(−a3 b4 )3
a8 b9
−ab3
y9
x2
56.
(−wy 2 )4
(wy 3 )3
w
y
2b
a
60.
(9p2 w)2
(6p2 w3 )2
9
4w4
64.
(w2 z 4 )3
(−wz 5 )2 (w4 z 2 )
r5
32w5
125
64d3
7.
10y
3
x2 y 2
81
11.
ab
10
15.
a3 b
− 2
c
4
a12 b4
c8
19.
−
3w
4z 4
3
−
y3
23.
r(r4 )
r10
1
k
27.
(a6 )(a4 )
a10
31.
(g 3 k)(gh5 k 2 )
g 2 hk 3
35.
(y 3 z 2 )(−y 5 z 3 )
(y 4 z 5 )
x2 y 2
39.
(ab)2
(ab)5
1
r5
43.
(a5 )(a2 )
(a5 )2
47.
(m3 )2
(m2 )3
51.
−x4 y 3
(x2 y)2
55.
(x2 y 3 )5
(−x2 y)6
59.
(4a3 b2 )3
2a2 b)5
63.
ab4 (−a3 b)2
(−ab2 )3
9
5
m5 r20
− 10
p
2
49d6
144a8
r 4 m7
m3
r2
(r2 m)(r4 m3 )
r 9 p3
(rp6 )(−r3 p4 )
−
1
a3
g
h5
1
r5
p7
ALG catalog ver. 2.6 – page 22 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−
4
4.
r 5
2w
3.
−
2
100y 2
9
3
a3 b3
1000
27w3
64z 12
1
r5
1
g 2 h4
1
a3 b3
1
a3
−a4
−y 4
81x4
y4
5
4
−
1
32d5
16r8
81p4
1
r2 w2
1
BF
Topic:
Variable exponents.
Directions:
6—Simplify.
1.
ax ay
5.
10x · 10x · 10x
9.
5(5x )
ax+y
(2a )b
c
21. (y a )b
xa · xb
6.
y a y 3a
xa+b
y 4a
10. 10y · 10
5x+1
13. y 3n · y 1−n
17.
103x
2.
y 2n+1
18.
2abc
(x4 )a
22. (ax )y
y ab
25. (x4 y)a (xy 2 )a
x5a y 3a
29. (m2k+1 )2 (mk+2 )3
2
x2 · xc
7.
m2k · mk · m4k
x2+c
11. (3c+1 )(3)
10y+1
14. x2a−b · x2a+b
3.
19.
x8a
(y m )3
23. (y 6 )a
axy
26. (a2 b2 )x (ax b3x )2
a4x b8x
30. (x2 y n−4 )3 (x3 y n+6 )2
r
8.
−x2n (−xn )
y r+4
20.
(k x )5
24. (x5 )b
y 6a
2
10n
a4n
a5n−2
k 10x
x5b
28. (x2a−1 )3
y 2n+2
31. (an+1 )3 · an−3
x3n
16. an−3 · a4n+1
n4
y 3mr
27. (y n+1 )2
yr y4
12. 10 · 10n−1
3c+2
15. n2−a · n2+a
x4a
m7k
4.
x6a−3
32. (y 2 )x+1 · y 1−2x
y3
x12 y 5n
m7k+8
33.
ax
a
37.
x3n
xn
41.
k x+2
k 2−x
45.
(cx )(cy )
cz
49.
x(x2a )
xa+1
53.
(3x+y )2
32x−y
57.
34. mx ÷ my
ax−1
x2n
y 2n
y n+2
k2x
cx+y−z
xa
3
33y
y 3n−6
38.
m4a
m
42.
x4n−2
x2n−5
46.
a
ax ay
mx−y
36.
y5
ya
y k+4
40.
a6x
ax+1
a5x−1
m
44.
y 3−2a
y 1−2a
y2
48.
5x
5(5y )
5x−y−1
52.
(y a+1 )(y)
ya
56.
(x2a )3 (xa+3 )
(xa+1 )2
60.
35. 3n ÷ 3
3n−1
39.
y k+5
y
x2n+3
43.
m3a−1
m3a−2
a1−x−y
47.
nx ny
n
51.
m2x
(mx )(m)
a5
55.
23n+1
(2n−1 )3
x2a+4
59.
m4a−1
50.
an+2
a(a1+n )
54.
(ax+1 )3
(ax )2 (ax−2 )
58.
x2a−1
xa−3
2
1
ALG catalog ver. 2.6 – page 23 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
nx+y−1
mx−1
16
(a4x )(ax )
(ax−2 )3
2
a4x+12
y 5−a
y2
(an )2 (an+1 )
(a2 )
x5a+1
3
a9n−3
BG
Topic:
Negative and zero exponents.
Directions:
6—Simplify.
1
16
1.
4−2
5.
120
1
9.
0
4
−
5
13. 12(4−3 )
55—Rewrite using only positive exponents.
1
27
2.
3−3
6.
(2.6)0
1
1
10. −90
3
16
14. 35 · 21−1
5
3
3
4
1
64
23. (−3)−4
1
81
24. (−6)−1
−
1
1000
27. −12−1
26. −10−3
37. (12−1 )0
1
38.
−
1
32
(−10)0
42. (30 · 12)−2
45. (6−3 )(63 )
1
46. 37 · 3−8
2−2
−8
−
1
32
−1
3
53.
4
50.
7
2−3
1
1
144
1
3
39. (30 )−4
51.
23
2−2
32
58.
3−5
3−3
1
9
59.
2−2
52
61.
5−1
100
1
5
62.
140
7−2
49
63.
60
−3
65.
(2−2 )−2
66.
(3−3 )−1
69.
70.
73.
77.
(−3)3
32
−2
1
9
2−2
23
1
+ 3−1
6
5
74.
4−1
1 + 2−1
2−1 + 3−1
2−1 − 3−1
5
78.
4−1 + 2−2
4−2 + 2−4
2−1
81. −a−2
−
1
a2
82. 3x0
3
40.
1
64
32
1
6
71.
75.
4
79.
ALG catalog ver. 2.6 – page 24 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−3
5−1
−3
3
56.
2
8
27
1
3
64.
2−5
20
1
32
" #−2
2 −2
68.
3
1
49
−
1
8
12
−3
100
−15
1
1
r5
1
1
9
8−1
2−3
1
− 4−1
15
4
48. (10−3 )(105 )
16
−3
1
125
0
44. 3−2 · 90
52.
2−1
+ 2−1
3−1
83. r−5
(−5)3
60.
24
(−2)3
3−1
1
100
−
1
243
36. 22 − 2−2
" #−2
1 −1
67.
7
27
−1
17
4
1
50
−4
1
55.
2
81
25
−
1
6
32. (10 − 5)−3
8
10−1
5
57.
16
1
121
43. (4 · 20 )−3
1
9
28. −3−5
1
47. 82 · 8−1
56
−2
5
54.
9
4
3
1
12
35. 2−2 + 22
−1
1
25
−
31. (4 + 7)−2
3
4
41. 100 · 5−2
49.
1
8
30. (1 + 1)−5
34. 1 − 2−2
−1
20. 33 · 6−2
1
625
10
3
12. −(0.7)0
−1
1
22. (−4)−3
33. 3 + 3−1
1
19. 34 (9−2 )
1
49
−
8.
0
2
3
16. 4(6−2 )
21. (−7)−2
1
81
1
5−2
1
200
18. (10−3 )(53 )
29. (5 − 2)−4
7.
1500
1
25
4.
15. 5 · 10−3
8
−
2−4
11. −(−12)0
−1
17. 2−1 · 42
25. −5−4
1
16
3.
1
8
−43
(−4)2
72.
76.
2−1
1 − 2−1
80.
−2
1
16
1
2−2 + 4−1
5−1
84. 10y −1
16
81
10
y
−2
4
25
BG
6y 2
x3
85. 6x−3 y 2
86. −cd−4 e−1
1
m10
89. (m−3 )(m−7 )
94. (z −2 )(z 9 )
97. (−2r−5 )(−3r7 )
101. −6a−5 b6 · 5a4 b−2
−
z7
98. (3n−1 )(−5n−3 )
6r2
2
r8
87. 2p0 r−8
1
c8
90. c−4 · c−4
1
y
93. y −1 · y 0
c
d4 e
−
−
102. 9x−1 y −4 · 4x4 y 4
15
n4
91. k −6 · k −1
1
k7
92. (w0 )(w−9 )
95. (a8 )(a−4 )
a4
96. x−7 · x
99. −x−3 · 7x0
36x3
7
x3
−
42x
y6
106. (nr−6 )(−2n0 r−2 )
−
2n
r8
1
x6
−
107. −7ay 3 · 5a−1 y −2
10c
104. (−3u−5 w0 )(5u5 w−6 )
8
c5 d
a
1
w9
100. 5c−4 · 2c5
103. (−4c−7 d4 )(−2c2 d−5 )
30b4
105. 14x−4 y −3 · 3x5 y −3
u3 w
z3
88. u3 wz −3
−35y
15
w6
108. (13k 5 m−3 )(2k −3 m−1 )
26k 2
m4
109. (3u−3 )−3
u9
27
110. (6t−5 )−1
t5
6
111. (2x−4 )0
113. (pr−1 )−1
r
p
114. (x2 y 0 )−5
1
x10
115. (m2 r)−3
y 12
16
117. (2x0 y −3 )−4
121. (6p−4 )(3p2 )2
122. (5t−2 )2 (10t)−1
54
125. (−3r−2 )−2 (r−3 )
a4 b2
49
118. (7a−2 b−1 )−2
r
9
129. (x2 y 3 )0 (5x2 y)−3
126. (−k 3 )(2k 3 )−4
130. (a0 b5 )−3 (a2 b8 )2
1
m6 r3
s4
10t3
119. (10s−4 t3 )−1
120. (−6w4 z −3 )0
123. (3z −2 )−3 (9z −3 )
z3
3
1
16k9
127. (2d−2 )−2 (−d2 )3
−
a4 b
131. (3w2 y −3 )−1 (6w2 y)−2
133. (20kn−3 )−1 (4k −1 n2 )−2
134. (4r2 t−2 )3 (6r−2 t3 )0
d10
4
128. (2s−2 )−3 (2s3 )−2
w−3
w−4
135. (x−3 z 4 )4 (2x−6 z 8 )−2
1
4
136. (3a3 c−2 )−2 (a−4 c0 )−3
c4 a6
9
138.
r−2
r4
141.
9c0
12c−2
3c2
4
142.
−4y 2
8y −1
145.
5x−6
−y −1
5y
x6
146.
12b−5
3a0
149.
27s0 t−2
6s−2 t4
150.
x3 y −4
x−1 y −4
153.
w−5 x−1 y 3
w−5 xy −2
y5
x2
154.
10ab−7 c−2
15a−2 b0 c−3
157.
2−4 a3 b−3
8−2 ab−4
4a2 b
158.
3−1 s−2 t
3−4 s3 t5
27
s5 t 4
161.
(−2n3 )−2
4n−5
1
16n
162.
(3k −4 )−1
−12k 2
−
165.
(2t−3 )−3
(2−2 t0 )2
166.
(3x2 y 0 )4
(3−2 x−2 y)−2
w
−
9s2
2t6
2t9
1
r6
−
y3
2
4
b5
x4
2a3 c
3b7
k2
36
x4 y 2
140.
k −8
k −5
2
x8
144.
9d−3
3d−3
m3
9k 2
148.
−15r0
30p−2
152.
w−1 z −3
w−2 z 2
156.
21x2 y 0 z −1
3x−3 y 6 z −2
160.
7c−2 d−3
2−3 c−5 d2
164.
5a5
(2a2 )−3
168.
(5−1 y −4 )0
(10y −2 )−2
139.
s6
s−1
143.
10x−4
−5x4
147.
2k −2
18m−3
151.
4c−2 d−2
2c0 d
155.
m−3 n−2 p2
m−4 n−2 p−1
159.
5−3 x2 y −4
10−2 xy −3
163.
6y −6
(3y 2 )0
167.
(c2 d)−2
(2−1 cd−3 )4
ALG catalog ver. 2.6 – page 25 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1
32
p
72
t6
137.
1
a14
132. (3p−2 r3 )−2 (2pr−2 )−3
64r6
k
320n
1
124. (2a3 )−2 (2a−4 )2
y
108w6
1
125x6 y 3
a3
b2
116. (a−3 b2 )−1
5
2t5
−
1
25y 12
112. (5y 6 )−2
1
s7
−
2
c2 d3
mp3
4x
5y
6
y6
16d10
c8
1
k3
3
−
p2
2
w
y5
7x5 z
y6
56c3
d5
40a11
100
y4
BG
169.
(23 )−1 x2 y −3
−4−2 x0 y −2
−
2x2
y
170.
3−4 b−5 c−1
(9bc2 )−2
c3
b3
171.
(−4m3 )2 (n−2 )2
(2−1 )2 m4 n−8
172.
(5−2 )−1 p3 r−5
(10−1 pr3 )−2
176.
2−5 (k −3 m2 )−1 n−7
(2−2 )3 k 4 (mn−1 )−2
p5 r
4
64m2 n4
173.
(3a−3 )−2 b2 c−4
(−6)−1 (a2 b−3 )2 c−6
−
177.
174.
(6x2 y −5 )−2 (2x2 y)4
(−5x−4 )−2 (3x−6 y 3 )2
175.
−3−2 c−4 (d0 e−2 )2
(6−1 )2 (cd)−2 e−4
4d2
c2
9
x5 z
2a2 b8 c2
3
178.
100x8 y 8
81
181.
(5−1 )2 x−5 (y −2 z)−3
15−2 (x0 y 3 z −1 )2
2
kn9
(m2 r2 )−4 (mr−6 )−3
(−m−5 r4 )3 (m4 r6 )−2
179.
−m12 r10
2−2 r2 s−1
2rs−3
−2
64
r 2 s4
182.
(u4 w3 )3 (2u−3 w−5 )4
(−2u−3 w)3 (uw−1 )−2
−
3−1 a−3 b2
6−2 a−3 b−2
−1
1
12b4
183.
180.
2u11
w16
(c6 d−3 )−3 (−3cd4 )−2
(c4 d0 )−5 (c−2 d)−4
d5
9c8
5−3 c−4 d−2
5−1 cd−4
−2
625c10
d4
184.
12−1 x3 y −3
2−3 x−1 y −6
−3
27
8x12 y 9
185. 4x2 +
3
x−2
189. (c + d)−1
193.
7 − 2s0
r2
186. −
7x2
1
c+d
−2
r4
25
2
− 6ab0
a−1
−1
190.
1
x+y
194.
2x0 y + 3y
x
187.
−8a
x
5y
195.
ALG catalog ver. 2.6 – page 26 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
6t0 − 4
2u
188.
y5
191. (1 + a−1 )−1
x+y
−1
2y 3
− y5
y −2
−3
a
a+1
u3
5
w−1
+
2w
2
3
w
192. (w−1 + z −1 )−1
196.
7m − 4mn0
12m
wz
w+z
−2
16
BH
Topic:
Using scientific notation I.
Directions:
56—Rewrite without scientific notation.
1.
5 × 104
5.
2.6 × 105
9.
6.1 × 101
2.
1 × 106
260000
6.
8.3 × 102
61
10. 1.55 × 104
0.008
14. 4 × 10−1
50000
13. 8 × 10−3
1000000
3.
3 × 103
830
7.
9.9 × 107
15500
0.00000019
18. 2.08 × 10−3
21. 0.02 × 108
2000000
22. 0.0052 × 104
0.00208
99000000
11. 7.02 × 105
15. 5 × 10−6
0.4
17. 1.9 × 10−7
3000
8.
4.5 × 101
16. 1 × 10−5
0.000005
0.0000000044
23. 0.000001 × 1011
52
9 × 107
90000000
45
12. 2.33 × 106
702000
19. 4.4 × 10−9
4.
2330000
0.00001
20. 3.07 × 10−2
0.0307
24. 0.00088 × 106
880
100000
25. 0.003 × 10−2
0.00003
26. 0.9 × 10−5
0.000009
27. 0.202 × 10−1
0.0202
28. 0.00056 × 10−3
0.00000056
29. 0.0805 × 103
80.5
33. 70000 × 10−3
70
37. 490 × 10−8
0.0000049
30. 0.0007 × 103
0.7
34. 650000 × 10−5
38. 1310000 × 10−10
6.5
31. 0.000099 × 105
9.9
32. 0.000004 × 104
0.04
35. 1000000 × 10−6
1
36. 902000 × 10−4
90.2
39. 500 × 10−4
0.000131
ALG catalog ver. 2.6 – page 27 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
0.05
40. 16.8 × 10−1
1.68
BI
Topic:
Using scientific notation II.
Directions:
57—Rewrite using scientific notation.
1.
50000
5.
420
9.
31500000
13. 90.6
5 × 104
4.2 × 102
3.15 × 107
9.06 × 101
17. 0.007
7 × 10−3
21. 0.000064
25. 0.0805
6.4 × 10−5
8.05 × 10−2
29. 2700 × 103
33. 133.8 × 102
37. 0.02 × 109
2.7 × 106
1.338 × 104
2 × 107
2.
600000
6.
93000
6 × 105
3.
3000
9.3 × 104
7.
15000000
10. 4710000000
14. 710.2
18. 0.05
4.71 × 109
7.102 × 102
5 × 10−2
22. 0.00011
1.1 × 10−4
2.28 × 101
23. 0.063
5 × 10−5
6.3 × 10−2
26. 0.00000334
3.34 × 10−6
27. 0.000127
30. 1050 × 102
1.05 × 105
31. 89000 × 104
35. 100.4 × 102
34. 92.02 × 105
9.202 × 106
38. 0.00084 × 107
42. 0.00035 × 103
1.4 × 10−3
3.5 × 10−1
3.9 × 10−2
15. 22.8
1.5 × 107
8.26 × 102
19. 0.00005
41. 0.00014 × 101
45. 390 × 10−4
11. 826
3 × 103
46. 5000 × 10−6
8.4 × 103
1.27 × 10−4
4.
9000000
8.
720000
9 × 106
7.2 × 105
12. 50400
5.04 × 104
16. 300.1
3.001 × 102
20. 0.0002
2 × 10−4
1.4 × 10−7
24. 0.00000014
28. 0.0000905
9.05 × 10−5
8.9 × 108
32. 688 × 103
6.88 × 105
1.004 × 104
36. 93.9 × 107
9.39 × 108
39. 0.0008 × 105
8 × 101
40. 0.0057 × 107
43. 0.0007 × 102
7 × 10−2
44. 0.000003 × 104
5.7 × 104
3 × 10−2
5 × 10−3
47. 92000 × 10−8
48. 61 × 10−2
6.1 × 10−1
9.2 × 10−4
49. 0.55 × 10−3
5.5 × 10−4
53. 0.000844 × 103
50. 0.092 × 10−1
54. 61.2 × 10−5
9.2 × 10−3
6.12 × 10−4
51. 0.001 × 10−2
55. 3.01 × 10−4
8.44 × 10−1
ALG catalog ver. 2.6 – page 28 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1 × 10−5
52. 0.28 × 10−6
3.01 × 10−4
56. 0.0566 × 101
5.66 × 10−1
2.8 × 10−7
BJ
Topic:
Simplifing expressions with scientific notation.
Directions:
8—Simplify, using scientific notation.
58—Rewrite with scientific notation, and simplify.
1.
610 × 20,000
3.
1800 × 430
5.
1,000,000 × 8,250,000
7.
1,000,000,000 × 390,000
3.9 × 1014
9.
75,000 × 5300 × 40,000
1.59 × 1013
10. 90,000 × 10,000 × 710,000
6.5 × 1011
12. 8400 × 1,300,000 × 500
1.22 × 107
7.74 × 105
8.25 × 1012
11. 325 × 100,000 × 20,000
13. (0.00055)(800,000)
4.4 × 102
15. (170,000)(0.000009)
1.53 × 101
17. 300 × 3,000,000 × 0.003
19. 0.0001 × 80,000 × 500
4 × 103
4 × 1026
23. (7 × 1017 )(4 × 107 )
2.8 × 1025
25. (5 × 104 )(41.2 × 10−2 )
27. (340 × 102 )(0.25 × 104 )
452,000 × 1000
4.
10,000 × 100,000
6.
45,000 × 126,000,000
8.
800,000 × 2,500,000
4.52 × 108
1 × 109
5.67 × 1012
2 × 1012
2.06 × 104
6.39 × 1014
5.46 × 1012
14. (23,000,000)(0.004)
9.2 × 104
16. (0.0006)(3,500,000)
2.1 × 103
18. 2500 × 0.00004 × 0.8
2.7 × 106
21. (8 × 1012 )(5 × 1013 )
2.
8 × 10−2
20. 0.01 × 0.0006 × 160
9.6 × 10−4
22. (4 × 109 )(1.5 × 106 )
6 × 1015
24. (3.5 × 10−5 )(2 × 1015 )
7 × 1010
26. (78 × 106 )(0.01 × 103 )
7.8 × 108
28. (0.03 × 10−4 )(120 × 105 )
8.5 × 107
3.6 × 101
29. (8 × 101 )(3 × 102 )(7 × 10−3 )
1.68 × 102
30. (5 × 103 )(5 × 106 )(5 × 109 )
31. (1.5 × 102 )(4 × 104 )(8 × 106 )
4.8 × 1013
32. (7 × 10−5 )(3.2 × 102 )(1 × 103 )
33. (2 × 10−4 )(3.7 × 10−7 )
7.4 × 10−11
34. (6 × 10−6 )(4 × 104 )
35. (1.1 × 1010 )(6 × 10−15 )
6.6 × 10−5
36. (2 × 10−8 )(8.5 × 10−1 )
37. 1,000,000,000 ÷ 250,000
4 × 103
38. 485,000,000 ÷ 5,000
39. 840,000,000 ÷ 2,400,000
3.5 × 102
40. 44,100,000 ÷ 9000
4.9 × 103
42. 42,000 ÷ 0.000014
3 × 109
41. 0.0256 ÷ 1600
43. 810,000 ÷ 0.36
45.
0.078
60,000
49.
2.4(0.0015)
360,000
1.6 × 10−5
1.3 × 10−6
1 × 10−8
46.
6
100,000
50.
0.18
22,500(0.004)
6 × 10−5
2 × 10−3
47.
360
0.00072
51.
28,600,000(0.003)
0.000033
1.7 × 10−8
9.7 × 104
5 × 105
4 × 10−11
48.
55
11,000,000
52.
0.0034
0.000017(0.00625)
2.6 × 109
53.
(1,800,000)(0.00035)
0.000014
54.
4.5 × 107
57.
(0.00121)(0.07)
(0.055)(35,000)
4.4 × 10−8
(40,000,000)(0.018)
12,000
55.
6 × 101
58.
(3000)(0.000024)
(0.0048)(0.0012)
2.24 × 101
2.4 × 10−1
44. 0.00026 ÷ 6,500,000
2.25 × 106
1.25 × 1020
0.024
(1,000,000)(160,000)
5 × 10−6
3.2 × 104
56.
135,000,000
(0.009)(6000)
60.
(5100)(0.00023)
(0.17)(460,000)
1.5 × 10−13
59.
1.25 × 104
ALG catalog ver. 2.6 – page 29 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(14000)(0.00065)
(0.0035)(260,000)
1 × 10−2
1.5 × 10−5
2.5 × 106
BJ
61.
4.8 × 103
1.5 × 101
65.
2.065 × 1015
5.9 × 107
69.
8.05 × 103
2.3 × 108
73.
8.1 × 104
2.7 × 10−4
77.
1.84 × 10−2
5.75 × 10−1
81. (15,000)2
62.
5.1 × 105
6.8 × 102
66.
3.36 × 1011
2.8 × 109
3.5 × 10−5
70.
4.807 × 1010
1.9 × 105
3 × 108
74.
9.8 × 10−1
7 × 10−6
78.
2.32 × 10−4
9.28 × 1010
3.2 × 102
3.5 × 107
3.2 × 10−2
2.25 × 108
85. (3 × 105 )4
8.1 × 1021
82. (3000)3
7.5 × 102
1.2 × 102
2.53 × 105
1.4 × 105
2.5 × 10−15
2.7 × 106
86. (1.4 × 1010 )2
63.
4.8 × 108
6 × 103
67.
2.21 × 109
1.7 × 108
71.
5.751 × 102
8.1 × 1010
75.
7.8 × 10−5
1.3 × 10−8
79.
4.86 × 105
4.05 × 10−5
83. (20,000)3
8 × 104
1.3 × 101
7.1 × 10−9
6 × 103
1.2 × 1010
8 × 1012
87. (2.5 × 104 )2
6.25 × 108
64.
7.6 × 1010
3.8 × 104
68.
3.198 × 109
2.6 × 102
72.
2.07 × 105
9.2 × 1013
2.25 × 10−9
76.
5.6 × 10−8
8 × 103
7 × 10−6
80.
5.52 × 10−7
1.84 × 10−6
2 × 106
1.23 × 107
3 × 10−1
84. (110,000)2
1.21 × 1010
88. (1 × 104 )6
1 × 1024
1.96 × 1020
89. (0.002)5
3.2 × 10−14
93. (1.3 × 10−9 )2
90. (0.000012)2
1.44 × 10−10
91. (0.0003)4
94. (4 × 10−2 )3
6.4 × 10−5
95. (8 × 10−3 )2
8.1 × 10−15
6.4 × 10−5
92. (0.000005)3
96. (2.5 × 10−6 )2
1.69 × 10−18
6.25 × 10−12
97. (3 × 102 )2 (5 × 10−4 )
98. (5 × 1010 )(2 × 10−2 )4
4.5 × 101
101.
2.4 × 107
(2 × 10−3 )3
1.25 × 10−16
99. (6 × 103 )(4 × 104 )2
8 × 103
3 × 1015
102.
(6 × 103 )2
9 × 10−2
100. (5 × 10−4 )3 (2 × 10−7 )
2.5 × 10−17
9.6 × 1012
4 × 108
103.
5.4 × 10−11
(3 × 104 )3
2 × 10−24
104.
(1.5 × 10−3 )2
5 × 10−4
4.5 × 10−3
ALG catalog ver. 2.6 – page 30 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
BK
Topic:
Mixed practice and review (scientific notation).
Directions:
0—(No explicit directions.)
1.
Write 73,500,000 in scientific notation.
2.
Write 100,000 in scientific notation.
3.
Write 5040 in scientific notation.
5.04 × 103
4.
Write 96.1 in scientific notation.
5.
Write 0.54 in scientific notation.
5.4 × 10−1
6.
Write 0.00319 in scientific notation.
7.
Write 0.0208 in scientific notation.
8.
Write 0.000001 in scientific notation.
9.
What is 9 thousandths in scientific notation?
7.35 × 107
2.08 × 10−2
1 × 105
9.61 × 101
3.19 × 10−3
1 × 10−6
10. What is one hundreth in scientific notation?
1 × 10−2
9 × 10−3
11. What is 27 billion in scientific notation?
2.7 × 1010
13. Write 6.2 × 104 in standard decimal notation.
62,000
12. What is 320 million in scientific notation?
3.2 × 108
14. Write 3.03 × 10−2 in standard decimal notation.
0.0303
15. Write
8 × 10−5
in standard decimal notation.
0.00008
16. Write 4.7 × 107 in standard decimal notation.
47,000,000
17. Write 1.55 × 10−3 in standard decimal notation.
18. Write 9.2 × 108 in standard decimal notation.
0.00155
920,000,000
19. Write 6.05 × 102 in standard decimal notation.
605
20. Write 3 × 10−6 in standard decimal notation.
0.000003
21. Rewrite in scientific notation, and simplify:
1,500, 000 × 4,400
0.00068 × 25,000,000
6.6 × 109
23. Rewrite in scientific notation, and simplify:
700,000 × 4,100,000,000
22. Rewrite in scientific notation, and simplify:
24. Rewrite in scientific notation, and simplify:
0.00021 × 0.00000008
2.87 × 1015
25. Simplify using scientific notation: (4 × 106 )(8 × 103 )
1.68 × 10−11
26. Simplify using scientific notation: (6 × 10−5 )(6 × 108 )
3.2 × 1010
3.6 × 104
28. Simplify using scientific notation: (3.2×104 )(7.5×103 )
27. Simplify using scientific notation:
(4.5 × 10−4 )(2.2 × 105 )
1.7 × 104
2.4 × 108
9.9 × 101
29. Simplify using scientific notation: (5×10−3 )2 (2×10−4 )
30. Simplify using scientific notation: (2 × 106 )3 (1.2 × 104 )
5 × 10−9
9.6 × 1022
31. Simplify using scientific notation: (1×1012 )(3×10−4 )3
32. Simplify using scientific notation:
(5.5 × 10−5 )(3 × 104 )2
2.7 × 101
4.95 × 104
33. Rewrite in scientific notation, and simplify:
140,000
10
0.0035 × 0.004 1 × 10
34. Rewrite in scientific notation, and simplify:
1000 × 40,000
8 × 100
5,000,000
35. Rewrite in scientific notation, and simplify:
900,000,000
2
600 × 5,000 3 × 10
36. Rewrite in scientific notation, and simplify:
420,000 × 0.0005
3 × 107
0.000007
37. Simplify using scientific notation:
7 × 10−6
5.6 × 10−9
38. Simplify using scientific notation:
1.25 × 103
39. Simplify using scientific notation:
4.2 × 104
1.4 × 10−7
3 × 1011
2.2 × 101
5.5 × 108
40. Simplify using scientific notation:
4 × 10−8
ALG catalog ver. 2.6 – page 31 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1.75 × 10−6
4.9 × 10−2
2.8 × 104
BL
Topic:
Word problems involving scientific notation
Directions:
0—(No explicit directions.) 139—Use scientific notation.
141—Write each answer using scientific notation.
1.
The diameter of a silver atom is about
0.000 000 000 25 m. Express that number using
scientific notation. 2.5 × 10−10
2.
3.
The average distance between the sun and Mars
is about 228,000,000 km. Express that number in
scientific notation. 2.28 × 108
4.
The world’s energy use in 1987 was equivalent to
9,650,000,000 metric tons of coal. Express that
number in scientific notation. 9.65 × 109
5.
The mass of the earth is about 6 × 1021 metric tons.
Express that number without scientific notation.
6.
An astronomical unit (the average distance between
the earth and sun) is 1.5 × 108 km. Express that
number without scientific notation. 150,000,000
8.
An Angstom, which is a unit of measure for light
waves, is 1 × 10−10 m. Express that number without
scientific notation. 0.000 000 000 1
7.5 × 10−4
6,000,000,000,000,000,000,000
7.
The mass of a proton is about 1.6 × 10−24 g. Express
that number without scientific notation.
0.000 000 000 000 000 000 000 0016
9.
The metropolitan area of Dallas, Texas, has a
population of about 1 million and covers 380 square
miles. Calculate the number of people per square
mile, and express your answer in scientific notation.
≈ 2.63 × 103
11. Japan has a population of about 124 million and
an area of 3.7 × 105 sq km. What is the population
density (number of people per square kilometer)?
The diameter of a red blood cell is about 0.00075 cm.
Express that number using scientific notation.
10. Bombay, India, has a population of about 11 million
and an area of only 96 square miles. Calculate the
number of people per square mile, and express your
answer in scientific notation. ≈ 1.15 × 105
12. The U.S. has a population of about 240 million and
an area of 9 × 106 sq km. What is the population
density (number of people per square kilometer)?
≈ 3.4 × 102
13. Momentum is a product of mass and velocity.
It is measured in units of kilogram-meters per
second (kg-m/s). What is the momentum of a truck
going 22 m/s, if its mass is 5 × 104 kg?
≈ 2.7 × 101
14. Momentum is a product of mass and velocity.
It is measured in units of kilogram-meters per
second (kg-m/s). What is the momentum of a bullet
travelling at 440 m/s, if its mass is 5 × 10−3 kg?
2.2 kg − m/s
1.1 × 106 kg − m/s
15. In an electrical circuit, the current (Amps) is equal
16. In an electrical circuit, the current (Amps) is equal
to voltage divided by resistance. Find the current in
to voltage divided by resistance. Find the current
a 1.5 volt flashlight, if the resistance is 6 × 103 Ohms.
in a 9 volt smoke detector, if the resistance is
1.2 × 102 Ohms. 7.5 × 10−2 A
2.5 × 10−4 A
17. The length of a light wave is calculated with the
formula λ = c/ν, where λ is the length (meters), c is
the speed of light (3 × 108 meters per second), and
ν is frequency (oscillations per second). What is
the length of an ultraviolet wave whose frequency is
4.8 × 1016 ? 6.25 × 10−9 m
18. The length of a light wave is calculated with the
formula λ = c/ν, where λ is the length (meters),
c is the speed of light (3 × 108 meters per second),
and ν is frequency (oscillations per second). What
is the length of an infared wave whose frequency is
2.5 × 1012 ? 1.2 × 10−4 m
19. A photon is a single “parcel of light.” Its energy,
measured in electron-Volts, is given by the formula
E = hν, where h is Planck’s constant (4.1 × 10−15 )
and ν is frequency (hertz). Find the energy of
a photon in a gamma ray, if the frequency is
5 × 1022 Hz. 2.05 × 108 eV
20. A photon is a single “parcel of light.” Its energy,
measured in electron-Volts, is given by the formula
E = hν, where h is Planck’s constant (4.1 × 10−15 )
and ν is frequency (hertz). Find the energy of a
photon in an FM radio signal, if the frequency is
9 × 108 Hz. 3.69 × 10−6 eV
21. The closest star to the earth (besides the sun)
is Alpha Centauri. It is about 4.3 light years
away. Find out how many kilometers that is, if one
light-year is 9.5 × 1012 km. 4.085 × 1013
22. The star Betelgeuse is in the constellation Orion. It
is 600 light years from Earth. Find out how many
kilometers that is, if one light-year is 9.5 × 1012 km.
ALG catalog ver. 2.6 – page 32 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
5.7 × 1015
BL
23. Light from the star Aldebaran in the constellation
Taurus takes 54 years to reach the earth. What is
the distance in kilometers, if one light-year is about
9.5 × 1012 km? 5.13 × 1014
24. The distance across the Milky Way galaxy is
estimated to be 80,000 light-years. What is that
distance in kilometers, if one light-year is about
9.5 × 1012 km? 7.6 × 1017
25. How many seconds does it take sunlight to reach
the earth, if the speed of light is 186,000 miles per
second and the average distance from the sun to the
earth is 9.3 × 107 miles? 5 × 102
26. How many seconds does it take sunlight to reach
Neptune, if the speed of light is 186,000 miles per
second and the average distance from the sun to
Neptune is 2.8 × 109 miles? ≈ 1.5 × 104
27. The speed of light is approximately 3 × 105 km/sec.
The average distance from the sun to Jupiter is
about 778,000,000 km. How long does it take for
sunlight to reach Jupiter? ≈ 2.6 × 103 sec
28. The speed of light is approximately 3 × 105 km/sec.
The average distance from the sun to Mercury, is
about 57,900,000 km. How long does it take for
sunlight to reach Mercury? 1.93 × 102 sec
29. Approximately how many years does it take light
from the star Bellatrix to reach the earth, if the
distance is about 2 × 1015 km and one light-year is
9.5 × 1012 km? ≈ 2.11 × 102
30. Approximately how many years does it take light
from the star Regulus to reach the earth, if the
distance is about 6.7 × 1014 km and one light-year is
9.5 × 1012 km? ≈ 7.05 × 101
31. The star Antares in the constellation Scorpius
is about 1.62 × 1015 km from the earth. How
many light-years away is that, if one light-year is
9.5 × 1012 km? ≈ 1.71 × 102
32. The center of our galaxy, the Milky Way, is about
2.8 × 1017 km from the earth. How many light-years
away is that, if one light-year is 9.5 × 1012 km?
33. The average distance between the sun and Pluto
(the farthest planet) is 5.9 × 109 km, and the speed
of light is 3 × 105 km/sec. Approximately how many
hours does it take sunlight to reach Pluto? ≈ 5.5
34. The average distance between the sun and Venus
(the second planet) is 1.1 × 108 km, and the speed
of light is 3 × 105 km/sec. Approximately how many
minutes does it take sunlight to reach Venus? ≈ 6.1
35. Radio waves travel at the speed of light
(3 × 105 km/sec); they are extremely fast, but they
are not instantaneous. This fact was apparent when
Apollo 11 landed on the moon in 1969. Find out how
long it took a radio signal to travel between the
moon and earth, a distance of about 384,000 miles.
36. Radio waves travel at the speed of light
(3 × 105 km/sec); they are extremely fast, but they
are not instantaneous. This fact is important
for space-age communications. For example, the
spacecraft Voyager I sent its final pictures back to
earth when it was 3.7 billion miles away. How many
hours did it take for the signals to arrive? ≈ 3.4
≈ 1.3 sec
≈ 2.95 × 104
37. The density of an object (measured in g/cm3 ) is
its mass divided by volume. How dense is ice, if a
20 cm cube has a mass of 7.2 × 103 g? 9 × 10−1 g/cm3
38. The density of an object (measured in g/cm3 ) is its
mass divided by volume. How dense is platinum, if a
5 cm cube has a mass of 2.8 × 103 g? 2.24 × 101 g/cm3
39. The density of an object is its mass per unit of
volume. Find the density of the planet Saturn
(in grams per cubic centimeter), if its mass is
5.7 × 1026 kg and volume about 8.5 × 1023 m3 .
40. The density of an object is its mass per unit of
volume. Find the density of the earth (in grams
per cubic centimeter), if its mass is 6 × 1024 kg and
volume about 1.1 × 1021 m3 . ≈ 5.5 g/cm3
≈ 6.7 × 10−1 g/cm3
ALG catalog ver. 2.6 – page 33 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
CA
Topic:
Order of operation.
Directions:
6—Simplify.
98—Perform the indicated operation(s).
1.
3+2·5
13
2.
4+3·7
5.
7−3·5
−8
6.
12 − 4 · 3
9.
−3 + 5 · (−2)
13. 6 · 5 + 3
−13
33
25
3.
6+2·4
14
4.
2+5·4
0
7.
9−2·4
1
8.
15 − 7 · 3
10. 4 − 5 · (−7)
39
14. −11 · 3 + 8
−25
17. 36 ÷ 4 + 2
11
18. 28 ÷ 4 + 3
21. 8 + 20 ÷ 4
13
22. −15 + 45 ÷ 5
25. −3 · 4 + 6 − 4 ÷ 2
11. −9 + 3 · (−2)
19. 40 ÷ 5 + 3
10
26. 7 · 8 − 5 + 6 ÷ 3
−8
53
20. 15 ÷ 3 + 2
11
30. 35 ÷ 7 − 2 + 4 · 3
15
27. 12 + 18 ÷ 6 − 3 · (−2)
7
9
28. 10 − 45 ÷ 5 + 4 · (−7)
−27
31. 42 ÷ 6 + 2 · 4 − 7
32. 7 · 3 + 8 − 12 ÷ (−4)
8
27
32
33. (8 − 4)3 + 12
41. 10 − 3(5 − 2)
45. 5 + 3(7 − 4)
34. (7 + 2)(−3) + 9
24
37. (11 − 4)5 − 15
38. (5 − 8)6 − 4
20
46. 6 + 9(5 − 8)
14
−18
−22
42. 14 − 7(4 + 2)
1
49. (26 + 4) ÷ (30 ÷ 2)
−28
28
36. (7 + 3)(−2) + 13
39. (7 + 2)3 − 5
22
40. (3 − 7)4 − 12
−28
43. 5 − 6(2 − 3)
11
44. 12 − 5(7 + 1)
−28
47. 4 + 3(7 − 10)
−21
50. (15 + 35) ÷ (80 ÷ 16)
2
35. (9 − 4)5 + 3
53. 25 ÷ 5 + (7 + 3)2
57. 4 · 5 + 2 · 3 − 21 ÷ 7
61. −2 · 7 + 9 · 5 − 16 ÷ 8
−5
29
−14
65. [2 + (3 − 5)6] ÷ (5 · 8 − 10)
67. [(5 − 7)3 + 8] · (5 − 3 · 4 ÷ 6)
56. 8(3 + 2) − 35 ÷ 5
−2
60. 4 · (−2) + 9 ÷ 3 − 6 ÷ (−3)
−3
62. 9 · (−3) + 5 · 3 + 35 ÷ (−7)
−17
64. −24 ÷ (−8) + 15 ÷ 3 − 8 · 2
−8
1
1
2
5
72. (5 + 6)2 + [12 ÷ (8 ÷ 4) − 3(7 − 9)] − 9 · 3
10
75. 23 + 52
−135
77. (32 − 4)3
125
78. (5 + 22 )2
81. 22 · 7 − 52
3
82. 25 + 4 · 52
79. 2(5 + 3)2
81
86. 47 − 5(2 − 4)3
−24
90. 4(24 ÷ 8 − 6)3
80. 7(8 − 3)2
128
83. 35 − 3(6 − 2)3
132
42
76. 42 − 24
33
−157
87
87. 2(3 − 4 · 2)2
−108
91. 5 (6 − 3)2 + 4 · 2
ALG catalog ver. 2.6 – page 34 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
50
85
33
−19
70. 21 ÷ (3 + 4) + [6(8 − 3) ÷ (1 + 9) − 1]
13
74. −5 · 33
89. 3(10 ÷ 5 − 4)3
19
68. [40 ÷ (10 − 2)] ÷ (3 · 4 − 2)
6
71. 4(3 + 2) − 5 [(7 − 4)2 + 8 ÷ (4 − 6)]
85. 45 + 3(7 − 8)5
55. (5 + 3)3 − 10 ÷ 2
66. [4 · (8 − 2 · 3) + 7] − (16 ÷ 2 + 6)
− 13
69. 5 · 3 − [(4 + 18) ÷ 11 · (8 − 5)] + 4
−28
52. (16 − 46) ÷ (5 · 3)
58. 20 ÷ 4 − 12 ÷ 2 + 3(−6)
63. 25 ÷ 5 + 7 · (−3) + 4 ÷ 2
48
12
23
59. 12 ÷ 4 + 2 · (−7) − 18 ÷ (−3)
−7
3
54. 3(8 − 6) + 42 ÷ 7
25
48. −8 + 5(2 − 6)
−5
51. (49 − 10) ÷ (52 ÷ 4)
10
73. 3 · 42
22
24. 14 + (−35) ÷ 7
0
21
29. 40 ÷ 5 + 3 − 8 · (−2)
4
16. −5 · (−2) + 12
−43
23. 6 + 18 ÷ (−3)
−6
−6
12. −8 + 4 · 3
−15
15. 9 · (−4) + (−7)
22
7
0
175
84. 22 + 2(5 − 7)4
54
88. 6(5 + 12 ÷ 6)2
294
92. −7 6 ÷ 2 + 3(5 − 3)2
−105
CA
93. −4 2(5 − 8)3 + 5 · 2 · 6
95. 32 · 5 + (2 − 4)
97.
2
− 15
94. 2 4 + 3(6 − 7)5 + 8 ÷ (−4)
−24
96. 63 − [4 · (3 + 2) − 15]2
66
16 ÷ (1 + 3)2 − 8 ÷ (−4) + 12
2
99. 15 − 12 ÷ (22 + 3 − 1) + 4
101.
26 − 6 · 2
10 + 16 ÷ 4
105.
2·4−3
3·7−6
109.
9 ÷ (5 − 8)
7−2·3
113.
12 − 8 ÷ 4
(7 + 1)5
117.
98.
15
6·3+2
5−9
106.
2+5·4
2 · 3 − 17
−3
110.
(7 − 4) ÷ 3
8−5·2
1
4
114.
4(5 − 7)
(6 + 2) ÷ 4
118.
[9 ÷ 3 + (−5)]4
(4 ÷ 2 · 3)2 ÷ 3
1
3
[18 ÷ 2 + 5] ÷ 7
1
18
[4 · 3 ÷ 2]2
24 ÷ (1 − 3)2 + 3 · (−2)
38
4
100. 12 + 25 ÷ (2 + 3) · (4 − 5)2
−21
102.
1
−2
0
17
103.
14 − 3 · 2
5 · 3 − 11
2
104.
5 · 2 − 10
4+6÷3
107.
(5 − 3)4
6 · 4 − 12
2
3
108.
(7 − 3)6
8+2·3
12
7
− 12
111.
6+5·2
4(3 + 5)
1
2
112.
8−7·2
3(4 − 2)
−1
−4
115.
5(6 − 2)
2(4 + 3)
10
7
116.
12 ÷ (2 − 6)
15 ÷ 5 + 2
119.
(3 − 5 · 23 + 1) ÷ 9
(5 − 6)3 + 6 ÷ 2
120.
[15 ÷ 3 + 2 · (−7)]2
4 · 33 − 6(7 − 4)
−5
−2
4
3
−2
121.
2 1 1
+ ·
3 3 2
122.
5
6
3 2 3 3
÷ −
4 3 8 4
3 2
1 17
129.
+
·
4 3
2 24
1 3
2 1
+
133. + ·
3 2
6 4
3 2 1
+ ·
5 3 5
2 1 1 13
÷ +
3 3 6 6
2 1
2 13
130.
+
·
5 4
3 30
7 1
3 5
134. + ·
−
8 4
2 8
126.
125.
9
8
124.
7
16
5 1 13 7
÷ −
6 4
6 6
5 3
1 2
131.
−
÷
6 4
8 3
4 1
3
1
135. − ·
+
5 3
10 2
127.
35
32
− 35
9
10
5 1 3
− ·
8 4 4
123.
11
15
0
7
1 3
− ·
10 5 4
11
20
5 2 3 19
÷ +
9 3 4 12
9
1
2 13
132.
−
÷
10 4
5 8
7
4
2 2
136.
− ÷
+
10 5
3 5
128.
8
15
1
− 20
137.
7 3
2 1
−
÷
·
8 4
3 3
138.
1
4
3
4 1
5
+
·
÷
7 21
5 2
139.
5
3
141.
2
2
1
+
6
3
145.
1 2
+
3 9
2
11
18
−
5
27
149. 2.3 + 0.5 · 4.6
10
81
4.6
3
4 5
1
+
−
·
10 5
3 8
140.
17
48
142.
3
1
3
+
2
2
146.
3
1 3 2
−
+
4
2 8
13
8
150. 7.1 + 0.4 · 3.5
1
− 64
8.5
5 5
3 2
−
·
÷
9 6
5 3
− 25
36
143.
2
1
2
−
3
3
147.
11 1
−
24 3
− 59
5 2
+
6 3
3
− 23
151. −6.2 + 0.72 ÷ 0.9
144.
2
3
5
−
8
4
148.
5
1
+
32 4
1
16
1 5
−
3 6
3
152. 3.4 + 1.44 ÷ 0.12
1
8
15.4
−5.4
153. 2.4 + 1.12
157.
154. 0.55 − 0.52
3.61
3.2 + 1.6
4.3 + 0.7 · 11.0
0.4
158.
161. (6.1 + 2.3)0.5 + 0.35 ÷ 0.5
9.5 − 3.8 · 2.5
6.3 ÷ 0.07
0
159.
1.43
5.7 − 2.3
5.8 · 0.5 − 1.2
2
162. 0.3(1.5 − 2.6) + 4.2 ÷ 6
4.9
163. 0.54 ÷ 0.6 − (3.5 + 2.9)1.25
155. 0.62 + 0.92
0.3
−7.1
156. 2.84 − 1.22
160.
4.7 − 2.4 · 1.5
3.2 + 1.2
7.3
164. 0.084 ÷ 0.21 + 0.75(5.8 − 7.4)
−0.8
165. (4.2 + 1.5 · 1.2) ÷ (2.1 − 0.6)
4
166. (2.8 − 1.8) · (4.3 + 2.1 ÷ 0.7)
7.3
167. (5.6 + 3.2 · 2.0) ÷ (4.8 ÷ 1.2)
3
168. (9.6 ÷ 3.2) · (4.1 + 3.9 ÷ 13)
13.2
ALG catalog ver. 2.6 – page 35 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
2.4
0.25
CB
Topic:
Absolute value.
Directions:
6—Simplify.
1.
|18|
5.
− |−37|
9.
98—Perform the indicated operation(s).
18
|−2.6|
1
13. − 3 6
−37
2.
|−11|
11
3.
|−20|
20
4.
|28|
6.
− |22|
−22
7.
− |14|
−14
8.
− |−12|
10. |1.5|
2.6
1.5
7
14. − − 2
−3 16
−2 11. −3 17. |−(−45)|
45
18. − |−(−88)|
21. −5 |−12|
−60
22. 8 |−6|
1 25. 2 − 10
26 29. − 65
23. −12 |8|
−72 30. − 24 2
5
−3
33. 26 − |−16|
10
34. |−8| − 13
37. −3 |8| + 10
−14
38. −6 − 5 |12|
41. 2.4 + |−3.6|
6
1 3
45. − 7 −
4
4
49. |14 − 9|
2 5
46. 1 + −3 3
6
−8
50. |9 − 14|
62. − |−67 − 16|
60
69. − |9 − (11 + 7)|
20
0
70. − |−(64 − 26) + 12|
−9
36. −5 − |21|
39. 23 − 4 |−7|
−5
40. 7 |−6| + 36
48.
23
4
2
3
− − 10
5
52. |−8 + 6|
2
9
59. − |−2 + 8|
−6
−26
44. 0.1 − 7 |0.3|
−23
78
−2
1
− 10
2
56. − |16 − 5|
−11
60. |−21 − (−20)|
−30
67. − |−(−51 + 20)|
24
− 23
0
55. |1 + (−10)|
50
−26
35. − |11| + 11
63. − |−8 + (−22)|
−83
66. |−(−38 + 14)|
22
−50 32. − −75 15
51. |−6 + 8|
5
63
2
28. − |−39|
3
48
1
1
47. −3 + 2
2
4
5 12
−1.08
24. 10 |−(−5)|
−96
43. −2 |5.5| − 12
1.4
58. |−15 − (−15)|
−20
61. |−30 + (−30)|
65. |−(45 − 67)|
−66
54. |23 + (−43)|
−6
57. − |−3 + 23|
−5
42. |−0.35| + 1.05
5
53. − |6 − 12|
105 31. −7 1 56
20. |−(−63)|
−12
27. 3.2 |−(−15)|
−15
−12
16. − |1.08|
−0.44
19. − |−(−12)|
−88
48
26. −6 |−2.5|
1
5
5
12. −1 6
2
3
15. − |−0.44|
− 72
28
−31
71. |−(28 − 48) − 75|
55
64. |−19 − 13|
1
32
68. − |−(94 − 49)|
72. |−23 − (4 − 15)|
−45
12
−26
73. − |3(2 − 8)|
74. |−5(10 + 2)|
−18
77. |(12 − 8) − (6 + 18)|
20
79. |(25 − 30) + (14 − 20)|
81. |(1 − 14)(5 − 7)|
26
75. − |−4(4 − 12)|
60
11
82. − |(3 + 6)(14 − 8)|
−2
86. −14 + 2 |−(6 + 4)|
76. |7(−9 + 6)|
78. − |(−7 + 23) − (16 − 11)|
−11
80. − |−(12 + 6) + (−4 + 21)|
−1
83. − |(15 − 5)(2 + 7)|
−54
85. 4 |−9 + 5| − 18
−32
21
84. |(2 − 16)(13 − 10)|
−90
6
87. 25 − 6 |5 + (−25)|
−95
ALG catalog ver. 2.6 – page 36 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
88. −10 |4 − 8| + 36
−4
42
CB
89. |−5| − |8|
90. − |15| − |−13|
−3
93. − |14| + |−28|
94. |−7| + |−17|
14
3 1
97. − + −1 4
4
101. − |2.5| − |−4.6|
−7.1
92. − |−20| + |−8|
20
95. − |16| − |−14|
24
7 1
98. − − 10
2
2
91. |−9| + |−11|
−28
−30
1 5
99. − −2 + 2
6
1
5
102. − |−0.47| + |0.83|
− 53
103. |10.2| − |−7.4|
2.8
96. |−42| − |−6|
−12
36
7 2
100. − − 12
3
− 54
104. |−1.06| + |−0.94|
2
0.36
105. |−9| · |−6|
106. − |−4| · |18|
54
109. |5 − 18| + |−14|
113. −6 |1.5| − |8|
27
117. 2 |17 − 10| − 3 |5|
110. |24| − |10 + 21|
114. 2 |12| − 8 |−2|
−17
−1
107. |12| · |−7|
−72
108. − |5| · |−21|
84
111. |−7| + |6 − 7|
−7
118. − |−33| + 4 |7 + 4|
11
112. − |4 + 21| + |−25|
8
115. −5 |0.6| + 2 |−2|
8
−105
1
119. −7 |8| + |24 − 18|
116. |−14| + 11 |−3|
0
47
120. 4 |13 + 3| − |−52|
12
−50
121. |4 − 9| + |6 − 11|
10
122. 9 |3 + 4| − |2 − 42|
23
123. |15 − 21| − 10 |3 − 6|
124. −7 |2 + 9| + 7 |2 − 9|
−24
125. 3|−4| − 14 129.
|−42|
3
133.
|30|
− |−3|
137.
|−15| − |3|
|−6|
141.
|6| − |−11|
|−8| + |−7|
2
14
−10
3
− 13
126. − 10 + 5 |−3| −25
28
127. 6 8 − 2|−7| −6
128. −3 8|−2| − 18
130.
− |14|
35
− 25
131.
|−30|
−48
− 58
132.
− |−84|
7
134.
|−33|
|−88|
3
8
135.
− |36|
|144|
− 14
136.
|108|
|−18|
138.
|−8| + |−10|
− |3|
−6
139.
|−20|
|14| + |−6|
140.
− |−32|
|−3| − |11|
4
142.
|−13| − |−4|
|2| − |−5|
−3
143.
|−7| + |−3|
|−7| − |−3|
144.
|8| + |−19|
|−13| − |4|
3
ALG catalog ver. 2.6 – page 37 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1
5
2
−12
6
36
CC
Topic:
Order on the number line.
Directions:
104—Fill in each blank with the correct symbol (<, > or =).
>
1.
5
5.
−3
<
9.
−6
<
13. 5
3
=
>
21. 10
25. 0.75
1
3
>
33. −3.5
37.
3
4
10
2
6.
2
0
10. 0
= 10
2
17. −3
29.
2.
20
3.
4
>
−5
7.
−4
>
−4
11. 0
=
14. − 63
>
<
>
15. − 20
5
−2
3
4.
7
<
11
8.
7
>
−6
9
− 10
12. − 12
=
16. 3
−4
<
0
= 12
4
=
18. 1.00
=
1.0
19. 7.1
= 71
10
20. − 36
10
−10
22. −15
<
15
23. − 23
< 2
3
24.
7
6
>
− 76
28.
4
3
=
1.3
= 3
4
26.
4
5
=
0.8
27. −0.6
0.3
30.
2
3
<
0.7
31. 0.16
>
=
−3.55
<
34. −7.22
38.
43
84
42. − |5|
8
=
>
− |−11|
46. |−12|
>
49. 24
=
42
50. 23
<
32
54. −5 · 6
=
=
53. (−2)(−8)
2·8
−7.2
> 1
2
45. 11
44. − |−10|
−12
47. |−7|
−7
48. − |8|
51. 34
3(−10)
65. 5 − 3
>
3−5
66. 12 − 15
<
>
75. 6 − (−2)
77. 5 + 2 · 3
−2 − (−6)
<
2+5·3
<
79. 9 + 6 ÷ (−2)
81. 2(3 + 5)
>
<
10 ÷ 5
>
6 − 9 ÷ (−3)
2(3) + 5
>
43
59. 3(−6)
=
<
3÷5
63. 2 ÷ (−8)
<
67. −4 + 2
15 − 12
52. 52
55. 2 ÷ (−4)
70. 4 − 9
−2 − (−6)
−2 + 4
−5 − (−1)
74. −2 + (−3)
<
(−3) + 2
76. −4 − (−4)
>
−4 − 4
>
3−4·2
=
>
−5(3) + 5(9)
84. −3(2 − 7)
85. 10 ÷ 2 + 3
>
10 ÷ (2 + 3)
86. −12 ÷ 4 − 2
89. 15 ÷ (3 + 2)
91. 8 + 2(−4)
=
=
18 ÷ 3 − 3
−15 ÷ (−3) − 2
−8 + 4(2)
=
80. 12 − 3 ÷ (3)
82. 7(8 + 14)
88. 15 ÷ 3 + 8
92. −5 + 3(−3)
8 − 6 ÷ (−2)
7(8) + 7(14)
3(2) + 7
>
>
−12 ÷ (4 − 2)
−20 ÷ 5 − 1
90. 18 + 10 ÷ (−2)
ALG catalog ver. 2.6 – page 38 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
>
68. −1 + 6
=
=
<
56. −12 ÷ 4
−8 + 3
=
=
7 − 12 ÷ (−2)
2(1 − 8)
|−10|
|−8|
25
64. −12 ÷ 3
−8 ÷ 2
=
<
>
72. −1 + (−3)
78. 4 − 3 · 2
<
−3.30
− 17
30
60. 4 · 3
<
=
−2 ÷ 4
<
2·9
83. −5(3 − 9)
87. −18 ÷ (9 − 3)
>
|−16|
62. 5 ÷ 3
73. −5 − 3
36. −3.03
=
8÷6
−4 − (−6)
−1.09
43. |16|
<
=
<
− |−5|
61. 6 ÷ 8
−5 + 7
35. −1.9
> 5
6
32. 0.84
40. − 35
58. 10 ÷ (−2)
=
< 1
6
− 23
6 ÷ (−2)
69. −3 + 5
− 23
<
>
>
=
−3.6
39. − 17
24
57. 6 ÷ 3
71. 8 − 6
1
−3.00
< 37
48
41. |−8|
<
=
15 ÷ (−5)
−6 · 2
<
>
−3 ÷ 12
−6 + 1
CC
93.
4(9 − 5)
5+3
95.
3(2)
5(3 + 9)
=
5(3 + 1)
19 − (4 + 5)
94.
3(6 + 2)
4+2·4
<
5(3)
6(2 + 8)
96.
2 + 12 ÷ 2
6(7 − 3)
97. 42 + 52
>
62
98. 82 − 32
=
72
99. 92 − 42
101. 22 + 42
<
(8 − 2)2
102. 72 − 32
>
(7 − 3)2
103. (2 − 5)2
=
30 + 6 ÷ 3
4(6 − 2)
=
>
8+2·4
3(7 + 9)
52
<
=
100. 52
52 − 22
42 + 32
>
104. (4 + 6)2
42 + 62
Irrational numbers
<
105. 3
√
2
<
113. −2
=
109.
117.
√
√
123. 10 −
√
125. 9 4
<
2
110. 5
>
√
− 4
114. −5
>
65
121. 4 +
106. π
π
8
>
10
√
50
>
16 −
<
7−
√
√
√
5
√
111. − 7
√
− 25
√
50
115. 6
119. −6
122. 7 +
15
124. 9 −
−π
>
√
=
85
√
4 9
√
√
127. − 4(− 16)
107. −3
=
<
118. 7
>
4
<
√
√
− 64
112. −3
<
=
√
35
√
120. − 77
2+
>
12 −
<
<
√
<
101
√
150
√
−4 16
√
√
− 4(− 9)
129. If x > 0, then −x
<
0.
130. If y < 0, then −y
>
0.
131. If −x < 0, then x
>
0.
132. If −y > 0, then y
<
0.
133. If a > 0 and b > 0, then ab
>
0.
134. If a < 0 and b > 0, then ab
<
0.
135. If a > 0 and b < 0, then ab
<
0.
136. If a < 0 and b < 0, then ab
>
0.
137. If c < 0, then
1
c
<
138. If c > 0, then −
0.
139. If c < 0 and d < 0, then
141. If a − b < 0, then b − a
143. If x − y = 0, then y − x
1
c
<
0.
c
d
>
>
0.
142. If x − y > 0, then y − x
0.
144. If |a − b| = 0, then b − a
=
0.
140. If c < 0 and d > 0, then −
ALG catalog ver. 2.6 – page 39 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
c
d
<
=
>
0.
0.
0.
−4
√
− 3
√
116. 3
60
√
128. − 36
>
36
√
24
√
126. −16 4
>
−7
108. −π
9
>
−9
CD
Topic:
Properties of Algebra.
Directions:
105—Name the property which justifies each statement.
1.
r+7=7+r
3.
m+n+8=n+m+8
5.
a(−20) = −20a
7.
a(x + y) = (x + y)a
9.
(2a)(x) = 2(ax)
assoc. mult.
10. 2(5x) = (2 · 5)x
assoc. mult.
11. 9k(k) = 9(k · k)
assoc. mult.
12. (3y)y = 3(y · y)
assoc. mult.
comm. add.
comm. add.
comm. mult.
comm. mult.
2.
k + 25 = 25 + k
4.
5 + (m + p) = 5 + (p + m)
6.
ac(8) = 8ac
8.
5a = a(5)
comm. add.
comm. add.
comm. mult.
comm. mult.
13. 4 + (16 + 7) = (4 + 16) + 7
assoc. add.
14. (8 + 17) + 3 = 8 + (17 + 3)
15. (c + 10) + 8 = c + (10 + 8)
assoc. add.
16. (3 + a) + a = 3 + (a + a)
assoc. add.
18. a + (b + c) = (b + c) + a
comm. add.
17. a + (b + c) = a + (c + b)
19. a(bc) = (bc)a
1
21. x
=1
x
1
=1
3
23. 3
20. a(bc) = a(cb)
comm. mult.
mult. inverse
mult. inverse
mult. inverse
24.
1
5=1
5
mult. inverse
26. 3.5 + (−3.5) = 0
add. inverse
27. y 2 + (−y 2 ) = 0
28. −x + x = 0
add. inverse
mult. identity
mult. identity
add. identity
add. identity
34. ab + 0 = ab
35. 5 + 0 = 5
add. identity
36. −8 + 0 = −8
37. a(0) = 0
mult. prop. zero
38. 0 · 9 = 0
40. 0 · 3c = 0
mult. prop. zero
41. If ab = 0, then a = 0 or b = 0
45. If
add. identity
mult. prop. zero
mult. prop. zero
42. If xy = 0, then x = 0 or y = 0
zero product
43. If x(x + 1) = 0, then x = 0 or x + 1 = 0
mult. identity
32. 3d(1) = 3d
33. y + 0 = y
39. (−5)(0) = 0
add. inverse
add. inverse
30. (1)(−3x) = −3x
mult. identity
31. (1)(45) = 45
comm. mult.
1
22.
a=1
a
25. 4 + (−4) = 0
29. b(1) = b
comm. add.
assoc. add.
zero product
m
m
= −10(3)
= 3, then −10
−10
−10
zero product
44. If (a − 1)b = 0, then a − 1 = 0 or b = 0
46. If
4
4
= −6, then (x) = −6(x)
x
x
48. If
2
3 2
3
x = 4, then
x = (4)
3
2 3
2
zero product
mult. prop. equality
mult. prop. equality
47. If
a
a
= b, then (c) = b(c)
c
c
49. If 10y = 50, then
10y
50
=
10
10
mult. prop. equality
div. prop. equality
51. If 3x = −27, then 3x ÷ 3 = −27 ÷ 3
div. prop. equality
50. If ac = b, then ac ÷ c = b ÷ c
52. If −7p = 14, then
ALG catalog ver. 2.6 – page 40 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−7p
14
=
−7
−7
mult. prop. equality
div. prop. equality
div. prop. equality
CD
53. If x + 18 = 33, then x + 18 − 18 = 33 − 18
54. If a + y = x, then a + y − y = x − y
subtr. prop. equality
subtr. prop. equality
55. If y − 2 = −8, then y − 2 + 2 = −8 + 2
56. If pq − r = 0, then pq − r + r = 0 + r
add. prop. equality
add. prop. equality
57. If x − a = y, then x = y + a
59. If
x
= 4, then x = 4y
y
61. If
a
= −5, then a = −15
3
add. prop. equality
mult. prop. equality
mult. prop. equality (and subst.)
63. If y − 3 = 18, then y = 21
add. prop. equality (and subst.)
58. If y − k = 15, then y = 15 + k
60. If
2
= 9, then 2 = 9d
d
62. If
y
= 2, then y = 8
4
reflexive
66. 4m − 7 = 4m − 7
67. a − 12 = a − 12
reflexive
68. d + 18 = d + 18
69. If 0 = x + 15, then x + 15 = 0
71. If 25 = a2 , then a2 = 25
mult. prop. equality
mult. prop. equality (and subst.)
64. If y − 1 = 10, then y = 11
65. 2y + 5 = 2y + 5
add. prop. equality (and subst.)
reflexive
reflexive
70. If x − 9 = 24, then 24 = x − 9
symmetric
add. prop. equality
symmetric
72. If 4(w + 1) = 0, then 0 = 4(w + 1)
symmetric
symmetric
73. If a = 6 + 9, and 6 + 9 = 15, then a = 15
trans.
74. If p = 30 − 8, and 30 − 8 = 22, then p = 22
75. If 2(10) = 20, and x = 2(10), then x = 20
trans.
76. If y 2 = 9, and 9 = 3 · 3, then y 2 = 3 · 3
77. 2(3 + d) = 6 + 2d
78. 8(w + 2) = 8w + 16
distr.
79. 3(−a + b) = −3a + 3b
81. 2w + 10 = 2(w + 5)
83. 4r − 12 = 4(r − 3)
87. −3(a − 2) = −3a + 6
86. −1(−x + y) = x − y
distr.
distr.
1
(2k + 6m) = k + 3m
2
distr.
91. a(c + 3) + b(c + 3) = (a + b)(c + 3)
93. If x = 24 − 16, then x = 8
subst.
95. If a = 15 ÷ (−3), then a = −5
97. 9 + (3 + 7) = 9 + 10
99. (2 + 6)a = 8a
subst.
subst.
distr.
subst.
distr.
distr.
84. 5x − 20y = 5(x − 4y)
distr.
85. 10(k − m) = 10k − 10m
89.
82. 3c + 3d = 3(c + d)
distr.
trans.
distr.
80. (−7)(r + p) = −7r − 7p
distr.
distr.
distr.
88. 6 [y + (−5)] = 6y + (−30)
distr.
90. 0.1m + 5 = (0.1)(m + 50)
distr.
92. (w − 5)(x + y) = (w − 5)(x) + (w − 5)(y)
94. If y = 7 · 9, then y = 63
subst.
96. If n = 12 + 13, then n = 25
98. (12 − 7)(3) = (5)(3)
subst.
100. 3b + (15 − 10) = 3b + 5
ALG catalog ver. 2.6 – page 41 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
trans.
subst.
subst.
distr.
CE
Topic:
Using distributive property.
Directions:
59—Rewrite each expression using the distributive property.
1.
4(a + b)
4a + 4b
2.
(c + d)9
9c + 9d
3.
(m + n)p
5.
7j + 7h
7(j + h)
6.
2r + pr
r(2 + p)
7.
wx + xy
9.
(1 + r)(10)
13. 4(c − 6)
4c − 24
17. ab − bc
b(a − c)
21. 4k + 4 · 2
25. 14 − 7x
29. 8d + 8
10 + 10r
4(k + 2)
7(2 − x)
8(d + 1)
33. −12(a + b)
−12a − 12b
37. −5(y − 1)
−5y + 5
41. −10f − 10g
−10(f + g)
45. 2(x + y + z)
10. 25(a + 1)
25a + 25
4.
k(x + y)
x(w + y)
8.
bc + 3b
11. a(b + 1)
ab + a
12. (x + 1)y
ax − ay
16. (c − d)(x)
cx − dx
18. 4g − gh
g(4 − h)
19. wx − wy
w(x − y)
20. km − 7m
m(k − 7)
23. 2 · 6 − 2r
2(6 − r)
24. 8 · 10 + 8d
22. 5a − 5 · 3
5(a − 3)
26. 20 + 10s
10(2 + s)
27. 9y + 27
30. 10 + 10c
10(1 + c)
31. 5 − 5x
34. (m + 4)(−9)
38. (−1)(a − b)
42. −ax − ay
−9m − 36
−a + b
−a(x + y)
−3 + d
40. (r − 5)(−6)
−6r + 30
−7(a − 1)
44. −5p + 25
wz + yz + 3z
52. k(m − n + p)
km − kn + kp
55. ax − ay + az
56. 2b + 2c − 2d
a(x − y + z)
20w + 40
−5(p − 5)
48. (w + y + 3)(z)
3p − 3r + 9
58. (4w + 8)(5)
2(y − 1)
−p − r
51. (−p + r − 3)(−3)
15(c + d − 2)
5(w − 6)
36. (p + r)(−1)
39. (3 − d)(−1)
43. −7a + 7
8(10 + d)
−4 − 4w
ab + ac + ad
54. 15c + 15d − 30
6r + 3p
32. 2y − 2
5(1 − x)
47. a(b + c + d)
50. (−1)(x + y − z)
8(k + m + 1)
28. 5w − 30
9(y + 3)
35. −4(1 + w)
−x − y + z
57. 3(2r + p)
xy + y
15. a(x − y)
4m + 4n + 4
53. 8k + 8m + 8
b(c + 3)
7y − 14
46. (m + n + 1)(4)
7a − 7b − 7
kx + ky
14. (y − 2)7
2x + 2y + 2z
49. 7(a − b − 1)
mp + np
2(b + c − d)
59. 6(2k − 3m)
12k − 18m
60. (−2)(10 − 7c)
−20 + 14c
61. 32y − 16
65. 6 m +
69.
3
4
1
2
16(2y − 1)
(12 − 4k)
66. 8
6m + 3
9 − 3k
73. (0.1)(50x − 10)
62. 9a − 18b
5x − 1
70.
2
5
5
4
−a
9(a − 2b)
10 − 8a
(10a + 20b)
4a + 8b
74. (6a − 10)(−0.5)
63. 2m + 4r
67. 15
71.
a + 2b
78. 10(1.3m + 0.6p)
2
3x − 1
(8y + 2)
10x − 15
12y + 3
75. (−0.1)(100c + 10d)
−3a + 5
77. (0.2a + 0.4b)(5)
3
2
2(m + 2r)
64. 10x + 5
68. 12
72.
5(2x + 1)
1
6c + 3
2c + 36
4
7 (21m − 35p)
12m − 20p
76. (0.5)(2k + 2)
k+1
−10c − d
79. (1.5 − 2.5y)(4)
13m + 6p
ALG catalog ver. 2.6 – page 42 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
6 − 10y
80. −100(0.02w − 0.1z)
−2w + 10z
CF
Topic:
Translating algebraic expressions.
Directions:
47—Write as an algebraic expression.
1.
m plus 24
5.
k minus 18
m + 24
k − 18
2.
r plus thirty
6.
w minus sixteen
r + 30
3.
p plus q
7.
20 minus a
p+q
20 − a
4.
Twelve plus y
8.
a minus b
12 + y
a−b
w − 16
9.
A number plus 15
n + 15
13. 12 minus a number
12 − n
17. The sum of n and 3
n+3
10. A number plus
twenty-seven n + 27
11. 23 plus a number
14. Twenty-four minus
a number 24 − n
15. A number minus 21
18. The sum of 15 and y
19. The sum of 3a and 2x
x+5
25. a increased by 6
a+6
16. A number minus
fourteen n − 14
n − 21
15 + y
21. Five more than x
12. Seventeen plus
a number 17 + n
23 + n
20. The sum of w and 4y
w + 4y
3a + 2x
22. Seventeen more
than w w + 17
23. Eight more than twice
k 2k + 8
24. Eleven more than half
of m m + 11
26. p increased by 14
27. Twelve increased by
a third of k 12 + k
28. Nineteen increased by
a fourth of r 19 + r
p + 14
3
2
4
29. A number decreased
by 15 n − 15
30. A number decreased
by nine n − 9
31. 12 decreased by
a number 12 − n
32. 23 decreased by
a number 23 − n
33. A number reduced
by 7 n − 7
34. A number reduced
by eighteen n − 18
35. 55 reduced by some
number 55 − n
36. 27 reduced by some
number 27 − n
37. A number increased
by thirteen n + 13
38. A number increased
by thirty-five n + 35
39. A number increased
by 0.12 n + 0.12
40. A number increased
by 6.25 n + 6.25
41. y diminished by 5
42. x diminished by 8
43. 14 diminished by the
quantity r 14 − r
44. 16 diminished by the
quantity 2m 16 − 2m
y−5
x−8
45. A number exceeded
by 21 n + 21
46. A number exceeded
by 18 n + 18
47. A number diminished
by 7 n − 7
48. A number diminished
by 11 n − 11
49. Twenty added to
a number n + 20
50. Sixteen added to
a number n + 16
51. A number added
to −2 −2 + n
52. A number added
to −6 −6 + n
53. 29 subtracted from y
54. Fifteen subtracted
from v v − 15
55. w subtracted from 26
56. r subtracted from m
57. A number subtracted
from 32 32 − n
58. A number subtracted
from 28 28 − m
59. Thirteen subtracted
from a number n − 13
61. 8 less than m
62. 12 less then a
a − 12
63. 3 12 less than q
q−3
1
2
64. 0.5 less than y
y − 0.5
66. r less than 22
22 − r
67. x less than 2y
2y − x
68. m less than 5p
5p − m
y − 29
65. w less than 17
m−8
17 − w
m−r
26 − w
60. 17 subtracted from
a number n − 17
69. The difference between
x and 9 x − 9
70. The difference between
a and 3b a − 3b
71. The difference between
24 and y 24 − y
72. The difference between
17 and p 17 − p
73. The difference between
a number and 32
74. The difference between
a number and 21
75. The difference
between 15 and
a number 15 − n
76. The difference
between 27 and
a number 27 − n
n − 32
n − 21
ALG catalog ver. 2.6 – page 43 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
CF
77. Seven times a number
7n
78. Four times a number
4n
79. A number multiplied
by 13 13n
80. A number multiplied
by −25 −25n
81. The product of 32
and w 32w
82. The product of 26
and c 26c
83. The product of p
and −8 −8p
84. The product of k
and −4 −4k
85. The product of 3, r,
and s 3rs
86. The product of 9, x
and y 9xy
87. The product of a, c
and −5 −5ac
88. The product of b, −7
and h2 −7bh2
89. Three–fifths of
a number 3 n
90. One–third of a number
91. Sixty percent of
a number 0.6n
92. Forty percent of
a number 0.4n
95. Three–fourths of the
quantity x 3 x
96. Two–thirds of the
product mn 2 mn
99. 32 divided by
a number 32
100. 27 divided by
a number 27
1
n
3
5
93. One–fourth of w
w
4
5
a
6
94. Five–sixths of a
4
97. A number divided
by 10 n
98. A number divided
by 16 n
10
n
16
101. The quotient of y
divided by 3 y
102. The quotient of z
divided by 11 z
3
106. 45 divided by the
product of m and n
22
cd
n
103. The quotient of 14
divided by k 14
14 + 3n
110. The sum of 12 and
five times a number
5z
107. Fifteen divided by the
sum of a and b 15
a+b
45
mn
109. The sum of 14 and
three times a number
104. The quotient of w
divided by 5z w
k
11
105. 22 divided by the
product of c and d
3
108. Nineteen divided by
the sum of x and y
19
x+y
111. The sum of a and four
times b a + 4b
112. The sum of d and
one–half of c d + c
2
12 + 5n
113. Twice the sum of a
and 3 2(a + 3)
114. Ten times the sum
x plus y 10(x + y)
115. Two–thirds of the sum
7 plus c 2 (7 + c)
116. One–fifth of the sum
of r and 2p 1 (r + 2p)
117. One–half of the
difference of x and y
118. Twice the difference of
a and b 2(a − b)
119. Three–fourths of the
difference between
a number and 16
120. Six times the
difference between
a number and 24
1
(x − y)
2
3
3
(n − 16)
4
121. 8 less than five times
a number 5n − 8
125. −2 subtracted from
a number n − (−2)
122. 12 less than twice
a number 2n − 12
126. The difference between
w and −4 w − (−4)
5
6(n − 24)
123. 25 more than half of
a number n + 25
2
124. 9 more than four
times a number
4n + 9
127. The opposite of the
sum of 27 and b
128. The sum of 32 and the
opposite of c 32 + (−c)
−(27 + b)
129. A number squared
130. The square of
a number n2
131. A number cubed
133. A number to the fifth
power n5
134. A number to the
eighth power n8
135. a to the sixth power
137. Twice the cube of
a number 2n3
138. Three times the square
of a number 3n2
139. Two times x to the
fourth power 2x4
140. Five times y cubed
141. The sum of a number
and its square n + n2
142. The sum of a number
and its cube n + n3
143. The difference between
y cubed and y y3 − y
144. The difference between
x squared and x
n2
n3
132. The cube of a number
n3
a6
ALG catalog ver. 2.6 – page 44 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
136. m raised to the tenth
power m10
5y 3
x2 − x
CF
145. Nineteen plus
a squared 19 + a2
146. Fifteen plus b cubed
147. The quantity x plus
twelve (x + 12)
148. The quantity y minus
four (y − 4)
149. The quantity 3a plus
five (3a + 5)
150. The quantity c
minus 2d (c − 2d)
151. Six times the quantity
m plus seven 6(m + 7)
152. Twice the quantity
p minus five 2(p − 5)
153. Four times the
quantity a plus b
154. One-half the quantity
x plus four 1 (x + 4)
155. The quantity y plus
three squared (y + 3)2
156. The quantity a minus
six cubed (a − 6)3
158. The square of the
quantity x minus eight
159. Three times x plus
y squared 3x + y2
160. p squared plus four
times p p2 + 4p
163. The opposite of a
number increased by 7
164. The opposite of a
number decreased
by 15
15 + b3
2
4(a + b)
157. The quantity m plus n
to the fourth power
(m + n)4
(x − 8)2
Ambiguous problems
161. The opposite of p
decreased by 10
−p − 10 or −(p − 10)
162. The opposite of m
increased by 4
−m + 4 or −(m + 4)
−n + 7 or −(n + 7)
−n − 15 or −(n − 15)
165. Three times a number
increased by 13
3n + 13 or 3(n + 13)
166. Twice a number
increased by 28
170. Six times a number
decreased by the
number
4n + 2n or 4(n + n)
6n − n or 6(n − n)
n
n−5
− 5 or
2
2
1
1
n − 8 or (n − 8)
4
4
2n + 28 or 2(n + 28)
169. Four times a number
increased by the
number
173. 5 less than a number
divided by 2
167. One–fourth of a
number decreased by 8
174. 8 less than a number
divided by 3
171. Two times a number
decreased by half the
number
2n −
1
1
n or 2(n − n)
2
2
175. 2 more than a number
divided by 5
n
n−8
− 8 or
3
3
n
n+2
+ 2 or
5
5
168. Five times a number
decreased by 20
5n − 20 or 5(n − 20)
172. One–half of a number
increased by twice the
number
1
1
n + 2n or (n + 2n)
2
2
176. 10 more than a
number divided by 4
n
n + 10
+ 10 or
4
4
Common units
177. The number of inches
in f feet. 12f
178. The number of feet in
k yards. 3k
179. The number of
centimeters in
x meters. 100x
180. The number of
millimeters in
y centimeters. 10y
181. The number of inches
in f feet and y yards.
182. The number of
millimeters in x meters
and y centimeters.
183. The number of cents
in q quarters and
d dimes. 25q + 10d
184. The number of cents
in n nickels and
p pennies. 5n + p
187. Five times the number
of cents in x dollars.
188. Six times the number
of cents in y nickels.
12f + 36y
1000x + 10y
185. The value (in cents) of
d dimes and n nickels.
10d + 5n
186. The value (in cents)
of p pennies and
q quarters. 25q + p
ALG catalog ver. 2.6 – page 45 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
500x
30y
CG
Topic:
Statements about real numbers.
Directions:
15—Solve.
111—Tell whether each statement is sometimes, always, or never true.
112—For what value(s) of x is the statement true?
113—Which of the following statements are true?
114—Which statements are true for all real numbers?
1.
x = |x|
0
2.
x = − |x|
0
3.
x − |x| = 0
5.
x = −x
0
6.
x−x=0
0
7.
x − (−x) = 0
9.
x(0) = 0
10. x(1) = x
IR
13. −[−(−x)] = −x
17. 0 ÷ x = 0
21.
x
1
÷ =1
x
1
25.
|x|
= −1
x
29. −x3 = x3
33.
37.
41.
45.
√
x=x
0
30. x2 = (−x)2
x2 = |x|
49. x > −x
|x|
=1
−x
42.
IR
46.
IR
39.
√
−x2 = |−x|
IR
43.
√
x2 = |−x|
47.
√
√
x5 = x2 x
(−x)2 = |−x|
√
√
−x3 = −x x
50. −x < x
x>0
x≤0
0
51. x <
x<0
|x| − x = 0
8.
−x − (−x) = 0
57. x ≤ x2
54. −x < |x|
IR
20. x ÷ (−1) = −x
24.
−1
1
=
x
−x
28.
− |x|
= −1
|x|
44.
IR
48.
x≥0
58. x > −x2
IR
61. x + 1 < x
62. x − 2 < x
Ø
65. |x + 3| = |x| + 3
x=
6 0, x 6= 1
73. (x − a) = −(a − x)
70. (x − 1)2 = x2 − 1
0
IR
x−a
= −1
a−x
x 6= a
IR
1
x
64. |4 − x| > x
x=1
IR
x<0
Ø
x≤2
68. |x − 4| = |x| − 4
x≥4
1
71.
√
x2 + 1 = x + 1
74. (x + a)2 = x2 + 2ax + a2 75. |x − a| = |a − x|
78. (x − a)2 = x2 − a2
0
x≥0
60. −x2 > (−x)2
x≥0
0
IR
72.
√
x2 − 4 = x − 2
IR
79.
√
x2 − a2 = x − a
x=a
ALG catalog ver. 2.6 – page 46 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
x=a
2
76. (x − a)(x + a) = x2 − a2
IR
77.
x≥0
56. −x > − |x|
x≤0
69. (x + 3)2 = x2 + 9
0
−1 < x < 0 or x > 1
IR
67. |x − 1| = |1 − x|
IR
(−x)2 = |x|
52. x >
63. |x − 2| < x
IR
66. |x − 2| = |x| + 2
x≥0
59. x2 ≤ x3
x=
6 0
p
√
x6 = x3
IR
IR
√
40. ( −x )2 = − |x|
0
1
x
55. − |x| ≤ |x|
x>0
x 6= 0
√
36. ( x )2 = x
x≥0
0
16. x − (−x) − x = x
0 < x < 1 or x < −1
53. x ≥ − |x|
0
32. −x2 = (−x)2
IR
√
x2 = x
p
x 6= 0
x>0
35.
0
0
x 6= 0
31. −x3 = (−x)3
IR
√
38. ( −x )2 = −x
x≤0
1
=x
x
− |x|
= −1
x
27.
x<0
√
√
34. − x = −x
IR
15. −x − [−(−x)] = 0
23. 1 ÷
4.
1
12. x
=1
x
19. x ÷ (−x) = −1
x 6= 0
26.
0, 1
x4 = x2
1
1
x
IR
x 6= 0
x<0
(−x)2 = −x
√
22. x =
x 6= 0
p
√
18. x ÷ x = 1
x 6= 0
0
11. x(−1) = −x
IR
14. x − [−(−x)] = 0
IR
0
80.
|x − a|
=1
|a − x|
x 6= a
CH
Topic:
Understanding the roots of equations.
Directions:
150—Tell whether the given number is a solution to the equation. Is it the only solution?
1.
2x + 12 = 0; −6
2.
3y − 5 = 19; 8
yes
3.
a+5
= −3; −11
2
4.
x−9
= 3; 21
4
yes
5.
5y + 3 = 3y + 2y; 0
Ø
6.
7a − 14 = 7(a + 2); 16
7.
−4(c + 3) = −4c; 4
Ø
8.
20 − 3n = 25 − 3n; 5
9.
3(p − 5) = 3p − 15; −8
yes
yes
11. 10(y + 2) − 10y = 20; 2
13. −7c + 4 = 2c − 5; 8
x = +10
23. −u2 = −36; 6
14. 10 − 3d = d + 12; 2
25. y 2 − 6y + 9 = 0; 3
27. r2 + 2r = −1; −1
Ø
29. y 2 − 12y + 20 = 0; 10
31. c2 + c − 20 = 0; 4
y = 2, 10
c = −5, 4
r = −2, 5
32. w2 + 8w + 15 = 0; −5
35. |a| = −a; −7
a≤0
36. |u| = u; 15
41.
√
√
n + 3 = 2; 1
u≥4
yes
Ø
u≥0
38. y − 8 ≤ −2; 7
y≤6
40. 3c − 1 < 20; 3
c<7
√
42.
c − 10 = −3; 1
x2 = x; −5
x≥0
44.
√
8 − y = 4; −8
45.
2
= 0; −2
p+2
Ø
46.
1
1
= ;5
u−3
2
47.
2
6
=
;1
x
x+2
yes
48.
1
2
=
;3
y−3
y−3
43.
Ø
30. r2 − 3r − 10 = 0; −2
34. − |n| = 12; 12
39. 2u + 7 ≥ 15; −4
p = ±1
yes
Ø
x > −2
a = ±9
28. n2 + 25 = −10n; −5
33. |x| = −3; −3
37. x + 5 > 3; −1
3
2
yes
26. x2 − 4x + 5 = 0; 1
yes
a=
yes
24. 1 − p2 = 0; −1
u = ±6
IR
d = − 12
16. 10a + 6 = 2(a + 9); 23
22. a2 − 18 = 0; −9
c = ±5
IR
12. 1 − 6w = 4w − 10w + 1; 5
k
20. = 2; −6
3
yes
Ø
10. 5x + 2 + 3x = 8x + 2; −1
18. |y − 6| − 10; −4
yes
19. − |2m| = −20; 10
21. c2 = 25; 5
IR
c=1
15. 2(x + 1) = 3x − 8; −10
17. − |x + 3| = −5; 2
IR
Ø
ALG catalog ver. 2.6 – page 47 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
Ø
yes
yes
Ø
w = −5, −3
DA
Topic:
Evaluating expressions (one and two variables).
Directions:
60—Evaluate for the given value(s).
1.
x + 7 for x = −10
2.
4 + y for y = 14
18
3.
−3
5.
4.
15 − c for c = 41
6.
−r + 12 for r = −9
7.
21
34
64
8.
12. −11 + 4r for r = −5
−1
14. −k 4 for k = 3
−81
−(z + 7) for z = 17
−24
11. 21 − 2p for p = 11
13
13. −y 3 for y = −4
−(w − 6) for w = −3
9
10. −3n − 5 for n = −6
15x + 4 for x = 2
n − 10 for n = −6
−16
17
−26
9.
−8 + a for a = 25
−31
15. (−c)5 for c = −2
16. (−x)2 for x = 9
81
32
17. 10a2 for a = 7
490
18. 4w3 for w = −5
19. −2n4 for n = 3
−500
21. (−2h)4 for h = −1
1
3
23. −(−10y)2 for y = −4
−32
25. 5(n − 12) for n = −4
3
10
27
8
27. −3(3 − a) for a = 11
−52
29. 9 − (y + 4) for y = 16
24. −(−5r)3 for r =
−1600
26. −4(p + 7) for p = 6
−80
96
−162
22. (−6x)5 for x =
16
20. −3k 5 for k = −2
28. 8(6 − w) for w = −5
24
88
30. −7 − (1 − 2z)
for z = −8 −24
31. 4x + (8 − x)
for x = −5 −7
32. a − (a + 15)
for a = 26 −15
33. − [−(3x − 5) + x]
for x = −9 −23
34. − [4y − (y + 18)] for
y = 12 −18
35. (u − 1)(u + 1)
for u = 8 63
36. (c + 2)(c − 2)
for c = 3 5
37. a2 − 4a + 8 for a = 5
38. w3 + w2 + w
for w = −3
39. 10(t2 + t) for t = −4
40. y(y 2 − 1) for y = 11
−11
13
41.
2
u for u = 99
3
3
42. − a for a = 48
4
66
120
−21
43. −
−36
45.
3d + 4
for d = 2
d
5
46.
4c
for c = −1
10 − 2c
1320
5r
for r = −36
6
44.
7t
for t = −18
2
48.
x−4
for x = −8
x+4
52.
a3 − 9a
for a = 3
a+3
−63
30
47.
y+1
for y = 3
y−1
51.
n(n − 2)
for n = −1
n2 − 4
2
3
− 13
49.
s2 + s
for s = −4
s+1
−4
53. |p + 7| for p = −13
6
57. −2 |−6x| for x = 5
−60
61. r − |r | for r = −10
−20
50.
y 2 + 3y + 2
for y = 5
y+2
−1
6
54. − |4 − u| for u = 18
−14
a
55. − for a = −22
2
0
56. |−11y | for y = 7
−11
y
58. 12 for y = −30
5
59. 3 |s − 11| for s = 8
62. x − |x| for x = 10
63. y |y | for y = −6
9
72
0
ALG catalog ver. 2.6 – page 48 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−36
60. −5 |k + 1|
for k = −11
64.
−50
|n|
for n = −27
n
−1
77
DA
65. −ab
for a = 6, b = −4
24
66. 8xy
for x = −3, y = 2
67. −c(3d) for c = −4,
d = −5 −60
68. 11(rs)
for r = 2, s = 10
71. −2ab + 8 for a = 3,
b = −8 56
72. −n + 7p
for n = −16, p = 4
220
−48
69. x + 3y for x = −12,
y = −3 −21
70. 10 − uw for u = 6,
w = 5 −20
44
73. −(k − j) for j = −5,
k = 11 −16
74. −(a + b) for a = −4,
b = −12 16
75. − [−(x − y)]
for x = 7, y = 14
−7
76. − [c + (−d)]
for c = −3, d = −8
−5
77. 2(rs − 1) for r = 2,
s = 8 30
78. −6(3e + f ) for e = 3,
f = −1 −48
80. −5(xy − 9)
for x = 5, y = −2
79. 12(w + 5z)
for w = −16, z = 3
95
−12
81. 12 − 3(q + 2r)
for q = −6, r = −4
82. −11 − k(h + 1) for
h = −5, k = 9 25
83. −6r + 2(s − 8)
for r = 8, s = −8
84. 2(4a − b) + 18
for a = 7, b = 12
50
−80
54
85. 7c − (c + 2d)
for c = 9, d = −10
86. 3(x − 5y) + 10y for
x = 6, y = 2 8
87. −6b − 6(a − b)
for a = −1, b = −4
74
88. 8r + 3(r + u)
for r = −5, u = 9
−28
6
89. 12n + 3p − 6np for
n = −2, p = 5 51
90. −2c − 9cd + d
for c = −3, d = −4
91. x(x + 13) − 4xz for
x = 5, z = 9 −90
92. 2k − jk(j − 4)
for j = 6, k = −4
95. (a + d − 3)(a + d) for
a = −7, d = 3 28
96. (x − y)(x − y + 5)
for x = −8, y = −1
40
−106
93. r − [−s − (r + s)]
for r = 15, s = 12
94. −p + [n − (p − n)]
for n = 14, p = −8
54
44
97. (w − z)(w + z)
for w = 11, z = −4
14
98. (a + b)(a − b)
for a = 4, b = 10
99. (nt + 1)(t − 3)
for n = −6, t = −2
−84
105
101. 6b ÷ 5a for a = −9,
b = 18 − 12
5
100. (c − 7)(c + d)
for c = −3, d = 14
−65
102. −4x ÷ 9y for x = 9,
y = −4 1
103.
−3r
2s
for r = 20, s = 6
−110
104.
−5
7m
4k
for k = −21, m = −8
2
3
105.
y
+x
3
for x = −5, y = −12
c
106. d −
2
for c = −6, d = 7
107.
10
−9
109.
k
j+5
for j = −7, k = 4
108.
p
n−8
for n = 2, p = 3
12
t+u
for t = 1, u = −9
110.
r−s
for r = −16,
−4
s = −12 1
111.
xy + 3
2x
for x = 3, y = 11
112.
6
a−b
for a = −5,
2b + 1
b = 7 −4
5
− 32
113.
c+d
c−d
for c = 6, d = 4
114.
5
np − 4
np + 4
for n = 3, p = −1
115.
w(8 − z)
wz + 1
for w = −2, z = 3
118.
2
a − 3b + 4
a−7
for a = −5, b = −2
116.
3(a − b)
for a = 8,
b−a
b = 11 −3
119.
2d + 6(c + 1)
3d − 8
for c = −6, d = 4
120.
− 11
2
121. −x2 y
for x = −4, y = −7
122. p2 q 2
for p = −3, q = 2
7x + z + 8
x−z
for x = 4, z = −6
3
5
− 12
−7
117.
− 12
−2
36
123. −ns3
for n = 9, s = −2
112
ALG catalog ver. 2.6 – page 49 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−3(r + s)
6r − s + 4
for r = −2, s = −10
18
124. b4 d2 for b = 2, d = 5
72
400
DA
125. (ab)3 for a = 5, b = 2
1000
126. −x(4y)3 for x = −6,
y = 12 48
127. 4r(3u)2
for r = 61 , u = 3
54
128. (2ac)2
for a = 3, c = −4
576
129. n2 − p2
for n = 6, p = −9
130. w2 + 4u3 for u = −2,
w = −10 68
131. −a2 + 7c4
for a = −11, c = 2
−45
134. (c + d)2 for c = 15,
d = −9 36
135. w2 y − wy 3
for w = −4, y = −1
−125
e2
−4f
for e = −12, f = −4
x2
y2
+
4
9
y = −6
136. 2ns2 + n2 s
for n = −6, s = 2
138.
−2x3
u
for u = 6, x = 3
139.
−9
(−s)3
n2
for n = 5, s = −10
140.
40
for x = 6,
142.
13
r2
s2
−
for r = 9,
6
10
s = 5 11
143.
m + p2
for m = −1,
3mp
p = 5 −8
5
149. |c − d|
for c = 4, d = 18
146.
6xz 2
x+z
for x = −4, z = −2
147.
(a + d)2
−d
for a = −12, d = 4
144.
2(c − 3)2
for b = 3,
bc2
c=6 1
148.
6
16
14
a4
−d2
for a = −3, d = 9
−1
−16
145.
24
−20
9
141.
160
−9
133. (r − t)3
for r = 8, t = 13
137.
132. −10x2 + 5y 2
for x = 4, y = 8
(e − 1)2
for e = −9,
e+f
f = −11 −5
−3(y − z)2
y2 + 5
for y = 2, z = −4
−12
150. |n + p|
for n = 1, p = −9
8
151. |w | − |y |
for w = −2, y = −8
152. − |s| + |r |
for r = −7, s = 6
1
−6
153. |−5p + r |
for p = −3, r = −11
4
157. −b |a − b|
for a = −5, b = 6
−66
154. − |3c − 2d|
for c = −5, d = 4
155. − |−5(r + s)|
for r = 6, s = −10
−23
88
−20
158. u |u + w |
for u = −3, w = 8
159. b + |a| for a = −9,
b = 4 13
160. k − |h| for h = −15,
k = −12 −27
163. 3 |j + k | − 10j
for j = 3, k = −7
164. 6d − 2 |c − 3d|
for c = 9, d = −3
−15
161. |7w − 4y | − |−y | for
w = −4, y = 8 52
162. − |−2t| + |5s − t|
for s = −2, t = −6
n+p
165. − n for n = −3, p = −9
−a for a = 10,
166. a−b
b = −5 2
−8
−4
156. |8(w − 3y)|
for w = 4, y = 5
−18
167.
3
ALG catalog ver. 2.6 – page 50 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
|x| + |y |
|−y |
for x = −6, y = 2
−54
168.
4
|y + z |
− |−z |
for y = 4, z = 8
− 32
DB
Topic:
Evaluating expressions (replacement set).
Directions:
61—Evaluate for the given replacement set.
1.
−x for {−8, −6, 0, 2, 4}
3.
x − 6 for {−5, 0, 3, 6, 14}
5.
5
7x for −2, − 47 , 0, 21
,3
7.
−
9.
x4 for {−2, −1, 3, 5, 10}
x
for {−9, −3, 0, 6, 15}
3
11. x3 for − 13 , −1, 0, 12 , 4
2.
x + 10 for {−15, −10, −5, 0, 5}
{−11, −6, −3, 0, 8}
4.
6 − x for {−4, −2, 0, 6, 12}
−14, −4, 0, 53 , 21
6.
−4x for − 34 , − 18 , 56 , 32
{3, 1, 0, −2, −5}
8.
x
for {−20, −6.5, 0, 2.4, 8}
2
{8, 6, 0, −2, −4}
{16, 1, 81, 625, 10000}
x 2
13. (2x)3 for −2, − 12 , 0, 1, 34
−64, −1, 0, 8,
15. −8x3 for −5, −1, 14 , 32 , 2
1000, 8, − 18 , −27, −64
16. 3x2 for −3, − 13 , 56 , 1, 10
3
x for −4, −2, 12 , 16, 96
4
−1, − 32 , 38 , 12, 72
18.
17.
3, 12 , − 10
3 , −6
12. −x2 for −6, − 14 , 1, 23 , 7
1
, −1, 0, 18 , 64
− 27
27
8
{10, 8, 6, 0, −6}
x+1
for {−11, −1, 0, 1, 101} −5, 0, 12 , 1, 51
2
21. 5x − 1 for −5, −3, 0, 15 , 3 {−26, −16, −1, 0, 14}
19.
{−10, −3.25, 0, 1.2, 4}
10. x2 for {−8, −0.1, 0.5, 1.2, 11}
14.
3
{−5, 0, 5, 10, 15}
{64, 0.01, 0.25, 1.44, 121}
1
, −1, − 49 , −49
−36, − 16
for {−6, −2, 3, 5, 12}
4
4, 9 , 1,
27, 13 ,
−2x
for {−21, −2, 0, 3, 15}
3
25
9 , 16
25
12 , 3, 300
14, 43 , 0, −2, −10
x−1
for {−1, 1, 2, 5, 6} 2, 0, 12 , 45 , 56
x
22. −3x + 4 for −1, − 13 , 0, 23 , 1 {7, 5, 4, 2, 1}
20.
23. 2(x + 7) for {−10, −7, −4, 1, 13}
{−6, 0, 6, 16, 40}
24. −3(x − 6) for {−2, 0, 4, 6, 9}
25. x + 2(x − 2) for {−2, −1, 0, 2, 4}
{−10, −7, −4, 2, 8}
26. 3x − (x + 8) for {−3, −1, 0, 4, 6}
{−14, −10, −8, 0, 4}
28. 3(x + 4) − 3x for {−2, 0, 4, 8, 14}
{12, 12, 12, 12, 12}
27. x − (x + 3) for {−12, −6, 0, 5, 20}
29. x2 + 5x for {−6, −5, −1, 0, 2, 5}
31. −x(1 − x) for {−6, −2, 0, 1, 4}
33.
x+1
for {−3, −1, 0, 2, 3}
x−1
35. (x3 )2 for {−2, −1, 0, 1, 2}
{−3, −3, −3, −3, −3}
{6, 0, −4, 0, 14, 50}
{42, 6, 0, 0, 12}
1
2 , 0, −1, 3, 2
{64, 1, 0, 1, 64}
37. |x| for {−12, −7.5, −0.44, 0, 9}
39. |x − 14| for {−5, 0, 3, 14, 25}
{12, 7.5, 0.44, 0, 9}
{19, 14, 11, 0, 11}
{24, 18, 6, 0, −9}
30. x(x + 3) for {−5, −3, −1, 0, 3}
{10, 0, −2, 0, 18}
32. (x + 1)(x − 1) for {1, 2, 3, 4, 5}
{0, 3, 8, 15, 24}
34.
x2 − 1
for {−4, 0, 2, 5, 6}
x−1
36. 2x for {1, 2, 3, 4, 5}
{−3, 1, 3, 6, 7}
{2, 4, 8, 16, 32}
38. −2 |−x| for −8, −5, 0, 12 , 6
40. |5 − 6x| for −2, 0, 56 , 1, 3
ALG catalog ver. 2.6 – page 51 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
{−16, −10, 0, −1, −12}
{17, 5, 0, 1, 13}
DC
Topic:
Evaluating expressions.
Directions:
62—Evaluate for w = 6, x = 3, and y = 10.
1.
2(wxy)
360
5.
5w − 3x + 2y
41
9.
2(x + y) − 4w
2
2.
(3x)(wy)
6.
y + 2w − 6x
540
4
10. 5(y − w) + 12x
56
3.
3xy − 11w
7.
12x − (w + 2y)
24
10
11. 5w + x(3y − 4w) − 11
4.
wy + 2wx
8.
25 − (y − x + w)
14. 8(5x − y) − 2(w + x)
17
17. 2w(3y − wx − 4x)
0
18. xy(2w + y − 5x)
5
22.
w−x
3
25.
y+8
w+x
2
26.
15 + w
y−x
29.
w−x
y−1
1
3
30.
x+5
xy
33.
3y − wx
2y
34.
7w + 2xy
2y − x
13
38.
y
2y
+ − 2x
w
x
1
42.
7(2y − 3x)
4w − 2
46.
w − 4 2y
+
w
x
37. 2x +
3
5
3w
y
−
2
5
41.
4(y + x − 8)
3w + 2
45.
x+5 w
−
y
y
49. y 2 − 5wx
1
5
53. w(3x)2 − 3y 2
57. 2y 2 − 3x2 w
186
38
61. (x + y)2 − (w + 2)2
210
1
3
4
15
50. w2 + 4xy
10
6
2
3
7
2
7
58. x2 y + 5w2
27
19. (4w + y)(4 − x)
23.
w
+y
x
27.
y−w
+x
2
31.
4w − y
4x
35.
6x − y + 8
7w − 4y
39.
w
w+x
−
8
y+8
43.
6y − w
(y − 8)(2x + 3)
47.
5
2w
·
y w+x
0
270
62. (w − 4)2 + (y − x)2
34
20. (y − w)(x + w)
24. y −
12
5
7
6
8
1
4
3
2
3
51. 2(wy − x2 )
156
54. 3wy − 5(2x)2
105
16. x + [4w − 3(y − 10)]
74
x
3
21. w −
14
15. 15w − [2(w − x) + y ]
22
59. wx2 − 2y + w2
63. 3(wx − y)2
192
28.
y
2
−
w
x
32.
w + 2x
y+w
36.
2wx + 3y − 3
4wx
40.
y + 5x
x
−
y
w
44.
(w + 1)(x + 2)
w(y − 3)
48.
4x − y
x
÷
8
wy
1
3
4
70
3w2
x2 y
69.
(w − 2x)2
y+5
6
5
0
66.
xy 2
5w2
70.
2(x + 2w)2
3y − 5
5
3
18
67.
6y − w2
x
71.
(w + x)2
9y + x2
ALG catalog ver. 2.6 – page 52 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
8
9
11
7
8
2
5
6
5
354
56. 2w2 + (xy)2
24
36
8
52. x(y 2 + 3w)
102
55. (wx)2 − 5wy
w
x
972
60. y 2 + 5x2 − 6xw
64. 4(2y − wx)2
16
53
65.
12
12. 6y + 8 − xw(2w − 9)
37
13. 5(y − w) − 3(y − 3x)
96
68.
2x2 + 7w
2y
72.
3(y − w)2
y2 − w2
3
3
4
37
DD
Topic:
Evaluating expressions.
Directions:
63—Evaluate for a = 5, b = 7, and c = −2.
10
2.
a−b−c
140
6.
5c · 2ab
1.
a+b+c
5.
(ac)(bc)
9.
−3a + 2b + 4c
13. b(c − a)
−9
0
3.
4b − ac
−700
7.
−3c(ab)
10. 6b − 4a − 11c
14. −c(a + b)
−49
17. 9a − 10(b + 5c)
75
38
210
11. −7ac + 6bc
44
15. 5a(b + c)
24
18. 3(2c + a) − 11b
3a + bc
8.
(−5b)(−ac)
1
12. 9b − 2abc
−14
−350
203
16. −2b(a − c)
125
19. 2(a − b + c) + 2ac
−74
4.
−98
20. 13c + a(2b + c)
34
−28
21. (a − b)(12 − c)
25. ab ÷ 10c
4b − a − c
a
33.
3b − 2c
a+b+c
b
c
−
a a
22. (−a + c)(3a − b)
26. ac ÷ bc
− 74
29.
37. −
−28
−56
23. (2b + c)(−5 + a)
5
7
27.
a−b
c
1
30.
ac + b
a+c
−1
31.
c − 2b
a+b
5
2
34.
ab + 1
−c
18
35.
4c − a + b
2b − c
−1
38.
a+b
+b
c+1
−5
39.
a+c b
−
c
c
42. (ac)2 − b2
51
43. −b(ac)2
5
41. a2 c + bc2
−22
45. (a − b)2 − c2
0
46. ab − (a − c)2
0
− 43
− 38
2
28.
a+b
a+c
32.
a + 4c
a − 5b
36.
a+b+c
−a
−2
40.
a−c 1
+
b−1
c
2
3
64
4
1
10
44. ac(b + c)2
−700
47. (b + c)2 − (a + 1)2
−14
24. (3a − b)(c + 2a)
250
48. a2 − (a + b + c)2
−75
−11
49. ac4 + bc3
53.
50. bc5 − a3
24
b2 − a 2
c2
54.
6
57. 6c(2a − b) + 3(a + b)
(b + c)2
ac
99
51. (ac)3 + (ab)2
− 52
55.
58. 7(b − c) − a(b + c)
38
52. a2 c3 − b2 c
56.
3
59. −5(ac + 1) + 9(2b − 3a)
0
62. (8c − b) − (−ab + ac)
−18
65. − [−(−15c + a) − (b + 6c)]
67. (a + b) − [−(b − a) + (c − a)]
73. |a − b|
36
81. |−a(b − c)|
85. − |−(b + c)|
a+b
93. c −3
6
45
−5
12
66. − [−2ab + (−b − c) − 3ac]
45
68. 3ac − [(ab + c) + (ac + b)]
−60
71. − |5a|
7
74. |b − c|
2
77. 4 |a + bc|
89. |c| − |a|
21
70. |−b|
12
75. |a − b + c|
9
−75
79. b |1 + ac|
82. |c(a + 2b)|
38
83. |2abc|
86. − |−(2a − c)|
90. |b| − |a|
|a − b|
c
72. − |−c|
−25
78. −a |b − 4c|
94.
64. − [(b − 16) − (a + c)]
−15
30
2
−1
−12
−1
−16
63. − [2a − (b − c) + 14]
22
−a2
3b + c2
−102
60. c(b − 9) + a(bc + 10)
36
61. −(−b − 7c) + (−a + 3c)
69. |6c|
(b − c)2
4a + b
225
−2
4
76. |a − b − c|
0
63
80. −c |a − 3b|
32
84. |−3a − bc|
1
140
87. |a − (b + c)|
0
88. |b − (c − a)|
91. − |ac| − |3b|
−31
92. |bc| + |2a|
95.
ALG catalog ver. 2.6 – page 53 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
|4b| + |c|
a
6
4c − b 96. − a 14
24
−3
DE
Topic:
Evaluating expressions.
Directions:
64—Evaluate for n = −3, p = 4, and r = −1.
1.
nr + pr
5.
(np)(−pr)
9.
5r − 6n + p
−1
−48
13. nr(2p + 3)
6.
−5npr
15
3.
10n + pr
−60
7.
2p · nr
14. −p(nr + p)
33
−4
25. n − [−(−p + r) − n]
−28
22. 7(n − p − r)
−4
−n
6pr
−2
38.
5n + r
p
5
42.
n + 9r
p−2
46.
5n 3r
−
p
p
50.
5n
n
·
n+p r+6
37.
p − 2r
n
41.
pr − 6
n+1
45.
n nr
+
p
p
49.
p+r
p
·
n
r−n
3
4
0
−2
12. 4pr − 5n + 6
15. −7(p + 2nr)
−70
16. 2p(n − 10r)
19. 11n − 3(p − r)
27. − [−(p − r) − (p + n)]
(p − r)2
3n + r
− 52
77. |p − r |
28. − [−(n + r) − (−n + p)]
3
32. −7nr(4n + p − 2pr)
31. 3pr(np + 8p + 9r)
p
r
− 12
40.
−n − r
−p + r
− 45
−6
43.
np + 2pr
n + 2r
44.
r − 5nr
np − p
1
48.
6
p
− nr −
n
r
52.
−10p(pr + 5r)
3n(p + 2n)
47. 7p +
−3
51.
9
66.
p2 r 3
2n
70.
(n + pr)2
n + 10
4
p
r
+
n n
27
8r(p + r)
n(3n + p − 7r)
55. −n2 pr3
17
8
3
7
56. np2 r2
67.
7r5
p−n
71.
n2 + p + r
p2 + r
−1
4
5
15
64. (n − p)2 − (r − 1)5
68.
n2 − p
3n + r
72.
n2 + p 2 − r 2
p
8
82. r |8n + p|
−20
83. p |1 − nr |
8
84. −p |15r − n|
85. |−10n − p|
26
86. |7n + 5p|
1
87. |2nr + p|
10
88. |9 − npr |
89. |−(p − n)|
7
90. − |−(3p + r)|
94. − |r | + |n|
−1
97. − |2r(p − n)|
−14
95. |2pr | + |n|
2
98. |n(p − r + 5)|
91. − |p − (r − n)|
30
−2
11
99. − |(p − r) − (n + 2)|
6
−6
−42
−11
2
5n + r 102. − p −4
103.
ALG catalog ver. 2.6 – page 54 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
|n| − |p|
r
1
−48
3
92. |n − (p + 2r)|
5
96. − |nr | − |3p|
−15
100. |(r + 8) − (p − n)|
−6
p − 2r 101. n 104.
81
− 12
80. |n − p + r |
0
20
−48
76. − |−6r |
−4
79. |n + p + r |
7
−174
−9
81. −6 |p + nr |
93. |n| − |p|
0
1
60. 6pr2 − n2
−46
63. (5r)3 − (p − n)2
75. − |−p|
21
4
36
59. p2 r − 10nr
84
62. (n + p)3 + (2r)4
78. |n − p|
5
−40
−55
n+p
n−r
108
74. |−7n|
3
24. (r − p)(8 − n)
39.
58. −7p · nr2
69.
20. −9p − 2(r − n)
−4
96
16
−6
56
36. n −
57. np(2r)3
−np2
nr
−48
5
7
54. n3 pr
65.
−36
n
+p
r
−240
−137
−3r(np)
35.
− 18
53. −5np2 r
61. (r − p)2 − 2n4
8.
−132
34.
9r
np
18
0
−54
33.
6p − 2nr
6
30. r(p − 4) − 2(6p + 3)
−35
24
4.
11. 11 − nr − 2p
7
29. 5(n + r) + n(p − r)
−34
23. (p + r)(n − r)
−42
26. −r − [−p − (−n + r)]
−11
73. |3r |
−6
18. −2p − (n + r)
2
21. −2(n + p − r)
nr − np
10. −3p + n − 9r
17
17. 5r − (n − p)
2.
|−7p − r |
n
−9
0
DF
Topic:
Evaluating expressions.
Directions:
65—Evaluate for x = −2, y = −3, and z = 5.
1.
4x − yz
5.
−y(x + yz)
9.
2(x − y) + 9z
7
−51
2.
z + 10xy
6.
8x(yz − 1)
65
256
10. −z(y + 4) − 3x
47
−225
14. xyz(xy − z − 5)
17. (x + 4y)(x − z)
98
18. (3y + z)(2y + x)
3x − (y + z)
−8
4.
y − (6z − x)
7.
5(x + y + z)
0
8.
−x(y − z + 18)
11. z − xy(z + 1)
1
13. 3yz(x − 4y − z)
3.
−120
32
12. yz + 6(x + z)
−31
15. −4xz + 9y − 2yz
−35
20
3
16. 10x − 8xy + 2xz
43
−88
20. 4(y + z) − 8(x − y)
19. x(y + 1) + y(z + 3)
0
−20
21. y − [5x − (6y + z)]
22. 3z − [−y − (x − 4z)]
−6
26.
z−1
xy
2
3
27.
2y + x
z+3
30.
xyz
x+y
−6
31.
xy − 11
5xz
−4
34.
4x − yz − 7y
xz + 8
−14
35.
12y + 8z
x+y−z
4
3
38.
x(9 + 2yz)
2z(1 − x − y)
7
10
39.
7z(6y − 4x)
−2(x − z)
42.
−3y − z
(y − 2x)(z + 4)
43.
x − 4y x − 4
·
y
xz
46.
y+z
z
+
4x
2−x
5y
z
−
4
2x
25.
y+9
x+z
29.
3yz
4z + x
33.
xz + 4y − 6
2xy − z
37.
−8(2z + 5y)
3x(x + y)
41.
(4z + x)(y + z)
−x + 3y + z
45.
z
z
−
xy
y
49. −x2 yz
23. xy [z − (2x − y)]
24. 5x [2y + x(z + 1)]
36
2
− 52
−18
5
2
50. xy · z 2
60
53. x3 + xy 2 − z 2
57. 10z − xy 3
−51
−4
61. z 2 − (x + y)2
−3
4
9
47. −x +
1
51. −xy 3 z
150
28.
y−4
xz − 4
32.
8xz − y
x + 3y
36.
3xz + z − 2
x−y+z
40.
−yz(7y + 1)
5xyz
−2
44.
2y
1
·
3y + z 2y − x
− 12
48.
y
xz
+
+y
3
z
−1
1
10
− 25
25
52. x4 yz 2
−270
1
2
7
− 92
−2
−1200
12
55. z 2 − (xy)2
−11
56. (xz)2 − 6y 2
46
58. −x4 y 2 + 2z
−134
59. 2z 2 − x3 y 2
122
60. 4xy 2 − x5 z
88
64. 3z(y − x)5
−15
63. x(y + z)4
22
66. 10y 2 + xy 2 − z 3
−53
−32
67. (x + z)2 + (y + z)2
13
− 38
−6
54. x5 − y 4 + z 3
62. (z − x)2 + y 3
0
65. 6x3 − 5xy + 3z 2
68. (z − y)2 − (−x − y)2
39
69.
y 2 − 3x
z2
73. |x + y |
70.
3
5
z2 − y2
x2
74. |x − z |
5
71.
4
−2
78. |1 − xyz |
81. |x + y − z |
10
82. |x + y + z |
85. |z | + |y |
86. |x| − |z |
8
89. z |y | − y |x|
21
93. |(12 − z) − (x + y)|
(x − z)2
y−4
75. |z − xy |
7
77. − |4y − xz |
79. y |2xz |
29
72.
1
76. |x + zy |
87. |z | − |xy |
−3
90. −4 |yz | + |4x|
−52
94. |(y + z(x − 1)|
18
− 53
17
80. −x |y + z |
−60
83. |x − (y − z)|
0
(x + y)2
yz
−7
84. |−z(x + y)|
6
88. |2x| + |yz |
−1
91. |6x − z | − 8y
95. 2y |x + z + 12|
41
−90
4
25
19
92. 6x − |z − 10y |
−47
96. −x |y + 4z − 2|
30
12
97.
|y − z |
− |x|
180
−10
−4
x+y
98. − z −1
99.
ALG catalog ver. 2.6 – page 55 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
|yz |
|x − 1|
5
x+y−z
100. −5
2
DG
Topic:
Evaluating expressions.
Directions:
66—Evaluate for a = −4, b = 8, c = 2 and d = −3.
1.
a−b−c+d
−17
2.
a+b−c−d
5.
2a + 3bc − d
43
6.
−a + 7c + 5cd
9.
6c + d(a + b)
0
5
−12
10. −9d + a(b − c)
3.
10ac − bd
7.
a + b − (c − d)
−56
−1
11. 3(c + d) − 2(a − b)
3
21
4.
−acd + 6b
8.
a − (b + c) − d
24
−11
12. −7(b + c) + 5(a − d)
−75
13. −4a(bc + bd)
14. abc(5c + 3d)
−128
17. b(3a − c + d + 10)
15. −ad(2bc + 7d)
−64
−132
18. −d(a + b − 4c + 9)
15
19. (a − d)(3b − 11c)
22. −(b − c) − (a + d)
1
23. −c − [−(a + b) + d]
−2
16. 10c(−13a + bcd)
80
20. (5d + b)(a + 2c)
0
24. a − [b − (c − d)]
−7
−56
21. (−a + c) − (−b + d)
5
17
25. − [b + (−a + c) − (−d)]
26. − [−(b − d) + (a + c)]
−11
29. −ac [a + cd(b + d)]
28. − [−(a + d)] + (b − c)
−17
30. cd [b − a(ac + bc)]
−272
33.
27. −(b − a) − [−d − (−c)]
9
−1
31. (ac − b)(cd + bc)
−160
32. (bc − ad)(c + ad)
56
−240
a+b+c
d
−2
d a
37. − +
c
b
1
41.
6c − d
2b − a
45.
−6(a + bc)
3 − 7d
3
4
49. ad3 − bc2
−3
34.
c
b + ad
38.
a+b
−d
c
42.
11a − bd − 6
c
−13
43.
9d − 5
−a + bc + 8
46.
c(a + d − 5)
6b
− 12
47.
−cd(b + 7)
4a + 1
35.
1
10
39. ad −
5
50. −ab − c3 d2
76
a+b
cd
− 23
36.
bd
a−c
8
40.
c d
−
b a
44.
7c + 5b
ad + d
48.
7(b + d)
3c − 9a
b
c
− 87
−6
51. (b − c)2 + (a − d)2
−40
37
4
− 12
6
5
6
52. (b + d)2 − (a − c)2
−11
53. (a + b)2 (c − d)
54. (a + b)(c − d)2
80
57. −6a − (cd)2 + b2
52
55. a(b + c + d)2
100
58. ac2 + (4d)2 + 3b
152
−196
59. 2bd − 5c3 + (a − 1)2
56. −d(a − b + c)2
60. 10a − (b + c)2 − 2d3
−63
61.
b2 − a 2
c + 2d
65. |a − b|
62.
−12
66. |c + d|
12
69. |ac + bd|
32
73. − |5(ac − d)|
−25
77. |(a + b) − (c − d)|
81. |−8d| − |a − b|
85. |a − b + c| + 10d
89.
|a| + |bd|
|c|
d 2 + a2
4b − c
14
1
12
−20
63.
5
6
1
(a − b)2
cd
−86
64.
67. − |c − b|
−6
68. − |a + d|
40
−16
71. c |ad + b|
74. |a(7 − bc)|
36
75. |cd − (a + 7)|
82. − |c − d| + |a − 1|
86. ab − |7c + d|
90.
|ad − c|
|b − 3|
2
−43
0
83. 5c |b − 2a|
b − 4d 91. − a ALG catalog ver. 2.6 – page 56 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−5
−7
8
76. |2(b + d) − 5c|
9
−5
80. |b + d| − |ac|
84. cd |3a + 11|
160
87. |(a + b)(c + d)|
− 34
72. −a |b + cd|
79. |c − a| − |b − d|
17
−ad
(a + b)2
−24
70. − |acd − b|
78. |b − (a − c + d)|
300
4
−3
−6
88. |(b − c)(a + d)|
ab − 1 92. d 11
0
42
DH
Topic:
Evaluating expressions (negative and zero exponents).
Directions:
66—Evaluate for a = −4, b = 8, c = 2 and d = −3.
1.
c−5
5.
−b−2
9.
(a − d)−5
13.
−3
d
9
1
32
1
− 64
21.
cd
ab
25. a3 b−2
6.
−d−4
14.
1
8
1
− 81
c −2
10
22.
1
5a
3b
0
33. a−1 + c−2
0
34. b−1 − c−3
0
38. a−1 + a0
− 14
16
9
−1
42. (c−2 − 1)−3
45.
1
c−4 − 1
− 16
15
46.
1
b−1 + c−3
49.
1 + a−2
a−2
17
50.
b−1
1 − b−1
a−2
1
4
8.
−c−3
c −3
d
1
16
− 18
12. (c − b)−2
1
16
16.
1
20. (d − c)0
ac −2
24.
9
bd
1
36
−2
d
a
− 27
8
3c
4d
16
9
1
−3
−8
28. 12c−1 d−2
−6
2
3
31. (5a−3 b2 )0
1
32. (b−1 c3 d)−4
35. a−1 + b−1
− 18
36. d−1 + d−2
39. c0 − c−3
− 64
27
4
1
7
4.
27. b2 c−5 d
− 12
30. (3a0 d−1 )−5
41. (a−1 − c−1 )−2
−a−1
23.
1
16
8
9
7.
1
− 27
19. (−3b)0
1
12
29. (b−2 c2 )−1
37. 1 − d−2
d−3
15.
25
26. a−1 bc−2
−1
3.
11. (a + b)−2
1
18. (b − a)−1
1
− 20
0
b−1
10. (c + d)−4
−1
−27
17. (5a)−1
2.
40. c−2 − 1
7
8
43. (c−1 + d−1 )−1
47.
1
1 − a−1
51.
b−1 + c−3
a−3
ALG catalog ver. 2.6 – page 57 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
4
5
−16
6
1
81
− 29
− 34
44. (1 + b−1 )−2
48.
1
d + d−1
52.
b−1
c−2 − a−1
64
81
3
− 10
1
4
DI
Topic:
Adding variable terms. See also categories DJ (combining like terms), EA and EB (adding and
subtracting polynomials).
Directions:
6—Simplify.
1.
a + 4a
5.
10d − 6d
9.
y − 7y
5a
4d
17. −4m − (−23m)
21. −8x − (−8x)
6.
20w − 11w
10k
37. 3y 5 + (−15y 5 )
53. −6ux + (−8ux)
57. 11bc − (−9bc)
9p
5a + 3a
8.
−5k + 8k
8a
12. 8n − 12n
−9c
3k
−4n
15. 16r − (−12r)
28r
16. 4w − (−7w)
11w
19. 2h + (−14h)
−12h
20. 5x + (−11x)
−6x
24. 14k − 14k
0
0
27. −7d3 − 14d3
−21d3
28. −m4 − 4m4
30. 12c2 − 11c2
c2
31. −6y 4 + 10y 4
4y 4
32. −10x3 + 11x3
35. 5x2 − 6x2
−a7
−x2
39. −5p2 + (−4p2 )
2x4
43. 4wz + 2wz
8ab
6hk
−3rs
54. −11cd + (−cd)
−12cd
58. −3wz − (−13wz)
20bc
−3p + 12p
4.
7w4
50. rs − 4rs
−14ux
7.
12y
23. −10r + 10r
0
46. −7hk + 13hk
−5xy
7y + 5y
26. 5w4 + 2w4
42. ab + 7ab
8pr
49. 14xy − 19xy
10a
38. −x4 − (−3x4 )
−12y 5
5cd
45. −pr + 9pr
−10z
34. −10a7 + 9a7
−6h2
3.
11. 2c − 11c
−x
22. 21p + (−21p)
0
5a8
33. −7h2 + h2
9w
18. −6a − (−16a)
19m
2k 2
41. 2cd + 3cd
9k + k
14. −6z − 4z
−11z
29. 9a8 − 4a8
2.
10. 4x − 5x
−6y
13. −3z − 8z
25. k 2 + k 2
98—Perform the indicated operation(s).
−9p2
6wz
−5m4
x3
36. 8w2 − 13w2
−5w2
40. 2a4 − (−a4 )
3a4
44. 3xy + xy
4xy
48. 8ab − 4ab
4ab
47. 10mn − 9mn
mn
51. −15ab + 9ab
−6ab
52. −21mp + 8mp
55. −7pr − 12pr
−19pr
56. −2ax − 4ax
59. −16km − (−10km)
−13mp
−6ax
60. −8rx − (−9rx)
rx
−6km
10wz
More than two terms
61. 5k + k + 2k
62. 3m + 2m + m
8k
65. 12p − 3p + 5p
69. −4a − 15a + 9a
66. 8x + 3x − 2x
14p
−10a
21x
74. 2k + 5k + 11k
77. 12k − 8k − 5k
−k
78. 9z − 2z − 6z
81. −7r3 + 10r3 − 2r3
r3
9x
70. 13x − 21x + 6x
73. 6x + 3x + 12x
63. 3c + 6c + 4c
6m
−2x
18k
z
82. −5n2 − 7n2 + 4n2
64. w + 2w + w
13c
4w
67. 2k + 15k − 9k
8k
68. 11z − 2z + 5z
71. 10c − 2c − 16c
−8c
72. −5y − 9y − 2y
−16y
14z
75. 15p + 9p + 2p
26p
76. 4h + 12h + 3h
19h
79. 5r + 9r − 13r
r
80. 8u − 11u + 2u
−u
84. 3z 4 + z 4 − 6z 4
−2z 4
83. 2c2 − 8c2 + 6c2
0
−8n2
85. 5d2 + (−11d2 ) + 3d2
86. 3k 3 + (−8k 3 ) − (−5k 3 )
0
−3d2
89. −7ay + 10ay − 3ay
87. x4 − (−2x4 ) + (−4x4 )
0
90. −2cd − 5cd + 8cd
−x4
cd
91. 6ab − 8ab − 3ab
88. −4y 2 − (−5y 2 ) + 9y 2
10y 2
−5ab
92. 9mn + 4mn − 6mn
7mn
93. 3bc − (−7bc) + (−18bc)
−8bc
97. 4y − 2y + 6y − 2y
94. 11uw + (−12uw) + 4uw
95. −2rz + (−2rz) + rz
−3rz
3uw
6y
98. −8k + 20k − 7k + 2k
99. −4x + 10x − 6x + 5x
7k
ALG catalog ver. 2.6 – page 58 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
5x
96. −8y 2 + 5y 2 − (−3y 2 )
0
100. 10p − 5p + 9p − 3p
11p
DI
101. 3a + 12a − 19a + 4a
102. 6x − 15x + 3x + 6x
0
103. 10y + 4y + 6y − 20y
0
104. −16k + 5k + 8k + 3k
0
105. −15c − 28c − (−11c) + 45c
107. −8p − (−4p) + 2p − 7p
−9p
109. 5w + (−18w) + 26w + 7w
111. 12c + 6c + (−10c) + 3c
13c
0
106. −5n + 9n + (−16n) + 9n
−3n
108. 15x − (−20x) + 8x − 17x
26x
110. 6p − 11p − (−9p) + 21p
20w
112. −12y + 5y + 17y + (−9y)
11c
113. −3p − p − (−9p) + (−11p) + 7p
y
114. 11x + (−7x) + (−4x) + 3x + 2x
p
115. 3m + (−5m) − 13m − (−6m) + 8m
25p
116. 5a + (−11a) − 2a − (−14a) + 3a
−m
5x
9a
Fractions and decimals
117.
4
1
y+ y
5
5
121. −
k
k
−
6
6
−
3
1
125. − d + d
4
4
2
129. 2a − a
3
133.
m
−m
4
137.
2w
w
+
3
6
118.
y
k
3
1
− d
2
5a a
+
6
2
−w
1
1
123. − z − z
2
2
−z
124. −
7
1
c− c
10
10
1
5
n− n
12
12
1
− n
3
127.
128. −
7r
r
+
8
8
5w
6
135. −2a +
5w
6
138.
3x 3x
+
7
14
9x
14
9
k
10
4
h
3
−
a
3
5
1
161. − x − x
6
8
146. −
3
1
r+ r
10
2
4
r
7
7k
3k
−
4
8
3
a
10
166.
c
7c 5c
173. − +
−
4
2
4
170. −
2c
174.
k
8
−
a
4
132.
7
− a
4
2
1
c+ c
5
10
3g
g
−
8
2
147. −
5x 2x
+
18
3
−
7
2
y− y
5
10
148.
3
y
10
152. −
4r
5r
−
5
6
0
171.
9a 4a
a
+
−
4
3
12
−n
175.
5
5
5
p− p+ p
9
6
18
−
2b
5
5p
2
11
2
p+ p
15
3
−
1
p
15
5x
3x
+ 2x
4
4
5
1
160. − p − − p
9
3
164.
4
r
30
1.3y
8p p
−
3
6
7
z
12
156. −
5
n
6
179. 4.7w − (−2.4w)
ALG catalog ver. 2.6 – page 59 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
5y
6
7
x
18
3k k
2 2
7b
b
159. − − −
10
2
183. 5.2y − 3.9y
−
−
167.
−9.2x
8
r
3
5z
z
144. − −
6 12
7
w
12
9.1x
3r
4
7g
8
3
8
n+ n
10
15
1
1
7
n− n− n
3
6
6
−
4
− c
5
1
a
2
−
−
y
6
x
1
1
a+ a
3
6
140.
155. −k +
9
x
5
2
r + 2r
3
136. −y +
1
c
2
163.
2x x
x
+ −
5
2
10
182. 2.6x − 11.8x
3
k
2
143. −
151.
2x
3
−
19
r
18
−
178. 3.6x + 5.5x
−4.1w
3
− w
4
1
r
5
17
9
w− w
6
4
0
4.3r
139.
5
1
154. −2c + c − c
3
3
3
12
158. − x − − x
5
5
5
5
162. − r − r
9
6
2
1
1
y− y− y
3
2
6
181. −9.6w + 5.5w
7
2
w− w
20
5
23
x
24
−
2
7
165. − a + a
5
10
142. −
150.
k
2
131. k +
7
− x
5
−
x 7x
−
9
9
w
6
3
x − 2x
5
5
1
w − 3w − w
2
2
2r
6r
157. − − −
7
7
177. 1.4r + 2.9r
120.
4w
3w
−
7
7
130. w −
153.
169.
3m
4
119.
134.
7
9
h− h
5
15
149. −
126.
x 2x
+
3
3
3m 3m
+
8
8
1
a
2
3m
4
3
3
141. − k − k
5
10
145.
122. −
4
a
3
−
1
1
a+ a
4
4
3
7
c+ c
4
10
168. −
172.
a
2
− p
9
29
c
20
11
11
m+ m
4
5
5
2
5
k+ k− k
6
3
2
−
11
m
20
−k
11w
7w
9w
−
+
4
4
2
0
176. −
7.1w
180. 2.8k − (−3.6k)
184. −1.7p + 8.2p
6.4k
6.5p
0
DI
185. 0.7k − 0.4k
186. −0.2w + 0.9w
0.3k
0.7w
187. −0.8x − (−0.2x)
188. −0.7m − (−0.4m)
−0.6x
189. −0.01y + 0.05y − 0.02y
0.02y
193. 22.7z − 25.1z
190. −0.22x − 0.43x + 0.85x
191. 0.26c − 0.97c + 0.31c
−0.4c
0.2x
194. 12.4c − 18.9c
−2.4z
197. 6.8x − 1.9x − 3.9x
x
−0.3m
−6.5c
198. −3.7a − 5.4a + 7.1a
195. −19.7p + 22.3p
−0.07a
2.6p
199. −2.1m + 2.7m − 1.6m
2a
ALG catalog ver. 2.6 – page 60 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−m
192. 0.03a − 0.09a − 0.01a
196. −5.3r + 15.2r
9.9r
200. 3.6q − 9.3q + 2.7q
−3q
DJ
Topic:
Combining like terms. See also categories EA and EB (adding and subtracting polynomials).
Directions:
6—Simplify.
7—Simplify, if possible.
98—Perform the indicated operation(s).
1.
6m + 2m + 3
8m + 3
2.
5p − 7 − 2p
3p − 7
3.
9 + 8a − 1
5.
−r3 − r − 2r
−r3 − 3r
6.
y + y 2 − 2y
y2 − y
7.
4a2 + 3a + 5a2
8a + 8
4.
4 + 7y − 10
8.
2x − 2x2 + 3x2
7y − 6
x2 + 2x
9a2 + 3a
9.
2x + 5 + 12x − 4
10. 11c − 3 − 4c + 8
7c + 5
11. −3 − 2y + 1 + 5y
13. −5m + 2 − 4m − 9
14. 3x − 5 − 9x + 6
−9m − 7
17. 18m − 2k + k + 2m
29. −9ab2 + 9b2 + 18ab2
9ab2
19. 2a + 7c + 8a − c
−2r − 3p
23. 14x + 6xy − 17xy + 8x
26. 6g − f + 7f − 18g + 3f
37. w + 4 − (−w) − 8 − (−10)
−5x + 5y
24. 7pr − 8r − 5pr − 3r
−11r + 2pr
−10k
30. −5kn − 2k 2 n + 11kn
35. 22p − 7p + 19 − 15p − 12
20. 11x − 4y + 9y − 16x
27. 12h − 14k − 8h + 4k − 4h 28. −9b + 9a + 2b − 2a + 7b
9f − 12g
31. c2 d − cd − 3c2 d
6kn − 2k 2 n
33. 18r − 12r − 17 + 21 − 3r − 4
−4a − 3
22x − 11xy
−3a + 4ab
+ 9b2
16. −a + 5 − 6a − 8 + 3a
10a + 6c
22. −5a − 6ab + 10ab + 2a
3x − 3y
8y − 21
−2k + 11
18. 4r − 8p − 6r + 5p
17n − 5kn
25. −x + 3y + 4x − 5y − y
15. 2k − 3 − k + 14 − 3k
−6x + 1
20m − k
21. 4kn + 13n − 9kn + 4n
12. 19 + 8y − 34 − 6
3y − 2
14x + 1
7a
32. 4x2 y − 8xy 2 − 2x2 y
−2c2 d − cd
34. 6h + 6 + 5h − 13 + 7 − 15h
3r
36. 4m − 11m − 27 + 7m + 15
7
41. 11p + (−5p) − 3r − (−2r) − 6p + r
45. −13a + 6 − 4c − 11 + 5a − 2c
49. −3w + 8x + (−y) − 9x + 4w + y
0
46. 1 − 2w + (−5z) − 11 − (−7w) + 4z
5w − z − 10
−10a + 10b
11r + 4w
54. −64 − 2m + 28 − 8a − (−m) + 6a + 36 + m
6
56. −9y − 4x − 15 + 7x + 46 + 5y − 3x − 8
8c − 1
57. −17a − 20 + 3b + 13 + 24a − 11 − 18b − 6 + 9b
−2a
−4y + 23
58. 8k − (−6w) − 14 + (−8w) − k + 3w − 9k + 10
7a − 6b − 24
−2k + w − 4
59. −2r − 5p + 3s − 4p + 4r + 9p − 7s − 2r + 10s
61. 9a − 8aw − 5a + 3w + 10aw − 4a
−2m + p − 5
52. 17r − (−2w) + (−2y) − 6r + 2w + 2y
27h + 25k
53. 12 + 4n − (−6k) − 6 + 8n + (−6k) − 12n
0
0
50. 10a + 15b − 15c − 5b − 20a + 15c
w−x
55. 4d − 12 + (−3d) + 8c − (−6) − d + 5
3d − 8
44. 12 + (−4a) + (−14) − 8a + 2 + 12a
48. −4m + 6p − 8 + 2m − 5p + 3
−9d − 2e + 26
51. 45h + 6j + 11k − 28h + (−6j) + 14k
7k + 17
42. 4x − 16y + 9x + 20y − 13x − 4y
0
−8a − 6c − 5
47. 15 − 7d − (−3e) − 2d + 11 − 5e
−12
40. 2d + (−15) − 5d − (−6d) + 7
−z + 2
43. −9 + 4w − (−15) − 6w − 6 + 2w
−4h
38. 6 − 6k − (−11) + 10k − (−3k)
2w + 6
39. 6 − (−4z) − (−8) + (−5z) − 12
2x2 y − 8xy 2
6s
3w + 2aw
63. 2h − 11k − 13hk − (−7k) + 5h + 4k
7h − 13hk
65. −2xy + 7xz − 5x − (−4xy) + 10x − xz
6xz + 2xy + 5x
60. c − 3a − 6c + (−4d) − 4a + 4d − (−5c) + 6a
62. 4cy − 10c + 6y − 2c + (−4cy) + y
64. −3mx − 9x + 18mx + 9 − 15mx
−12c + 7y
−9x + 9
66. 2ah − 6ak − 10a + 4ak − 7ah + 12a
ALG catalog ver. 2.6 – page 61 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−a
−5ah − 2ak + 2a
DJ
67. 20c − 12cd − 45d + 21cd − 32c + 32d
68. 20np − 4n − (−13nw) − 5np + 3n − 16nw
−12c + 9cd − 13d
15np − 3nw − n
69. −7cw + 6c + 4w − 4c + (−5w) + 6cw + w − 2c
71. 16a + 8ax − 23x − 9a − 9ax + 18x − 7a + 5x + ax
73. 3p2 − 3 + p2 + 2 − 6p2
70. 8p − (−3mp) + 5m − 6p + mp − 2m − 2p − 4mp
−cw
0
74. 4 − (−c3 ) + (−8) − 2c3 + 5
−2p2 − 1
75. 3w2 + 19 − (−10w2 ) − 10 − 9w2
77. −9x + 4x3 − (−3x) − 2x3 + 6x
79. −9y 2 − 5y + 4y 2 − 2y + 7y
2x3
81. 7y 2 − 2y 3 + y 3 − 6y 2 + 3y 2 − 3y 3
83. 2k 2 − (−2k 3 ) − 3k 2 + (−k 3 ) + 9k 3
87. −3x2 − 7 + 4x2 + 4x + 8
−m4 − 9m
2c3 + c
3x2 + 10x − 7
95. 11n − 10n2 + 8 + 5n − 14 − 3n2
−13n2 + 16n − 6
97. y 2 + 18 − 7y − 23 + y − 3y 2 + 6y + 5 + 2y 2
99. 5x2 − 7x3 + x + 2x3 − 5x2 − 4x + 5x3
0
5p3 + 3p2 − 6
103. −5k 2 − 12k + k 3 − 4k 3 + 15k − 3k 2 − 2k
− 8k 2
6c
4w4 + 3w2
−3r4 + r3
−p2 − 4p + 8
2y 2 − 2y − 4
90. −8w3 − 6w − 2w + 6w3 + 4w
−2w3 − 4w
92. 3k + 2k 4 − 9k − 6k 4 − (−7k)
−4k 4 + k
94. 4a2 + (−10a) + 5 − a2 + 9a − 6
3a2 − a − 1
96. −5 − 5c + 7 + 12c2 + 14c − 7c2
5c2 + 9c + 2
98. 9n4 + n3 + 5n2 + 4n3 − 10n4 − 5n3 − 5n2
−n4
100. w − w2 + 14 − 3w2 − 6 − 6w + 4w2 + 7w − 8
−3x
101. 4p3 + p − 6 − (−p3 ) + (−4p) + 3p2 + 3p
−3k 3
80. c2 − (−8c) + (−c2 ) + 5c − 7c
88. 6y 2 − 6 + 2y − 4y 2 − 4y + 2
x2 + 4x + 1
93. −3x2 + 7x − 7 + 3x + 6x2
−4n
86. 3p2 − 7p + 8 + 3p − 4p2
4a2 − 2a + 7
91. −c3 − (−5c) + 8c3 − 4c − 5c3
78. 14n3 − 5n − 11n3 + n − 3n3
84. −r3 + r4 − 6r4 + 3r3 + 2r4 − r3
10k 3 − k2
89. 4m4 − 4m + (−10m4 ) − 5m + 5m4
2m3 − 2
82. 8w4 + (−4w2 ) + 6w2 − (−w2 ) − 4w4
−4y 3 + 4y 2
−3
−c3 + 1
76. 16 − 8m3 + 4m3 − 4 + 6m3 − 14
4w2 + 9
−5y 2
85. −8a + 17 + 4a2 + 6a − 10
72. 2br − 13r + 8 + 3r − 7br + 6r + 5br − 11 + 4r
3m
102. 6m2 − 5 + 2m + 8 − 7m − 4m2 − 9
2w
2m2 − 5m − 6
104. 15 − 4r2 − 6r + 9r2 − 19 + 7r + 8 − 2r + r2
6r2 − r + 4
+k
105. y 2 − y 3 − 4y 2 − y 4 + 2y 3 + 6y 4 + 3y 2
5y 4 + y 3
106. −w2 + w − (−5w2 ) − 3w3 − 7w − 4w2 + w3
−2w3 − 6w
107. 6b2 + 2b4 − 8b + 5b2 + b4 − 11b2
3b4 − 8b
108. −8m + (−m3 ) + m2 + 12m − (−4m3 ) − 4m
109. mp2 − 4m2 p + (−6m2 p) − 2mp2
−mp2 − 10m2 p
110. −3ac − (−ac2 ) + 9ac + 6ac2
111. 2r2 q − 6r2 − (−8r2 ) − 7r2 q
113. x3y 2 − (−x2 y 3 ) + (−2x2 y 3 ) − x3 y 2
115. x4 y 2 + (−9x2 y 4 ) − (−3x2 y 4 ) − 4x4 y 2
−3x4 y 2 − 6x2 y 4
7k 5 m3 + k 5
116. 2x2 y 2 + xy 3 − (−2x2 y 2 ) − 5xy 3
121. 10a2 − ac + 7c2 − 4a2 − 3ac − a2 − 4ac + c2
3x2 y
4x2 y 2 − 4xy 3
118. 17a2 b3 + 11a2 − 14a2 b3 − 4b3 − 3a2 b3
11a2 − 4b3
120. −d5 + c2 d5 − c2 + 2d5 − 4c2 d5 − d5 + c2
9p3 r3
−3c2 d5
122. −20m3 − 18mn + 2n2 + 12mn + 14m3 − 4n2
5a2 − 8ac + 8c2
123. −rx2 − 5r2 x − 5rx2 − 3r2 + r2 x + 6rx2
6hk 2 − 3h2 k
114. xy 4 + 2x2 y + (−3xy 4 ) + x2 y − (−2xy 4 )
−x2 y 3
117. 20m3 + 6k 5 m3 − 9m3 + k 5 + k 5 m3 − 11m3
119. 5p3 − 4r3 + 9p3 r3 + 4r3 − 5p3
7ac2 + 6ac
112. −5hk 2 + (−6h2 k) + 3h2 k + 11hk 2
−5r2 q + 2r2
3m3 + m2
−6m3 − 6mn − 2n2
−4r2 x − 3r2
124. 4h2 k − 4hk 2 + 2k 2 + 7hk 2 − 3h2 k − 3hk 2
ALG catalog ver. 2.6 – page 62 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
h2 k + 2k 2
DK
Topic:
Multiplying monomials. See also categories BB, BC and BE (laws of exponents),
and EC (multiplying monomials and polynomials).
Directions:
6—Simplify.
1.
5(−2x)
−10x
5.
k 3 (k 2 )
9.
(−2y)(−6y)
k5
13. −r · 15r3
−15r4
21. x2 y 2 (xy 4 )
2.
−6(3y)
6.
c4 c7
−18y
c11
10. −5x · 4x
12y 2
17. (10n2 )(6n3 )
3—Multiply.
60n5
25. (−5mp)(−m2 p)
5m3 p2
3(11a)
33a
4.
−4(−8a)
7.
y2 · y5
y7
8.
(r3 )(r6 )
11. 5p(−7p)
−20x2
14. (−a)(−12a5 )
12a6
18. (−2p)(25p4 )
−50p5
22. c4 d · c2 d3
x3 y 6
3.
c6 d 4
26. (−3x2 y 4 )(8xy 3 )
15. (−7w4 )(−3)
−27a6 b5
21w4
r9
12. 8w · 9w
72w2
16. −6(8t6 )
−48t6
19. 5y 3 (−7y 5 )
−35y 8
20. 4m2 · 11m8
44m10
23. km3 · k 3 m3
k 4 m6
24. (a7 b3 )(ab5 )
a8 b8
27. (2r4 w3 )(−11rw2 )
−24x3 y 7
29. −3a5 b2 · 9ab3
−35p2
32a
28. 7h4 k 2 · 8hk 3
56h5 k 5
−22r5 w5
30. −12k 6 m(−k 3 m)
31. c3 d2 · 18c7 d3
18c10 d5
32. (4xy 5 )(−7xy)
−28x2 y 6
12k 9 m2
33. (2xy 2 z)(−5x2 y 2 z)
34. 3w4 x2 y · 9wx3 y 5
35. (−6a5 bc7 )(9ab3 c2 )
27w5 x5 y 6
−10x3 y 4 z 2
37. x3 · x2 · x4
38. y 5 · y · y 7
x9
41. (2c3 )(−c)(13c2 )
−26c6
36. (−6m2 p3 r2 )(−6m2 pr2 )
−54a6 b4 c9
36m4 p4 r4
39. p · p4 · p3
y 13
42. (3y)(12y 2 )(y 2 )
36y 5
40. a7 · a3 · a2
p8
43. (4x2 )(−5x)(−2x)
40x4
a12
44. (−7c3 )(−2c)(−5c4 )
−70c8
45. −1(−cx2 )(−5c4 x)
46. −a(3b3 )(−8ab5 )
24a2 b8
47. −c(−9c6 d)(4d2 )
36c7 d3
−5c5 x3
48. −5(a2 y 5 )(12a6 y)
−60a8 y 6
49. (−2km3 )(4k 3 )(−8k 2 m)
64k 6 m4
50. (−9ax)(−3a2 x3 )(−4x)
51. (6w3 y 4 )(−2y 2 )(5w5 y)
−108a3 x5
53. (2xy 3 )(w5 x2 )(5wy)
10x3 y 4 w6
−60w8 y 7
54. (m5 r2 )(3m)(−2r2 p7 )
28p8 r9
55. (3xy 5 )(2yz 2 )(5x6 )
−6m6 p7 r4
52. (4r4 )(−7p7 r3 )(−pr2 )
30x7 y 6 z 2
56. (−14a2 )(−2b3 c)(−a3 bc2 )
−28a5 b4 c3
Power to a power
57. (4w)3
58. (−2q)5
64w3
61. (−y 3 )2
62. (r3 )4
y6
65. (−2d3 )5
66. (3p2 )3
−32d15
27p6
81m12 p8
70. (−8x4 y)2
73. −5x(2x)4
−80x5
74. (3y)2 (−9y)
81. (−2a3 b4 )3
15n10
−8a9 b12
85. (2a2 b)3 (5ab3 )
40a7 b6
64x8 y 2
−81y 3
78. (−2c2 )3 (−3c2 )
82. (7y 3 z 4 )2
24c8
49y 6 z 8
86. (5w4 x2 )(2wx2 )4
60. (−5a)2
81c4
63. (−x2 )3
r12
69. (3m3 p2 )4
77. (15n4 )(−n3 )2
59. (3c)4
−32q 5
64. (k 2 )5
−x6
25a2
k 10
67. (−7a4 )2
49a8
68. (2y 3 )4
71. (2r2 p4 )5
32r10 p20
72. (−5ac3 )3
75. (−5a)2 (6a)
150a3
79. (−3p4 )(−2p2 )4
83. (−3n5 p3 )4
−48p12
81n20 p12
87. (3cy 2 )3 (2c2 y)
54c5 y 7
16y 12
−125a3 c9
76. −9r(−2r2 )3
72r7
80. (−x2 )3 (4x3 )
−4x9
84. (−4m6 r)3
−64m18 r3
88. (4pr2 )(5p2 r3 )2
100p5 r8
80w8 x10
89. −p2 r3 (−p4 r)4
−p18 r7
90. −ab3 (−a2 b)2
−a5 b5
91. −km(−2km3 )5
32k 6 m16
ALG catalog ver. 2.6 – page 63 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
92. x3 (−x2 y 3 )4
x11 y 12
DK
93. (−8ab)2 (a2 b)3
94. (2xy 2 )5 (x2 y)2
64a8 b5
32x9 y 12
95. (−p2 r)3 (3pr)4
96. (−2cd)3 (−3c3 d)2
−81p10 r7
97. (−2x3 y)(y 4 z)(−4x5 z 2 )
98. (−3a4 c2 )(−6bc3 )(a2 b3 )
8x8 y 5 z 3
99. (5k 2 n)(n2 w3 )(−2kw3 )
18a6 b4 c5
101. (7u2 w)(−3u)2 (−u4 w5 )
100. (m2 r3 )(−3m3 pr4 )(7mp2 )
−10k 3 n3 w6
102. (−2p3 )3 (pr5 )(12r2 )
−63u8 w6
−72c9 d5
−21m6 p3 r7
103. (4x)2 (−2x3 y)(−5y 3 )
−96p10 r7
104. (−6a3 b)(−a2 b4 )3 (9b2 )
160x5 y 4
54a9 b15
Fractions and decimals
105. (10d2 )( 35 d5 )
106. (− 73 a5 )(−12a3 )
6d7
109. (− 54 a2 c6 )(20ac2 )
110. (24km3 )(− 18 k 3 m)
−25a3 c8
113. ( 25 y 2 )2
28a8
107. (6k 3 )(− 52 k 2 )
−15k 5
111. ( 49 x5 y 2 )(54xy 3 )
24x6 y 5
108. (− 34 m4 )(24m5 )
112. (−80rw5 )(− 25 r3 w2 )
−3k4 m4
114. (− 34 b2 )3
4 4
25 y
117. (− 32 a3 h4 )3
121. ( 14 x4 )( 23 x3 )
115. ( 21 c3 )5
6
− 27
64 b
3 8
10 d
126. ( 45 y 4 )(−20y)( 21 y 5 )
−5a6
− 34 y 8
127. (24x2 y)( 23 y)( 56 xy 2 )
120. ( 25 r2 p3 )3
130. (0.1ab4 )(60a2 b)
6a3 b5
134. (−0.3x4 wy 2 )(0.6xw5 y)
−1.2h4 r3 p7
138. (0.9r2 p3 )2
0.125h3 k 15
132. (−0.8h7 k 2 )(30hk)
141. (1.5ab2 )(0.4ab)(25b4 )
15a2 b7
143. (−0.2pr)(−6p2 )(−0.3pr2 )
−0.36p4 r3
−24h8 k 3
135. (0.6abc2 )(0.5a2 bc3 )
136. (0.2kmr3 )(1.2k 2 mr3 )
0.3a3 b2 c5
0.81r4 p6
139. (1.2x5 z)2
0.24k 3 m2 r6
1.44x10 z 2
140. (0.4a2 b4 )3
142. (10c2 )(−0.4cd2 )(−0.7d7 )
2.8c3 d9
144. (8w4 y)(0.01w2 )(−5w4 y)
−0.4w10 y 2
ALG catalog ver. 2.6 – page 64 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1 5
− 10
a
3c4 d4
131. (0.5y 4 z 3 )(−18yz 3 )
−0.18x5 w6 y 3
8 6 9
125 r p
128. (− 34 cd)(−10c2 )( 25 cd3 )
−9y 5 z 6
133. (2.4h3 rp4 )(−0.5hr2 p3 )
49 10
100 w
124. (− 35 a2 )( 61 a3 )
30x3 y 4
3w4 z 5
137. (0.5hk 5 )3
16 8 4
81 m k
3 7
y )
123. (− 52 y)( 10
−8y 10
129. (−0.2w3 z)(−15wz 4 )
7 5 2
w )
116. (− 10
1 15
32 c
119. (− 23 m2 k)4
1 14 6
25 c d
122. (− 38 d2 )(− 45 d6 )
1 7
6x
125. (−6a2 )( 12 a)( 53 a3 )
32r4 w7
118. ( 15 c7 d3 )2
9 12
− 27
8 a h
−18m9
0.064a6 b12
DL
Topic:
Dividing monomials. See also categories BD and BE (laws of exponents),
and EG (dividing monomials and polynomials).
Directions:
6—Simplify. 9—Simplify (assume no denominator equals zero).
76—Find the quotient.
1.
10x
2x
5.
−15y
45
9.
70p4 ÷ p3
13.
−y 6
−2y 6
17.
9c5
6c2
21.
k
3k 5
25.
−32y 4
16y 3
5
−
y
3
6y
12y
1
2
3.
−20a
6a
6.
24w
−3
−8w
7.
65x
5
2
c
10. 24c ÷ 12c2
70p
−
k8
15.
−30r3
−5r3
3c3
2
18.
72a3
8a
9a2
19.
1
3k 4
22.
10y 4
25y 5
2
5y
26.
36w8
−12w5
1
4x4
−3w3
30. 20n4 ÷ (−30n3 )
7
a3
33. 35a2 ÷ 5a5
−14k 12
−42k 9
−
34. 100y 6 ÷ 200y 3
k3
3
38.
41. −108h7 ÷ (−9h7 )
12
−24r5
−72r4
1
2n2
11. 44 ÷ 88n2
−6k 8
−6
−2y
10
3
13x
14.
1
2
29. −6x ÷ 24x5
37.
2.
−
2n
3
y3
2
r
3
4.
35m
−5m
8.
8
48k
13n4
−1
49.
pr3
p2 r
53.
12x2 y
4xy
57.
75c4 d6
100c3 d4
61.
−8a3 b3
30a2 b4
−3a6
3a6
50.
n4 m
nm
54.
3cd
18c4 d
3cd2
4
58.
14wx5
49w3 x
2x4
7w2
4a
15b
62.
32kn8
−6kn3
−
r2
p
3x
−
65. 6pr4 ÷ 72p2 r4
69. −2x3 z ÷ 22xz 4
1
12p
−
16.
−5c7
−75c6
c
15
2n2
22n4
1
11n2
20.
100h7
20h3
5h4
23.
−8w
32w6
−
24.
56x2
−7x9
−
27.
120r4
24r5
28.
3p10
21p4
p6
7
1
4w5
5
r
31. −33w6 ÷ 3w5
4c2
3
35. 36c4 ÷ 27c2
−18x8
−80x12
−11w
1
5x4
32. 105y 4 ÷ (−70y 4 )
40.
2
7p7
43. −16p ÷ (−56p8 )
n3
1
6c3
16n5
3
51.
x2 y
x2 y 3
55.
10rw
2rw3
59.
−12mp8
36mp10
63.
17c12 d2
17c7 d8
70. 18n3 w2 ÷ (−12nw2 )
1
y2
5
w2
−
1
3p2
c5
d6
67. 2y 5 z 4 ÷ 18y 5 z
8m7 r2
−
8
x7
36. 63k 5 ÷ 28k 9
47. −22y 4 ÷ (−22y 4 )
−1
66. 80m8 r10 ÷ 10mr8
x2
11z 3
y3
16
−36w4
−12w
−
3
2
9
4k 4
3w3
44. −18x11 ÷ (−66x6 )
3x5
11
46.
−13n4
1
6k
12. 6y 3 ÷ 96
4y 6
45.
−7
6
39.
42. −140y 10 ÷ (−35y 4 )
4—Divide.
z3
9
71. 15ab9 ÷ (−85a2 b8 )
3n2
2
ALG catalog ver. 2.6 – page 65 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−
3b
17a
1
48. −15h7 ÷ (−1)
52.
ab2
a4 b
56.
5p2 q 2
20pq 2
60.
64h5 k 2
−4hk
−16h4 k
64.
7x4 y 3
35x5 y 5
1
5xy 2
15h7
b
a3
p
4
68. 12a2 b6 ÷ 30ab7
72. −80k 4 p ÷ 50k 4 p
2a
5b
−
8
5
DL
73.
−5k 3 m8
−10k 2 m9
77.
32x4 y
16y
2x4
81.
r7 w2
−7rw9
−
k
2m
r6
7w7
85. −84cd ÷ (−7c2 d)
12
c
74.
−14dh3
−2dh12
78.
4p3
20pr4
82.
2h3 k 6
−hk 2
75.
−24x2 y
−8xy 5
p2
5r4
79.
−3m
21m3 n
−2h2 k 4
83.
20a
5a2 b3
7
h9
86. (−16rp3 ) ÷ (−2rp3 )
8
3x
y4
76.
−4w7 z 2
−6w4 z 2
1
7m2 n
80.
8r2 w3
−12w3
84.
y2
5x3 y 2
−
4
ab3
87. (−6k 2 n) ÷ (−24kn6 )
−1
−6a2 b
−1
6a2 b
92.
−2c5 d2
−2c5 d2
1
2hk
95.
−3x2 y
−x
3xy
96.
−r3 p5
r3
x
y
99.
−4a4 b
8b
−
a4
2
100.
−z
−4wz 2
1
4wz
103.
x4 y 2 z
x8 y 2 z 2
1
x4 z
104.
d2 h8 k 3
dh4 k
dh4 k 2
107.
96m2 rt5
−12mrt5
−8m
108.
−39b3 x2 y
78bxy 3
2
17acd
112.
4pr2 w2
64pr3 w3
1
16rw
6m
p
116.
−0.6b2 y 5
2.4by 5
−
120.
−5.4h
−1.8h3 k
3
h2 k
124.
0.21a3 c
−0.03a2 c2
−2p2 r2
−6
−12hk
m
3
98.
5xy
−5y 2
102.
−m3 pr2
mp2 r2
106.
2a3 bc
10abc6
a2
5c5
3mp
4
110.
81x2 y
−9x2 z 2
−
9y
z2
111.
−6ad
51a2 cd2
1
3a2 x
114.
−0.9cd4
−0.3cd
3d3
115.
1.8m2 p3
−0.3mp4
c
3y 5
118.
4.8w3 x
−1.2wx2
−
119.
0.5ad2
3.5a2 d2
12p2
r2
122.
12ab4
0.3ab5
123.
−0.04x2 y 8
0.2xy 7
97.
−m2 n
−3mn
101.
a5 b7 c2
−a9 b2 c3
105.
28xyz
7x2 yz 3
109.
−9m2 pr
−12mr
113.
0.2a3 x
0.6a5 x2
117.
−1.3c2
3.9cy 5
121.
−6p3 r2
−0.5pr4
−
b5
a4 c
4
xz 2
−
−
1
5x3
91.
90. 2p2 r2 ÷ (−1)
94.
45vw
−5
2r2
3
3a
11b2
−9vw
93.
−
88. −15a10 b7 ÷ (−55a9 b9 )
k
4n5
89. −5xy 5 ÷ (5xy 5 )
2w3
3
−
40
b
m2
p
4w2
x
ALG catalog ver. 2.6 – page 66 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−
−
1
7a
−
xy
5
1
−p5
−
b2 x
2y 2
b
4
−
7a
c
DM
Topic:
Combined methods (monomials). See also category BE (laws of exponents).
Directions:
6—Simplify.
1.
8(3w2 ) − 3w(5w)
98—Perform the indicated operation(s).
2.
9w2
2a(−5a) − (−10a2 )
0
3.
y(2y 2 ) + 5y 2 (−6y)
14m2 (2m3 ) − 3m(9m4 )
4.
−28y 3
5.
(3x3 )(2x) + 7x4
6.
13x4
8y 5 − 5y 2 (3y 3 )
−7y 5
7.
m5
2a3 (−4a3 ) − 10a6
−4k 6 (3k) + 2k 2 (9k 5 )
8.
−18a6
9.
(2n)3 − n(3n)2
10. 2d(3d)3 + (5d2 )(2d)2
−n3
11. (−2a)5 + (4a)2 (2a3 )
14. −4n4 (5n)2 + n3 (5n)3
−36p10
21. 3y(2y)2 + 4y 2 (y 3 ) − (2y)3
22. −5a(2a3 ) + (6a2 )2 − (5a)2
27. a3 (a2 )2 + (2a)4 − a(3a)3
a7 − 11a4
26. (5n3 )(3n)4 + (−2n2 )(3n2 )3
31. (2x)5 (2xy 2 ) − (6x3 y)2 + (−3x2 y)3
8x4 (−6x3 )
12x5
37.
28n6
2n(6n8 )
41.
(−x2 y 4 )(−x3 y 10 )
x9 y
−4x2
7
3n3
y 13
x4
18a(3ab4 )
36a2 b2
49.
14p2 r4
−2pr3 (7pr2 )
53.
(10p5 w8 )(6pw2 )
24p6 w7
57.
3c2 k · 14c3 k 2
7c4 k · 6c2 k
61.
14w5
(−2w2 )3
65.
(3m2 )2
(6m)2
3b2
2
−
m2
4
7
4w
−
k
c
1
r
5w3
2
405n7 − 54n8
32. (3a)2 (ab)2 − (2a2 b)2 + (5ab2 )3
28x6 y 2 − 27x6 y 3
34.
(−5y 2 )(10y 4 )
−25y 7
38.
12x10
9x(−10x4 )
42.
c4 d 3
(−c6 d)(−c5 d10 )
−
46.
−15p2 w8
9pw(7p5 )
2
y
35.
2x5
15
39.
1
c7 d8
43.
−2a3 (6a)
8a4
50.
10c7 d4
(15cd)(4c4 d2 )
54.
5ab3 (8a2 b2 )
−12a4 b
40.
16r8
−10r3 (2r5 )
44.
−g 4 h11
(−g 2 h4 )(−gh8 )
5x4
2y
48.
22n10 r4
3n4 (33nr9 )
2n5
9r5
3
40m3
52.
−24u4 w2
u3 w(6uw2 )
−
8k
9h
56.
(2xy)(−6x2 y)
10xy 4
60.
8rx3 · 7r2 x
28x2 · 4r3 x4
3
2
64.
2a5 (−15a2 )
(−10a2 )2
2
y2
68.
(4d3 )3
(2d2 )5
3k 7
1
10k 3
(−6k 4 )(−5k 6 )
(−mx5 )(−m4 x3 )
−m9 x2
51.
6mn
3
(4m n)(20m)
−
10b4
3a
55.
−2h7 k 2 (20hk 3 )
−45h9 k 4
58.
(−12yz)(6yz 4 )
(−y 3 )(−54yz 5 )
−
59.
(25a3 b2 )(−3ab7 )
(5a2 b4 )(15a2 b5 )
62.
(5k 3 )2
25k 6
63.
(−3r2 )3
−3r(6r5 )
66.
(−5c3 )3
(−5c3 )4
67.
(2y)4
(−2y 2 )3
1
−
1
5c3
ALG catalog ver. 2.6 – page 67 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
c2
4
4
5
−
−
g
h
x6
m4
c2
6d
4
3y 2
5a4 b2 + 125a3 b6
(3c4 )(3c3 )
36c5
3
2
−x3 y(10x2 )
4xy 2
5w7
21p4
u8 w6
36.
−
47.
−
−9w7 + 25w6
30. (4u3 w2 )(4uw2 ) + (u2 )4 (w3 )2 − (2uw)4
−200p2 r3
−
45.
−32k 6 + 4k 5
28. −w(2w2 )3 + (5w3 )2 − w(w3 )2
29. (5p)2 (−2r)3 + (4p2 r)3 − (8p2 )(8p4 r3 )
12x10 y 7
26a4 − 25a2
24. 2k 3 (−2k)3 + 4k(k 4 ) − (4k 3 )2
−27x3
3r9 − 32r7
15k 4 m4
20. 3xy(−2x3 y 2 )3 + 3x6 y 5 (2x2 y)2
14c7 d5
25. (3r4 )(−r)5 − (4r)(2r2 )3
33.
176x10
18. (−3k 2 m2 )2 + (3km2 )2k 3 m2
4y 5 + 4y 3
23. (4x2 )(5x) − (3x)(5x)2 + (3x)3
16. (2x2 )5 + 4x4 (6x3 )2
−9y 9
6a4 b4
19. (2cd2 )2 (5c5 d) − (6cd2 )(c2 d)3
−107r4
15. −18y(5y 8 ) + y(3y 2 )4
25n6
17. (2a3 b)(−5ab3 ) + (2ab)4
12. −7r2 (−3r)2 − 2r(−4r)3
0
74d4
13. (−p2 )3 (4p4 ) − (2p2 )5
6k 7
−
−
−1
2
d
4
w
−
6x2
5y 2
1
2x2
−
3a3
10
DM
69.
(−2c3 d)3 (−3cd5 )
−6c2 d2 (4cd)2
−
73.
70.
(4x4 y)(−5xy 5 )2
10x6 y 3 (−2xy)3
−
5y 5
4x3
m2 rx2 (mrx)5
(mr3 x)3 (mx4 )
71.
m3
r3
72.
c6 d4
4
k 4 g 2 p(k 2 gp2 )3
(k 2 g 3 p3 )2 (k 4 gp3 )
k2
g 2 p2
(−kn5 )(3k 2 n)2
(−6k 3 n2 )2 (−2kn3 )
74.
(2xy)3 (−5x2 y)2
(35xy 5 )(10x2 y 8 )
78.
(−6bc4 )(2b2 c2 )5
(−12b2 c7 )2
4x4
7y 8
75.
(8c2 d)2 (2cd3 )
(3d3 )(4cd2 )3
2c2
3d4
76.
(9a4 b3 )(12ab7 )
(−3ab)3 (2a2 b)2
4b7
3
79.
(2pr)4 (9p3 r6 )
(−6pr5 )2
4p5
80.
(−10h4 k 3 )3
(5hk 4 )2 (−12h2 k)
−
b5
a2
1
8k 2
77.
(5xy 3 )2 (4x2 )2
(30x5 y 2 )2
4y 2
9x4
−
10h8
3
81.
12h2
21h6
14h5
6h2
4
3h
85. (3m2 p)3 ÷ 18m5 p3
82.
93.
97.
4
x2 y 4
6h2
−k
2 k5
12h4
−20x2 y 3 12xy 2
·
24x2
15y 3
−
101.
−
−1
· (12xy)2
(6x2 y 3 )2
−
9y 8
−10y 3
−25y
45y 6
86. −15pr8 ÷ (−5pr4 )2
3m
2
89.
94.
98.
−8c 121c7
·
33c3
6c2
83.
3a2 b2
2
44c2
9
87. (4ab)3 ÷ (4ab)4
1
4ab
84.
3n
w2
2 4k 3
3h4
91.
1
10bc3
w2
3n
95.
8x6
3h2
−
9h4
8
1
(3h2 k)3
w2
3n
3
20ab3
−16a
·
−12a2 b2 20b3
(−4y 3 )2 −6y 8 z 7
·
8y 4 z 2
(2yz 2 )3
−
3a6
88. (−3x2 y)5 ÷ (3x2 y)3
4
3b2
99.
· (−10bc3 )
−3h2
4x2
−1
3
−12r2 w2 −16r3 w
10rw2
12rw4
92.
5x8 8y 3
·
8y 3 5x8
96.
103.
3y 7
2z
ALG catalog ver. 2.6 – page 68 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−30k 3 m3 (−3km)3
·
(9km2 )2
5k 6 m8
2
k 2 m6
2a3
c2
4 1
5c2
12a5
2
25a2
9c4
100.
8r3
5w3
102.
a5 54a3
·
18a
a
−9x4 y 2
2xy 2
3
(3ab)3 (−2a3 b2 )2
·
6a2 b4
12a5 b
−
3
5p
90. (6hk 3 )2 ·
3k 3
1
2
−
104.
15hk
12k 3
16hk 2
−10hk 2
2h
k2
−3u2 (u3 w2 ) (−2u2 )3
·
(6uw2 )2
u4 w 2
2u5
3w4
DN
Topic:
Applications of monomials. See also category EK (polynomials).
Directions:
0—(No explicit directions.)
1.
The side of a square is
1
6 w.
48—Write each answer as a monomial in simplest form.
What is the perimeter?
2.
The side of a square is 7ab. What is the perimeter?
28ab
2
3w
3.
Find the area of a square whose side is 5cd.
5.
2
3 x.
4.
Find the area of a square whose side is
The dimensions of a rectangle are 5y and 3y. What
is the perimeter? 16y
6.
The dimensions of a rectangle are x and
is the perimeter? 7 x
7.
Find the area of a rectangle whose dimensions are
5xy and 35 y. 3xy2
8.
9.
The side of a square is 10xy. What is the area and
perimeter? 100x2 y2 ; 40xy
10. The side of a square is
perimeter? 9 k2 ; 3k
11. A rectangle has dimensions 3x and
area and perimeter? 3 x2 ; 7x
1
2 x.
25c2 d2
What is the
2
13. The side of a cube is 2x. What is the volume?
8x3
3
4 x.
4 2
9x
What
2
Find the area of a rectangle whose dimensions are
6ab and 4bc. 24ab2 c
3
4 k.
What is the area and
16
12. A rectangle has dimensions 9a and a. What is the
area and perimeter? 9a2 ; 20a
14. The side of a cube is 5y. What is the volume?
125y 3
15. The side of a cube is 4a. What is the surface area?
16. The side of a cube is 3n. What is the surface area?
96a2
54n3
17. The perimeter of a rectangle is 20pr and the width
is 3pr. What is the length? 7pr
18. The perimeter of a rectangle is 7h and the length
is 3h. What is the width? 1 h
19. The area of a rectangle is 21h2 k and the width
is 3hk. What is the length? 7h
20. The area of a rectangle is 12ab and the length is 6ab.
What is the width? 2
21. The perimeter of an isosceles triangle is 14a and the
base is 5a. What is the length of each side? 9 a
22. The perimeter of an isosceles triangle is 7xy and the
base is xy. What is the length of each side? 3xy
23. The area of a triangle is 20x2 and the height is 5x.
What is the base? 2x
24. The area of a triangle is 30mn2 and the base is 3n.
What is the height? 5mn
25. The length of a rectangle is four times the width.
What is the perimeter? 10w
26. The width of a rectangle is half of the length. What
is the perimeter? 3`
27. The length of a rectangle is three times the width.
What is the area? 3w2
28. The width of a rectangle is one-fourth of the length.
What is the area? 1 `2
29. The base of triangle is two-thirds of the height.
What is the area? 1 h2
30. The base of a triangle is two times the height. What
is the area? h2
31. The height of a triangle is six times the base. What
is the area? 3b2
32. The height of a triangle is four-fifths of the base.
What is the area? 2 b2
2
3
2
4
ALG catalog ver. 2.6 – page 69 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
5
EA
Topic:
Adding polynomials. See also categories DI and DJ (adding variable terms, combining like terms).
Directions:
6—Simplify.
1—Add.
1.
14k + (7m − 9k)
3.
(−4b + 9h) + (−14h)
5.
(4x − 5) + (9x + 14)
7.
(12 − 3a) + (−7 − 4a)
9.
(−9a2 − 10b) + (a2 + 4b)
5k + 7m
−4b − 5h
13x + 9
5 − 7a
13. (5c + 6d) + (2c + 8d)
27. (−y 2 + y − 1) + (6y − 7)
(−7y − 2) + (2y + 17)
−5y + 15
8.
(−1 + 5r) + (−5 − 6r)
−6 − r
9k 4 − 19m
3x3 − 4y
5x − 2y
−3a + 8b
18. (7r − 3rp2 ) + (−6r + 9rp2 )
−9xy − z
r + 6p2 r
20. (19h2 − 4hk) + (2h2 − 12hk)
−3uw − 19u
21. [4a + (−11b)] + (−10a + 15b)
25. (2r − 5) + (r2 + 3r − 6)
6.
16w + 7z
16. (a + 11b) + (−4a − 3b)
19. (−17uw − 4u) + (14uw − 15u)
23. [8a + (−x)] + (−7a − 3x)
(8w + 7z) + 8w
14. (4x + 8y) + (x − 10y)
5m − 3p
17. (8xy − 3z) + (−17xy + 2z)
4.
7a − c
12. (x3 + 3y) + (2x3 − 7y)
−7c3 + 11d2
7c + 14d
15. (2m − 7p) + (3m + 4p)
(12a − c) + (−5a)
10. (11k 4 − 4m) + (−2k 4 − 15m)
−8a2 − 6b
11. (−5c3 + 2d2 ) + (−2c3 + 9d2 )
2.
22. (2c − 10d) + [4c + (−d)]
−6a + 4b
21h2 − 16hk
6c − 11d
a − 4x
24. (−14c + 5y) + [9c + (−5y)]
−5c
r2 + 5r − 11
26. (x3 − 7x2 + 9x) + (4x2 − x)
x3 − 3x2 + 8x
28. (7w2 z + 15z) + (w2 z − 5wz − 16z)
−y 2 + 7y − 8
29. (3x2 + 7x + 9) + (3x2 − 7x)
31. (7cd − 4c) + (−4cd + 4c − 19)
4km + 10k + 15m
21r3 − 4r2 − 4r
37. (−2m + 3p − s) + (4m − 7p + 5s)
39. (5a − 2b + 7c) + (3a + b − c)
2m − 4p + 4s
43. (y 4 − y 2 − 2) + (−3y 4 + y 2 − 5)
−5x2 − 3x
5x − 3y
47. (2pr − r + 6p) + (−8pr + 3r + 2p)
34. (−7a2 + 9a − 16) + (a + 20)
−7a2 + 10a + 4
36. (2by − 5b + 19y) + (by − 7y)
3by − 5b + 12y
38. (6k − r + 9) + (−8k + r − 10)
−2k − 1
7w − 10x − 3
42. (7c3 − 2c2 + c) + (8c3 + 5c2 + 7c)
−6pr + 2r + 8p
−8c − cd − d
48. (2a + 4ay − 8y) + (−2a − 6ay + 8y)
−2ay
−3a2 − a + 10
50. (10x2 − 6x − x3 ) + (9x + x3 + 5x2 )
51. (−3c3 − 2 + 4c) + (5c + 3c3 − 7)
9c − 9
52. (8 + 2n4 − n2 ) + (−n4 − 1 + 6n2 )
a2 x2 − a + x
−9m2 − 5m
46. (−2c + cd − 5d) + (−6c − 2cd + 4d)
49. (5a − a2 + 7) + (−2a2 + 3 − 6a)
53. (−2a2 x2 + 3a − x) + (−4a + 2x + 3a2 x2 )
15c3 + 3c2 + 8c
44. (−2m3 − 7m2 + m) + (2m3 − 2m2 − 6m)
−2y 4 − 7
45. (−xy + 2x − 4y) + (xy + 3x + y)
12a
40. (w − 6x + 12) + (6w − 4x − 15)
8a − b + 6c
41. (−3x2 − 6x + 5) + (−2x2 + 3x − 5)
10y 3 − y 2
32. (−9a2 + 12a − 10) + (9a2 + 10)
3cd − 19
33. (−km + 6k) + (5km + 4k + 15m)
35. (r2 + 5r) + (21r3 − 5r2 − 9r)
30. (6y 3 − 2y) + (4y 3 − y 2 + 2y)
6x2 + 9
8w2 z − 5wz − z
15x2 + 3x
n4 + 5n2 + 7
54. (5u2 w − 8w + uw2 ) + (4w − 4uw2 − 6u2 w)
−u2 w − 4w − 3uw2
55. (5xy − 7y + 9x) + (3x + 4y − 4xy)
57. (2c + 9cd − d) + (5c2 − 8c + 4d)
xy − 3y + 12x
5c2 − 6c + 9cd + 3d
56. (−10n + 15p − 3np) + (13np − 6p + 10n)
58. (3ab − 3b + 8) + (−8a − 6ab + 4b)
ALG catalog ver. 2.6 – page 70 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
9p + 10np
−3ab + b − 8a + 8
EA
59. (−4p2 − 2p + 7) + (−p3 − 2p2 + p)
61. (y 3 − 2y − 4) + (−y 4 − y 3 + 3y 2 )
−p3 − 6p2 − p + 7
−y 4 + 3y 2 − 2y − 4
63. (a2 b − 9a + 5b) + (−2ab + 3a2 b − 5b)
4a2 b − 2ab − 9a
65. (−5by 2 + 3b2 y − 5y) + (−2by 2 − 5by + 7y)
60. (7r3 − 8r + 2) + (−10r3 − 3r2 − 2r)
−3r3 − 3r2 − 10r + 2
62. (−k 4 − 6k 3 + 9k) + (k 5 + k 4 + 2k 3 )
k 5 − 4k 3 + 9k
64. (4by + 7b2 − 10y 2 ) + (3b − 3by − 8b2 )
by − b2 + 3b − 10y 2
66. (−3w2 + w2 y − 4y 2 ) + (10w2 − 3wy 2 + 4y 2 )
−7by 2 + 3b2 y − 5by + 2y
7w2 + w2 y − 3wy 2
67. (4hk − hk 2 − 3k 2 ) + (7hk 2 + 2h2 k + k 2 )
68. (2a2 b + 4ab − ab2 ) + (−5a2 − 2a2 b − 3ab2 )
6hk2 + 2h2 k + 4hk − 2k 2
−5a2 + 4ab − 4ab2
69. (7a3 − 5a2 + 14a − 13) + (−2a + 15)
7a3 − 5a2 + 12a + 2
70. (8c2 − 15c) + (−10c3 + 9c2 + 15c − 17)
−10c3 + 17c2 − 17
71. (13p4 + 4p3 + 6p2 − 2p) + (−9p4 − 8p2 )
4p4
+ 4p3
− 2p2
72. (7w + 14y) + (−2w + 3x − 15y − 14z)
5w − y + 3x − 14z
− 2p
73. (x3 − 5x + 2) + (x4 − 2x3 + 5x − 4)
74. (2c2 + 4c − 10cd) + (−6c2 − 4c + 8cd − d2 )
x4 − x3 − 2
−4c2 − 2cd − d2
75. (−5a3 + a2 − 2a + 7) + (3a2 + 4a − 12)
76. (k 2 − 4km + 3m − 8m2 ) + (3km − 3m + 6m2 )
−5a3 + 4a2 + 2a − 5
k 2 − km − 2m2
77. (c3 − 5c2 d + d3 − cd2 ) + (−2d3 + 5c2 d + cd2 )
c3 − d3
78. (a2 b + 8ab2 − 4b2 − 7) + (8b2 + 3ab2 + 11)
a2 b + 11ab2 + 4b2 + 4
79. (8x + y − 9y 2 ) + (5xy 2 − 2y 2 + x − y)
81. (2x2 + 4) + (3x2 + 7x) + (x + 9)
9x − 11y 2 + 5xy 2
5x2 + 8x + 13
83. (17x − 12y) + (−23y − 15) + (10x − 19)
27x − 35y − 34
85. (x + 7y + 21z) + (7x − 3z) + (12y + 4z − 10)
80. (−a4 + 2a3 − 7a) + (−a4 − 2a3 + 8a − 4)
82. (5a − 9) + (−8b − 1) + (−7a + 4b)
2w3
− 2w2
13x3 − 11x2 + 37x − 75
88. (12x3 + 13) + (8x2 − 13x + 19) + (−8x2 − 32x)
12x3 − 45x + 32
90. (5p2 − p + r) + (−7p2 − 3p − 4r) + (5p2 − p + 3r)
−14a + 7b − 25c
91. (x2 + 7x + 9y) + (4x2 + 3xy + 7y) + (7x2 + xy + 2x)
12x2
3p2 − 5p
92. (cd − 4c + 6) + (4cd − 9d − 9) + (5c + 14d + 3)
5cd + c + 5d
+ 9x + 16y + 4xy
Fractions and decimals
93. 34 y 2 + 23 y − 1 + 54 y 2 − y + 45 2y2 − 13 y − 15
95. 12 c + 35 d + 2cd + − 32 c − 35 d − 32 cd −c − 12 cd
97. 12 m2 + 53 m − 14 + 12 m2 − 13 m + 12 m2 + 34 m + 14
3 2
1
3
99. 56 r2 + 3r − 13 + 23 r2 − 32 r + 16
2r + 2r − 6
101. (−3.4x − 0.6y + 2.8z) + (2.6x + 0.8y − 4.2z)
+ 58 p2 − 2p + 13 p2 − 6p + 1
96. 45 xy + x + 16 y + 15 xy − 23 x − 76 y xy + 13 x − y
1
1
98. 25 x − 12 xy + 23 y + 10
x + 32 xy − 12 y
2 x + xy + y
7 2
1
100. 34 w2 + 14 w − 12 + 18 w2 − 34 w − 32
8w − 2w − 2
94.
2
3
104. (4y 2 − 0.4y + 1.2) + (−2.9y 2 − 0.7y − 1.3)
1.1y 2 − 1.1y − 0.1
106. (−5w4 + 2.3w2 − 2.7) + (0.3w4 − 3.9w2 + 4)
c2 + 3.8c − 4
107. (0.6a3 + 0.1a2 − 0.4a) + (0.2a3 + 2.9a2 − 1.4a)
− 4p +
−2.4pr + 2.8p2 r − 7.5r2
1.8ab + 0.2a − 0.1b
105. (1.8c2 + 4.9c − 3.6) + (−0.8c2 − 1.1c − 0.4)
3 2
8p
102. (0.3pr − 2p2 r − 3.5r2 ) + (−2.7pr + 4.8p2 r − 4r2 )
−0.8x + 0.2y − 1.4z
103. (−2ab + 0.6a − 0.15b) + (3.8ab − 0.4a + 0.05b)
5c3 − 7c2 − 8c
86. (9x − 33) + (−11x2 + 19x − 23) + (13x3 + 9x − 19)
− 4w − 37
89. (9a + b − 12c) + (a − 15c + b) + (5b − 24a + 2c)
−2a − 4b − 10
84. (2c3 − c) + (−3c2 − 7c) + (3c3 − 4c2 )
8x + 19y + 22z − 10
87. (2w3 − 5w − 15) + (−6w2 + w − 15) + (4w2 − 7)
−2a4 + a − 4
−4.7w4 − 1.6w2 + 1.3
108. (−7.2x + 1.6xy + 2.9y) + (−0.8x − 4.3xy − 0.4y)
0.8a3 + 3a2 − 1.8a
ALG catalog ver. 2.6 – page 71 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−8x − 2.7xy + 2.5y
EB
Topic:
Subtracting polynomials.
Directions:
6—Simplify.
1.
−(4c − 12)
5.
−(−n2 − n + 9)
2—Subtract.
−4c + 12
2.
−(−5r2 − 2r)
6.
−(ab − 4a − 5b)
x − (x2 − 3x)
−x2 + 4x
−(6p2 − 7)
7.
−(−c2 − cd + d2 )
−6p2 + 7
c2 + cd − d2
10. y 2 − (−3y 2 − y)
11. ab − (2a + 3ab)
4y 2
13. (a − c) − (c − a)
3.
−ab + 4a + 5b
n2 + n − 9
9.
5r2 + 2r
4.
−(−2xy + y)
8.
−(5y 2 + y − 7)
−5y 2 − y + 7
12. 2m − (−m2 − 5m)
−2ab − 2a
+y
14. (x + 7) − (7 − x)
2x
15. (p − 8) − (p + 8)
2xy − y
m2 + 7m
16. (4 + w) − (4 − w)
−16
2w
2a − 2c
17. (9n + 7) − (4n − 5)
18. (6x − 17) − (−7x − 2)
19. (−8y + 7) − (3y − 2)
13x − 15
5n + 12
21. (−3x − 5y) − (8x + 2y)
22. (4a + b) − (6a − b)
−11x − 7y
23. (−10m + 5p) − (8m − 3p)
27. (4cd − 8c) − (−3cd − 8c)
28. (2r2 − 6) − (5r2 − 10)
7cd
31. (−5u + 4w) − (−5u − 4w)
−2a + 2b
3c + 7d
26. (−2y 4 + 18) − (−2y 4 + 17)
22p3 − 2p2
29. (−12rx − 8x) − (12rx − 8x)
−7u − 3
24. (c + 11d) − (−2c + 4d)
−18m + 8p
25. (12p3 − p2 ) − (−10p3 + p2 )
20. (−2u − 2) − (5u + 1)
−11y + 9
−3r2 + 4
30. (7x2 + y 2 ) − (7x2 − y 2 )
−24rx
32. (6a2 − 7a) − (−6a2 − 7a)
8w
33. (−a2 − 2a) − (−3a − 5)
−a2 + a + 5
34. (4r − 7) − (r2 − 8)
35. (x3 + 3x2 ) − (−x2 + x)
x3 + 4x2 − x
36. (−y 2 + y) − (−2y 3 + y 2 )
37. (7cd − 6d) − (2cd − 16c − 4d)
−a2 − b
41. (−x2 y − 5xy − 16y) − (7x2 y + 15y)
2y 2
12a2
−r2 + 4r + 1
2y 3 − 2y 2 + y
38. (−6pw + 8p + 14) − (5p − 21)
16c + 5cd − 2d
39. (−a2 b + a2 − b) − (−a2 b + 2a2 )
1
−8x2 y − 5xy − 31y
−6pw + 3p + 35
40. (2u2 + 3u − 6) − (−8u + 10)
2u2 + 11u − 16
42. (6a4 − 7a2 ) − (−a3 + a2 − a)
6a4 + a3 − 8a2 + a
43. (4km2 + 3m2 − 6) − (2km2 − 15)
2kmn2 + 3m2 + 9
44. (−4xy − x2 ) − (x2 y − 7x2 + 9xy)
6x2 − x2 y − 13xy
45. (3y 2 − 7y + 4) − (−2y 2 + 8y − 6)
5y 2 − 15y + 10
46. (−7 + 2c − c2 ) − (−5 − 6c + 2c2 )
−3c2 + 8c − 2
47. (2a2 − 5a − 4) − (3a2 − a + 1)
−a2 − 4a − 5
48. (−6x2 − 3x + 1) − (4x2 − 5x − 1)
49. (−5a + 3b − c) − (3a + b − 4c)
−8a + 2b + 3c
50. (m − 12mp − 7p) − (−2m − mp + p)
51. (−10w − 5z + 9) − (11w + 8z − 7)
−21w − 13z + 16
53. (−13n5 + 4n4 − n3 ) − (−2n5 − n4 + 2n3 )
−10x2 + 2x + 2
3m − 11mp − 8p
52. (−3rt + 6r − 7) − (−5rt − 3r + 8)
2rt + 9r − 15
54. (4k 4 − k 3 + 6) − (−3k 4 + 2k 3 − 7)
7k 4 − 3k 3 + 13
−11n5 + 5n4 − 3n3
55. (−c5 + 3c3 − 7c) − (c5 − 3c3 + 8c)
−2c5 + 6c3 − 15c
57. (−5x3 − 4x2 y 2 + 11y 3 ) − (10x3 + 3x2 y 2 − 3y 3 )
56. (2b2 y + by + by 2 ) − (b2 y − 2by − by 2 )
58. (8a4 − 12a2 b2 − 7b4 ) − (a4 − 8a2 b2 − 11b4 )
−15c3 − 7x2 y 2 + 14y 3
7a4 − 4a2 b2 + 4b4
59. (−3r2 t + 8rt2 − 9r2 t2 ) − (−11r2 t − 7rt2 + 5r2 t2 )
8r2 t + 15rt2
60. (−14k 2 m + 9km − km2 ) − (16k 2 m − 4km + km2 )
− 14r2 t2
61. (6xy − 4x2 + 10y) − (−3x2 − 10y + 6xy)
b2 y + 3by + 2by2
−30k 2 m + 13km − 2km2
−x2 + 20y
62. (−9a2 + 8a3 − 11a) − (9a3 − 12a + a2 )
ALG catalog ver. 2.6 – page 72 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−a3 − 10a2 + a
EB
63. (−3d2 − 4cd + 8) − (−10 − 4cd + 5d2 )
64. (−13w4 + w3 + 11w2 ) − (2w3 + w2 + 4w4 )
−8d2 + 18
−17w4 − w3 + 10w2
65. (p3 + 4p2 − 6p) − (−2p2 + 4p − 9)
66. (−y 2 + 5xy − 4x) − (6y 2 + 3y − 2x)
p3 + 6p2 − 10p + 9
−7y 2 + 5xy − 2x − 3y
67. (2cd + 6c + 13) − (3cd − 2d + 24)
68. (−ab + bc − 2a) − (2ab − 2b + 5a)
−cd + 6c + 2d − 11
69. (−3xy 2 + 3xy − y 3 ) − (−6xy 2 − y − 2y 3 )
−3ab + bc − 7a + 2b
70. (2m2 x − 4mx + 4m2 ) − (−2mx2 + m2 x − 8m2 )
3xy 2 + 3xy + y 3 + y
m2 x − 4mx + 12m2 + 2mx2
71. (−a3 − 2b3 + ab2 ) − (−a3 + b3 − a2 b)
72. (w2 − 2w2 x − wx2 ) − (−3w2 x + w3 − 4x3 )
−3b3 + a2 b + ab2
−w3 + w2 + w2 x − wx2 + 4x3
73. (−8ab − 9c + 3cd) − (4bc + 9cd − 14ab)
74. (13wx − 3wz + 17yz) − (9wx − xy − 5yz)
6ab − 9c − 6cd − 4bc
4wx − 3wz + 22yz + xy
75. (3x2 + 9xy + 7xy 2 ) − (5x2 y 2 + 4xy 2 + 3y 2 )
3x2
+ 9xy
+ 3xy 2
− 5x2 y 2
76. (2m2 + 6m2 r − 5s2 ) − (m2 r − 5mr + 7rs2 )
− 3y 2
2m2 + 5m2 r − 5s2 + 5mr − 7rs2
77. (6x2 + 11x) − (4x3 − 7x2 − 11x + 2)
78. (−c3 + 3c2 − 8c + 14) − (c3 − 10)
−2c3 + 3c2 − 8c + 24
−4x3 + 13x2 + 22x − 2
80. (−2x2 − 3x − xy − y) − (2x − xy)
79. (7a + 14x) − (−2a + 3w − 15x − 14y)
−2x2 − 5x − y
9a + 29x − 3w + 14y
81. (−rp − 15r2 p + 18r − 14) − (−16 − rp − 10r)
82. (wx2 + 5x2 + 6w) − (−x2 + 5wx3 + 7wx2 + 9w)
−15r2 p + 28r + 2
−6wx2 + 6x2 − 3w − 5wx3
83. (−4x − 5x2 y 2 − 7xy + 12y 3 ) − (8x − x2 y 2 + 5xy)
−12x − 4x2 y 2
− 12xy
84. (2a3 + 4a3 b − 19b3 ) − (11a3 + 17a3 b − 14ab − 9b3 )
+ 12y 3
−9a3 − 13a3b − 10b3 + 14ab
85. (−y 3 − 4y 2 + 9y − 14) − (7y 3 + 4y 2 − 12y − 9)
86. (28a3 + 23a2 − 34a − 14) − (23a3 − 15a2 + 25a − 27)
−8y 3 − 8y 2 + 21y − 5
5a3 + 38a2 − 59a + 13
87. (−7c3 + c2 + 7c + 9) − (−3c3 + 5c2 + 11c + 7)
88. (8r3 − 3r2 + 8r − 13) − (6r3 − 8r2 − 15r + 4)
−4c3 − 4c2 − 4c − 2
2r3 + 5r2 + 23r − 17
89. (11x − 19x3 + 10x2 − 29) − (−17 + 11x − 19x2 + 12x3 )
−31x3 + 29x2 − 12
90. (−4ab + 16bc + 7cd − 25ad) − (cd − 17bc − 18ad − 12ab)
91. (9m + 6m3 + 5m2 + 14) − (8 + 7m + 11m2 + 8m3 )
−2m3 − 6m2 + 2m + 6
92. (10k 2 − 7k 4 − 6k 3 + 18k) − (−7k + 9k 2 − 18k 3 − 4k 4 )
93. (−13hk − 15k) − (10hk + 19k) − (−17hk − 12k)
94. (23x2 − 27) − (−31x2 − 41) − (22x2 + 14)
95. (3wy 2 + 7y 2 ) − (w2 − 9y 2 ) − (−2wy 2 + 4y 2 )
96. (4ab − 5ab2 ) − (2a2 b + ab2 ) − (5ab − 7a2 b)
8ab + 33bc + 6cd − 7ad
−3k 4 + 12k 3 + k 2 + 25k
−6hk − 12k
32x2
5wy 2 − w2 + 12y 2
−ab − 6ab2 + 5a2 b
97. (−11a2 + 9a − 23) − (9a − 33) − (13a2 + 9a − 19)
−24a2 − 9a + 29
98. (8rp − 13r + 19p) − (−8rp + r − 32p) − (−12r + 13p)
99. (−7z 3 − 3z 2 ) − (z 3 + 7z 2 + 11z) − (7z 3 + 4z 2 + 10z)
100. (4x2 − 7x) − (2x2 − 5x + 15) − (−6x2 + x − 15)
16rp − 2r + 38p
−15z 3 − 14z 2 − 21z
8x2 − 3x
ALG catalog ver. 2.6 – page 73 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
EB
101. (c2 − 3cd + 4cd2 + d3 ) − (7cd + c2 − 10d3 + cd2 )
102. (−h4 − 4h3 − 7h5 + h) − (−3h − 5h4 − 4h2 + h5 )
−10cd + 3cd2 + 11d3
−8h5 + 4h4 − 4h3 + 4h2 + 4h
103. (5a3 − a2 − a + 1) − (−5a3 − a2 − a + 1)
10a3
104. (6y 3 − 5y + 2y 2 − 1) − (3 − 7y − y 2 − y 3 )
7y 3 + 3y 2 + 2y − 4
Fractions and decimals
105. 23 x2 − 34 x + 1 −
107.
1
3
2m + 7r
2m + 67 r +
2 2
3a
−x−
4
5
−x2 + 14 x +
106.
9
5
+ 2mr − − 32 m − 37 r − 53 mr
+ 83 a −
1
3
3
2
− − 35 c2 − 13 c +
−
2 2
3a
− 18 a +
1
6
1
2
1 2
2c
+ 2c − 2
1
1
2a − 2
113. (0.4r2 + 2.3r − 6.5) − (−0.7r2 − 1.6r − 0.5)
1.1r2
2
3
−
− 3y −
1
3
1
2
1
5 ab − 3 a − 6 b
1 2
2y
+y+1
−ab + 53 a + b
5
6w
− 12 wx + 32 x − − 16 w − 32 wx + 12 x
112.
3 2
4y
+ 14 y +
3
5
−
1 2
8y
− 34 y −
2
5
5 2
8y
w + wx + x
+y+1
−2.03n4 − 8.1n2 + 16.7
116. (−5.2p + 1.7pw + 2.9w) − (−0.05p − 4.3pw − 0.1w)
−5.15p + 6pw + 3w
+x
117. (−2.6a − 0.2b + 1.8c) − (−2.6a + 0.8b − 3.2c)
1 2
4y
114. (−2n4 − 4.1n2 + 8.7) − (0.03n4 + 4n2 − 8)
115. (0.1x3 + 0.2x2 − 0.6x) − (0.2x3 + 2.8x2 − 1.6x)
− 2.6x2
− 2y +
110.
+ 3.9r − 6
−0.1x3
3 2
4y
108. − 45 ab + a + 65 b −
11
3 mr
1 2
109. − 10
c + 53 c −
111.
5 2
3x
−b + 5c
118. (0.5rs − 8r2 s − 2.5s2 ) − (−3.5rs + 6.8r2 s − 5s2 )
4rs − 14.8r2 s + 2.5s2
119. (−7yz + 0.04y − 0.03z) − (4.8yz − 0.1y + 0.05z)
120. (2c2 − 0.3c + 1.2) − (2.9c2 − 0.7c + 1.3)
−11.8yz + 0.14y − 0.08z
ALG catalog ver. 2.6 – page 74 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−0.9c2 + 0.4c − 0.1
EC
Topic:
Multiplying monomials and polynomials.
Directions:
3—Multiply.
1.
x(x + 4)
5.
(−2)(5 − 11a)
x2 + 4x
22a − 10
2.
(w2 − w)(w)
w3 − w2
3.
(4k 2 + 1)(3)
12k2 + 3
4.
7(2y − 5)
6.
−14(2c − 1)
−28c + 14
7.
−y(2y + 6)
−2y 2 − 6y
8.
(−f )(−f 2 + 3f )
14y − 35
f 3 − 3f 2
9.
(2g + 16)(3g)
10. 8z(−z 2 − 3z)
6g 2 + 48g
−8z 3
11. 10c(4c2 − 7)
12. (3d3 + d2 )(6d)
40c3 − 70c
− 24z 2
18d4 + 6d3
13. 5m3 (3m2 − 4m)
14. (f 2 )(3f 3 + 5)
15. (g 3 )(−2g + 8)
15m5 − 20m4
3f 5 + 5f 2
−2g 4 + 8g 3
−20n3 − 14n2
19. (3b2 + b)(−9b4 )
20. −v 4 (−8v 2 + 5v)
17. −h2 (−7h2 − 1)
7h4
18. (4j 3 − 7j 2 )(−3j 3 )
+ h2
−12j 6
21. (5ab)(−3a + 12b2 )
22. 2xy 2 z(9x2 y 3 − z)
−15a2 b + 60ab3
26. −a2 b(−a2 b − ab3 )
+ x2 y 4
a4 b2
29. 2m2 n3 (m + 2mn2 )
37. −r(−r2 + 3r − 1)
− 3r2
20p3 r3
a4
− 2a3
14y 3 − 7y 2 − 70y
45. −d3 (d4 + 5d3 − 8d)
46. (a2 )(2a2 + 4a + 7)
+ 8d4
2a4
49. (−k 2 − k + 6)(5k 2 )
+ 4a3
32. 11a2 b(3a3 b2 + 11)
42x5 y 2 − 54x4 y 4
33a5 b3 + 121a2 b
2x3
+ x2
12p4 + 15p3 − 3p
47. −b4 (−6b2 − b + 1)
6b6
+ b5
48. (−3t3 + t2 − 7)(t3 )
− b4
−3t6 + t5 − 7t3
51. 2p3 (4p3 − 3p2 + 5)
52. −5w4 (w4 − w2 + 2w)
−7n5 + 21n4 − 7n3
8p6 − 6p5 + 10p3
−5w8 + 5w6 − 10w5
54. −rst(−3r − ds2 + rt)
57. (−3rs + r2 − 4s3 )(−rs3 )
3r2 s4 − r3 s3 + 4rs6
58. (a2 b)(a4 − ab2 + 1)
−x3 y 2 z + x3 y 3 z − x2 y 2 z 2
8x2 y 2 z − 20x2 yz 2 + 4xy 2 z 2
2j 3 k − 14j 2 k 2 + 24jk 3
15c3 d5 − 20c3 d4 − 55c2 d4
67. (−m2 + mn − n2 )(−4m2 n)
71. −c(3c5 + 4c3 − 9c + 1)
44. 3p(4p3 + 5p2 − 1)
−10d4 + 4d2 − 8d
56. (cdf )(c2 + cd + cdf )
69. y(y 3 + 2y 2 + 3x − 6)
−5n5 + n3 − 4n
43. (5d3 − 2d + 4)(−2d)
u3 w2 + 4u2 w2 − 5uw4
65. (3cd − 4c − 11)(5c2 d4 )
40. (5n4 − n2 + 4)(−n)
− 7x
55. (u2 w + 4uw − 5w3 )(uw)
63. 2jk(j 2 − 7jk + 12k 2 )
−18y 3 + 54y 2 + 30y
39. x(2x2 + x − 7)
+ 7a2
a3 b + 2a2 b2 + ab3
61. 4xyz(2xy − 5xz + yz)
36. −6(3y 3 − 9y 2 − 5y)
50. (n2 − 3n + 1)(−7n3 )
−5k 4 − 5k 3 + 30k 2
59. −x2 y 2 z(x − xy + z)
−20u2 w2 x + 10uw2 x3
31. (6x4 y 2 )(7x − 9y 2 )
+ 8a2
42. −7y(−2y 2 + y + 10)
28. (4uw − 2wx2 )(−5uwx)
− 30p4 r
44k + 11km − 77m
5a3 + 10a2 − 35a
53. ab(a2 + 2ab + b2 )
27. −10p2 r(−2pr2 + 3p2 )
−16x2 + 8x − 8
+r
− 5d6
3c2 d4 + 4c5 d3
35. (4k + km − 7m)(11)
38. (a3 − 2a2 + 8a)a
41. (5a)(a2 + 2a − 7)
−d7
−5m3 n − 3m2 n2
34. (−8)(2x2 − x + 1)
50c − 35d + 10
8v 6 − 5v 5
24. (cd3 )(3cd + 4c4 )
7c2 d4 e3 − 28cd2 e4
33. 5(10c − 7d + 2)
− 9b5
23. mn(−5m2 − 3mn)
+ a3 b4
30. (7cd2 e3 )(cd2 − 4e)
2m3 n3 + 4m3 n5
r3
−27b6
18x3 y 5 z − 2xy 2 z 2
25. (x2 y − xy)(−xy 3 )
−x3 y 4
+ 21j 5
16. 2n2 (−10n − 7)
4m4 n − 4m3 n2 + 4m2 n3
y 4 + 2y 3 + 3xy − 6y
−3c6 − 4c4 + 9c2 − c
3r2 st + drs3 t − r2 st2
c3 df + c2 d2 f + c2 d2 f 2
a6 b − a3 b3 + a2 b
60. p2 r3 (2pr + 4p2 − 3r)
2p3 r4 + 4p4 r3 − 3p2 r4
62. (−5m + p3 − 2mp)(−3mp)
64. (3a2 b + 2ab2 − 1)(9ab)
27a3 b2 + 18a2 b3 − 9ab
66. 6hk 3 (2h2 − 3k + 2hk)
12h3 k3 − 18hk4 + 12h2 k 4
68. −4x2 w2 (3x2 w + 2xw − 4w)
70. −3(t3 − 2t2 − 7t + 5)
−12x4 w3 − 8x3 w3 + 16x2 w3
−3t3 + 6t2 + 21t − 15
72. 2(−5d4 − 8d3 + 2d + 4)
ALG catalog ver. 2.6 – page 75 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
15m2 p − 3mp4 + 6m2 p2
−10d4 − 16d3 + 4d + 8
EC
73. (−4w)(w3 + 8w2 + 3w + 1)
−4w4 − 32w3 − 12w2 − 4w
74. 6k(2k 3 − k 2 + 7k − 1)
12k 4 − 6k 3 + 42k 2 − 6k
75. (4x5 − 2x3 + x + 3)(8c)
32x5 c − 16x3 c + 8xc + 24c
76. (−n4 − 2n3 + 6n2 + 7)(−3n)
77. (f 3 − 6f 2 + 3f − 9)(f 2 )
f 5 − 6f 4 + 3f 3 − 9f 2
78. (−2z 3 + 5z 2 + 3z + 4)(−z 3 )
2z 6 − 5z 5 − 3z 4 − 4z 3
79. (u4 )(7u4 − 7u3 + u2 − 2u)
7u8 − 7u7 + u6 − 2u5
80. q 2 (−5q 6 + 4q 4 + 3q 2 − 10)
81. −4t3 (−t5 − 2t4 + 5t2 + 6t)
4t8 + 8t7 − 20t5 − 24t4
82. (3h2 )(4h4 + 2h3 − 6h2 − 6h)
83. 2d3 (5d3 + 2d2 + d − 8)
10d6 + 4d5 + 2d4 − 16d3
85. (mnp)(2m + 3n2 − 2np − 1)
2m2 np + 3mn3 p − 2mn2 p2
84. (−7m4 )(2m4 − 3m3 − m + 1)
86. −xy(3xz + 2y − y 2 − 3yz)
a3 b + ab2 c + a3 bc − ab3
89. −c2 d3 (−cd3 + c2 d − cd + c3 )
c3 d6 − c4 d4 + c3 d4 − c5 d3
91. (−mn4 )(2mn + m2 − 8n2 − 1)
88. (−c2 d + c − d + d3 )(cd)
94. (2k 2 m2 n)(−9kmn − 6mn + kn2 − 1)
101. −
103.
3
xy 10x + 2y 2
5
36a5 b5 − 9a2 b3 c + 3ab4 − 3a3 b3
98. −6
−4m2 − 6m
2
18w3 − 12w
9
96. −3ab2 (−12a4 b3 + 3abc − b2 + a2 b)
− 20xy 3
Fractions and decimals
3 2 1
k −
6k3 − 4k
97. 8k
4
2
2
3
m+
5
5
8
w
3
−4w3 +
6x2 y +
6 3
xy
5
5
105. − uw (12u − 6uw + 18w − 30)
6
100. 21
1 2
bc 28bc + 4b − 8c + 16c2 7b2 c3 + b2 c2 − 2bc3 + 4bc4
4
3
a
a
1
2
109. −20a
− +
−4a5 + 10a3 − 2a2
5
2 10
107.
111. 6ab
5y 2
113.
6
c
4
115. −
a
b
1
+
−
b
3a 6ab
3y 2
10
3
+ 2b − 1
2 3 3
r −
3
2
−4r3 + 9
9 3 1
x + x
7
3
27x3 + 7x
a
(2a2 − 20a)
4
a3
− 5a2
2
104.
c
cd2 + 2d
2
d
c2 +
2c
d
2
106. − a2 x −5a2 x − 20ax − 10x + 5
5
2a4 x2 + 8a3 x2 + 4a2 x2 − 2a2 x
3 3
k 10k 3 + 2k 2 − 24k + 12 15k6 + 3k5 − 36k4 + 18k3
2
1
3
5
3
+
+
24y + 18y 2 + 15y 3
110. 24y
y2
4y
8
108.
3k 2
km 5m
112. −4km −
−
+
4m
2
2k
5 4 3 3
y + y − 10y 2
2
4
8r
114.
3
c3
5c
− 2c2 −
2
6
116.
9
+ y − 12
10
−2c2 + 8c +
6a2
102.
−10u2 w + 5u2 w2 − 15uw2 + 25uw
a4 b5 + a5 b2 − a3 b3 + a3 b2
−18k 3 m3 n2 − 12m3 n2 k 2 + 2k 3 m2 n3 − 2k 2 m2 n
95. (3xy 2 − 7x2 − x + 4)(−5xy 3 )
−3x2 yz − 2xy 2 + xy 3 + 3xy 2 z
92. w3 xy 2 (xy + 2wx − w2 + 4wy)
30r4 t2 − 18r3 t3 + 6r2 t4 − 24r4 t3
99. −10m
−14m8 + 21m7 + 7m5 − 7m4
w3 x2 y 3 + 2w4 x2 y 2 − w5 xy 2 + 4w4 xy 3
93. (5r2 − 3rt + t2 − 4r2 t)(6r2 t2 )
+ 5x2 y 3
12h6 + 6h5 − 18h4 − 18n3
−c3 d2 + c2 d − cd2 + cd4
90. (ab4 + a2 b − b2 + b)(a3 b)
−2m2 n5 − m3 n4 + 8mn6 + mn4
+ 35x3 y 3
−5q 8 + 4q 6 + 3q 4 − 10q 2
− mnp
87. ab(a2 + bc + a2 c − b2 )
−15x2 y 5
3n5 + 6n4 − 18n3 − 21n
ALG catalog ver. 2.6 – page 76 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
8ab
15
9r3
3r2
5
−
+
2
8
5 2
3
a + 5ab +
6
4
3k 3 + 2k 2 m2 − 10m2
24r4 − 4r3 +
5r
3
4 3
8
2
a b + a2 b2 + ab
9
3
5
EC
117. 0.3x(10x2 − 1.5)
118. 0.8y 2 (2y + 3.5)
3x3 − 0.45x
119. −1.5ab(0.3a − 4b)
−0.45a2 b − 6ab2
121. 0.7a(a2 − 0.2a + 6)
123. 1.6w(2w2 + w − 0.3)
0.7a3 − 0.14a2 + 4.2a
3.2w3 + 1.6w2 − 0.48w
125. −1.1c2 d(−0.2c2 + 8cd + 1.1d)
127. 0.2y 4 (0.7y 2 + 4.5y − 8)
0.22c4 d − 8.8c3 d2 − 1.21c2 d2
0.14y 6 + 0.9y 5 − 1.6y 4
1.6y 3 + 2.8y 2
120. 0.6n(0.7n3 − 20n)
0.42n4 − 12n2
122. 1.2s2 (5s3 − 1.2s2 + 3s)
6s5 − 1.44s4 + 3.6s3
124. 0.3k 3 (20k 2 − 3k + 0.4)
6k 5 − 0.9k 4 + 0.12k 3
126. −2.5rx2 (4rx − 0.1x + 0.6x2 )
−10r2 x3 + 0.25rx3 − 1.5rx4
128. 1.8mp3 (2m − 0.3mp + 1.5p)
3.6m2 p3 − 0.54m2 p4 + 2.7mp4
Variable exponents
129. xa (xa − x)
130. y x (y 2x + y 2 )
x2a − xa+1
131. a2n (an + a − 1)
a3n + a2n+1 − a2n
133. y m−1 (y 2m + y 2 − y)
135. 3xk y(2xy + xk )
y 3m−1 + y m+1 − y m
6xk+1 y 2 + 3x2k y
y 3x + y x+2
132. m3p (mp − m3 + 3)
m4p − m3p−3 + 3m3p
134. an−2 (an+2 + a2n + a2 )
136. 4an−1 b2 (ab − 2a2 bn )
ALG catalog ver. 2.6 – page 77 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
a2n + a3n−2 + an
4an b3 − 8an+1 bn+2
ED
Topic:
Multiplying binomials. For other muliplication problems, see categories EE (binomial squares),
EF (differences of squares), and EG (polynomials).
Directions:
3—Multiply.
1.
(k + 5)(k − 1)
k2
5.
(b + 20)(b + 1)
(t − 5)(t + 2)
(x − 8)(x + 1)
x2
+ 4k − 5
b2 + 21b + 20
9.
2.
6.
3.
(w + 1)(w + 6)
w2
− 7x − 8
(c − 1)(c − 35)
7.
10. (x + 5)(x − 3)
r2 + 24r − 25
11. (r − 3)(r − 4)
t2 − 3t − 10
x2 + 2x − 15
r2 − 7r + 12
13. (c − 14)(c − 3)
14. (k + 2)(k + 17)
15. (a − 11)(a + 4)
c2
− 17c + 42
17. (a + 3)(a − 16)
k2
+ 19k + 34
18. (b + 14)(b − 5)
a2 − 13a − 48
b2 + 9b − 70
21. (w + 12)(w + 5)
22. (d − 3)(d − 25)
w2 + 17w + 60
25. (d − 7)(d + 16)
d2
+ 9d − 112
29. (c − 11)(c − 9)
c2 − 20c + 99
33. (n − 25)(n + 6)
n2 − 19n − 150
37. (5 − p)(4 + p)
64 + 12y − y 2
53. (16 − r)(6 + r)
96 + 10r − r2
23. (q + 32)(q − 2)
27. (m + 17)(m + 6)
w2
m2
− 16w − 57
30. (p + 6)(p + 16)
34. (q + 30)(q − 6)
36 − 13r
12. (g + 7)(g + 3)
g 2 + 10g + 21
16. (d + 13)(d − 2)
d2 + 11d − 26
20. (a + 4)(a + 10)
a2 + 14a + 40
24. (u + 4)(u − 20)
u2 − 16u − 80
28. (x − 4)(x − 18)
x2 − 22x + 72
32. (t + 24)(t − 5)
36. (k − 10)(k − 20)
h2 + 55h + 250
k 2 − 30k + 200
39. (2 + w)(4 + w)
40. (3 − p)(2 − p)
8 + 6w + w2
6 − 5p + p2
43. (5 + x)(6 − x)
44. (10 − r)(4 + r)
30 + x − x2
40 + 6r − r2
47. (9 + y)(7 + y)
48. (20 + w)(4 − w)
63 + 16y
+ y2
51. (5 − a)(13 − a)
68 + 21k + k 2
54. (17 + y)(3 − y)
k 2 − 14k − 15
35. (h + 50)(h + 5)
+ r2
50. (4 + k)(17 + k)
(k + 1)(k − 15)
t2 + 19t − 120
32 + 18b + b2
46. (9 − r)(4 − r)
8.
c2 − 7c − 144
q 2 + 24q − 180
38. (5 + a)(2 − a)
+ 23m + 102
31. (c − 16)(c + 9)
p2 + 22p + 96
42. (16 + b)(2 + b)
49. (16 − y)(4 + y)
h2 − 18h + 45
26. (w + 3)(w − 19)
41. (4 − m)(8 − m)
70 − 3x − x2
19. (h − 15)(h − 3)
q 2 + 30q − 64
10 − 3a − a2
45. (7 − x)(10 + x)
− 7a − 44
d2 − 28d + 75
20 + p − p2
32 − 12m + m2
a2
(y − 3)(y − 1)
y 2 − 4y + 3
+ 7w + 6
(r + 25)(r − 1)
c2 − 36c + 35
4.
65 − 18a + a2
55. (8 + p)(12 + p)
51 − 14y − y 2
96 + 20p + p2
80 − 16w − w2
52. (18 − x)(3 + x)
54 + 15x − x2
56. (11 − a)(4 − a)
44 − 15a + a2
58. (2 − w)(26 − w)
59. (12 − x)(11 + x)
60. (8 − y)(12 + y)
72 + 38a + a2
52 − 28w + w2
132 + x − x2
96 − 4y − y 2
61. (13 + h)(12 − h)
62. (11 + k)(18 − k)
63. (9 − c)(13 − c)
64. (18 + z)(5 + z)
156 − h − h2
198 + 7k − k2
57. (2 + a)(36 + a)
65. (15 − n)(8 − n)
120 − 23n + n2
69. (50 + k)(5 − k)
250 − 45k − k2
66. (70 + a)(2 + a)
117 − 22c + c2
67. (26 + r)(10 − r)
140 + 72a + a2
70. (3 + w)(60 + w)
260 − 16r − r2
71. (6 − c)(20 − c)
180 + 63w + w2
ALG catalog ver. 2.6 – page 78 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
120 − 26c + c2
90 + 23z + z 2
68. (5 + a)(30 − a)
150 + 25a − a2
72. (45 − w)(8 + w)
360 + 37w − w2
ED
73. (y + 2)(5 + y)
y 2 + 7y + 10
77. (a + 3)(11 − a)
−a2
+ 8a + 33
81. (7 − x)(x − 12)
−x2 + 19x − 84
85. (21 + x)(x − 6)
x2
+ 15x − 126
74. (x + 3)(10 − x)
75. (u − 5)(7 − u)
−x2 + 7x + 30
78. (c − 4)(14 + c)
c2
−u2 + 12u − 35
79. (k + 18)(3 + k)
k2
+ 10c − 56
82. (6 + h)(h + 15)
83. (5 + a)(a − 16)
h2 + 21h + 90
86. (9 − r)(16 + r)
−r2
+ 21k + 54
a2 − 11a − 80
87. (4 − w)(w − 24)
−w2
− 7r + 144
+ 28w − 96
76. (w − 6)(8 + w)
w2 + 2w − 48
80. (y − 12)(5 − y)
−y 2 + 17y − 60
84. (11 − k)(k + 8)
−k 2 + 3k + 88
88. (15 + c)(c + 7)
c2 + 22c + 105
89. (−y − 11)(y + 2)
90. (−a + 1)(a − 19)
91. (−m − 3)(m − 12)
92. (−r + 12)(r + 4)
−y 2 − 13y − 22
−a2 + 20a − 19
−m2 + 9m + 36
−r2 + 8r + 48
93. (p − 3)(−p + 17)
94. (k − 25)(−k − 3)
−p2
+ 20p − 51
97. (−c − 15)(−c − 1)
c2 + 16c + 15
101. (−m + 9)(−10 − m)
m2
+ m − 90
105. (−2 − y)(y − 19)
−y 2 + 17y + 38
109. (2x + 1)(x + 1)
2x2 + 3x + 1
−k2
95. (w + 7)(−w + 8)
−w2
+ 22k + 75
98. (−x + 2)(−x + 12)
+ w + 56
99. (−a − 1)(−a − 21)
x2 − 14x + 24
a2 + 22a + 21
102. (−p − 6)(7 − p)
103. (−x − 13)(5 − x)
p2
x2
111. (3y + 1)(y + 8)
2c2 + 14c + 20
3y 2 + 25y + 8
114. (4y − 7)(y + 2)
115. (x + 1)(5x − 1)
4y 2 + y − 14
5x2 + 4x − 1
121. (3w + 5)(2w + 9)
6w2 + 37w + 45
125. (5c − 4)(8c + 7)
118. (t − 8)(3t − 2)
104. (−h + 1)(−11 − h)
108. (−26 − a)(a + 4)
−k 2 − 5k + 36
7a2 − 5a − 2
2m2 − 15m + 28
r2 − 18r + 45
107. (−4 + k)(−k − 9)
113. (7a + 2)(a − 1)
117. (2m − 7)(m − 4)
100. (−r + 3)(−r + 15)
h2 + 10h − 11
−w2 + 23w − 60
110. (c + 5)(2c + 4)
−y 2 − y + 110
+ 8x − 65
− p − 42
106. (−20 + w)(−w + 3)
96. (y − 10)(−y − 11)
−a2 − 30a − 104
112. (d + 6)(3d + 2)
3d2 + 20d + 12
116. (h − 9)(2h + 4)
2h2 − 14h − 36
119. (2r − 5)(r − 10)
120. (z − 12)(3z + 2)
3t2 − 26t + 16
2r2 − 25r + 50
3z 2 − 34z − 24
122. (6a + 1)(4a + 1)
123. (2n + 4)(5n + 3)
124. (3u + 7)(8u + 1)
24a2 + 10a + 1
10n2 + 26n + 12
24u2 + 59u + 7
126. (2r + 1)(2r − 9)
127. (4k − 5)(10k + 3)
128. (5p + 1)(4p − 1)
40c2 + 3c − 28
4r2 − 16r − 9
40k 2 − 38k − 15
20p2 − p − 1
129. (7y − 2)(6y − 1)
130. (3p − 10)(2p − 5)
131. (8d − 1)(3d − 4)
42y 2 − 19y + 2
6p2 − 35p + 50
24d2 − 35d + 4
133. (10a + 3)(2a − 5)
134. (7t − 11)(3t − 2)
135. (6c − 11)(6c + 7)
20a2 − 44a − 15
21t2 − 47t + 22
36c2 − 24c − 77
138. (4a + 8b)(a + 5b)
139. (7h − k)(2h − 3k)
137. (c + 2d)(8c − 3d)
8c2
+ 13cd − 6d2
141. (2m + 3n)(5m + 2n)
10m2 + 19mn + 6n2
145. (7kx − 6)(kx + 1)
7k2 x2
+ xk − 6
4a2
+ 28ab + 40b2
142. (8a − 3x)(3a + 4x)
14h2
143. (2x + 3y)(5x − 2y)
24a2 + 23ax − 12x2
146. (3 − 8pr)(6 − pr)
18 − 51pr
10x2 + 11xy − 6y 2
147. (cd − 12)(2cd + 3)
+ 8p2 r2
149. (k 2 − 5)(k 2 − 2)
150. (2 + c4 )(11 − c4 )
k 4 − 7k 2 + 10
22 + 9c4 − c8
− 23hk
+ 3k 2
2c2 d2
− 21cd − 36
151. (6 + r3 )(4 + r3 )
ALG catalog ver. 2.6 – page 79 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
24 + 10r3 + r6
132. (2a − 5)(4a − 11)
8a2 − 42a + 55
136. (2y + 9)(5y − 8)
10y 2 + 29y − 72
140. (4w − 11z)(3w + z)
12w2 − 29wz − 11z 2
144. (5p − 7r)(4p − 3r)
20p2 − 43pr + 21r2
148. (6 + ab)(1 + 6ab)
6 + 37ab + 6a2 b2
152. (h6 + 10)(h6 − 7)
h12 + 3h6 − 70
ED
153. (1 − u3 )(12 + u3 )
12 − 11u3
154. (x5 + 8)(x5 + 3)
− u6
x10
+ 11x5
155. (a10 − 3)(a10 + 11)
a20
+ 24
157. (4 + 9m4 )(3 − m4 )
158. (3u8 − 1)(u8 + 7)
12 + 23m4 − 9m8
3u16 + 20u8 − 7
161. (12a + c2 )(a + 3c2 )
162. (b3 − 3d)(5b3 − 2d)
12a2
+ 37ac2
+ 3c4
5b6
Fractions and decimals
2
1
165. y +
y+
3
3
y2 + y +
169.
4
r−
5
r2 −
166.
2
9
177.
2
r+
5
170.
2
8
r−
5
25
a2 −
181.
3k +
6k2 +
189.
178.
1
5c +
3
182.
2
9
2k +
8h4
3
2
167.
1
2
186.
1
3y +
2
190.
1
6
r−
1
m−
3
171.
2 2 1
1
p + p−
9
6
4
194.
1
3
179.
3
3n −
4
1
3
2
3
1
x+
4
1
2x −
5
195.
1 2 1
2
h − h−
4
6
9
3
2
180.
2
k−
5
1
2
k−
5
25
184.
7
w−2
3
p−
p2 −
1
(p − 1)
6
7
1
p+
6
6
2
p+
5
3p2 + 2p +
188.
4
3p +
5
8
25
3
3
x−
2
8
1
1
192. 3a −
6a +
4
2
18a2 −
196.
m
4
+
1
8
n 3m 2n
−
3
4
3
3m2
mn
2n2
+
−
16
12
9
197. (m − 4)(m + 0.5)
198. (y + 2)(y − 0.5)
199. (w − 0.5)(w − 2)
200. (k + 0.5)(k + 4)
m2 − 3.5m − 2
y 2 + 1.5y − 1
w2 − 2.5w + 1
k 2 + 4.5k + 2
201. (a − 0.2)(a − 0.8)
202. (c + 0.4)(c + 0.6)
203. (y + 0.9)(y + 0.1)
a2
− a − 0.16
c2
y2
+ c + 0.24
+ y + 0.09
3
1
3x −
4x +
4
2
12x2 −
1
18
x2
2y 2
xy
−
−
4
6
9
1
2
a−
2
176. (w − 3) w +
3
1
2
a−
3
15
x 2y x y −
+
2
3
2
3
3
4
1
k+
5
k2 −
1
4
1
1
191. 4r +
2r −
3
6
8r2 −
1
1
h+
2
3
172.
3
8
x+
5
5
1
1
10a −
4a +
3
5
1
10
a+
w2 −
3
(x + 1)
5
x+
a2 + a −
1
1
k−
183. 5k +
2
2
187.
168.
3
c−1
2
40a2 +
w+
4r2 − 19p4 r + 12p8
4
9
5k 2 − 2k −
5y +
164. (4r − 3p4 )(r − 4p4 )
− 3m6
1
3
x−
2
16
x2 +
3
16
1
2
h−
2
3
4
3
3
x−
4
c2 +
18 + 13y 3 + 2y 6
1
175. (c + 2) c −
2
1
2
y−
3
9
1
5x +
2
1
4
r−
5
5
2y −
w−
x−
1
(r + 1)
5
1
3n −
4
− 23h2 m3
w2 − w −
9
r+2
2
10x2 −
2
1
p−
3
2
2
1
m−
3
9
10y 2 −
c−
160. (9 + 2y 3 )(2 + y 3 )
163. (8h2 + m3 )(h2 − 3m3 )
3
4
9n2 − 3n +
1
1
p+
3
2
r2 +
5
1
k+
2
4
1
2y −
3
6y 2 −
193.
1
2
2
m+
3
r2 +
3
1
a−
4
4
20c2 − 2c −
185.
45 − 18n2 + n4
2k 14 − 19k 7 + 35
1
174. (r + 4) r +
2
1
(a − 1)
4
2
4c −
3
m2 +
5
y+1
2
a+
1
2
c−
156. (3 − n2 )(15 − n2 )
− 33
159. (k 7 − 7)(2k 7 − 5)
− 17b3 d + 6d2
c2 − 2c +
1
173. (y − 2) y −
2
y2 −
+ 8a10
204. (r − 0.3)(r − 0.7)
r2 − r + 0.21
205. (x + 0.7)(x − 0.3)
206. (m − 0.6)(m + 0.4)
207. (k − 0.4)(k − 0.8)
208. (w + 0.3)(w + 0.2)
x2 + 0.4x − 0.21
m2 − 0.2m − 0.24
k 2 − 1.2k + 0.32
w2 + 0.5w + 0.06
209. (k + 0.6)(k − 1.5)
210. (x − 1.5)(x − 0.4)
211. (a + 0.5)(a + 0.8)
212. (c − 0.6)(c + 0.5)
k 2 − 0.9k − 0.9
x2 − 1.9x + 0.6
a2 + 1.3a + 0.4
c2 − 0.1c − 0.3
ALG catalog ver. 2.6 – page 80 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
ED
213. (2k + 0.1)(5k − 0.3)
214. (3n + 0.8)(2n + 0.7)
215. (4p − 0.3)(5p + 0.5)
216. (10y − 0.6)(3y − 0.2)
10k 2 − 0.1k − 0.03
6n2 + 3.7n + 0.56
20p2 + 0.5p − 0.15
30y 2 − 3.8y + 0.12
219. (0.6w + 7)(0.3w + 4)
220. (0.2k + 3)(0.4k − 5)
217. (0.5a − 3)(0.8a − 6)
0.4a2
218. (0.8x − 5)(0.3x + 11)
0.2.4x2
− 5.4a + 18
0.18w2
+ 7.3x − 55
0.08k 2 + 0.2k − 15
+ 4.5w + 28
221. (0.1c + 0.4d)(0.3c − 0.2d)
0.03c2 + 0.1cd − 0.08d2
222. (0.7r + 0.2s)(0.4r + 0.6s)
223. (1.2p − 0.3q)(0.2p + 0.3q)
0.24p2 + 0.3pq − 0.09q 2
224. (0.5w − 1.1y)(0.5w − 0.7y)
0.28r2 + 0.5rs + 0.12s2
0.25w2 − 0.9wy + 0.77y 2
Variable exponents
225. (rn − 4)(rn + 1)
r2n − 3rn − 4
227. (y 2a + 8)(y 2a + 3)
y 4a + 11y 2a + 24
229. (c4x − 7d)(c4x − d)
c8x − 8c4x d + 7d2
231. (an + b2 )(7an − 6b2 )
233. (2pm − rn )(3pm + rn )
235. (ab + ac )(ab + ac )
7a2n + an b2 − 6b4
6p2m − pm rn − r2n
a2b + 2ab+c + a2c
226. (2 − y x )(10 − y x )
20 − 12y x + y 2x
228. (6 + c3k )(5 − c3k )
30 − c3k − c6k
230. (11d − m3k )(6d + m3k )
232. (2xa + 3y b )(xa + 5y b )
234. (ay − 2y )(ay − 2y )
236. (x2 + xy )(2x2 − xy )
ALG catalog ver. 2.6 – page 81 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
66d2 + 5dm3k − m6k
2x2a + 13xa y b + 15y 2b
a2y − 2y+1 ay + 4y
2x4 + xy+2 − x2y
EE
Topic:
Binomial squares. For other multiplication problems, see categories ED (binomials),
EF (differences of squares), and EG (polynomials).
Directions:
3—Multiply.
1.
(p + 3)(p + 3)
p2
5.
2.
m2
+ 6p + 9
(k − 20)(k − 20)
k 2 − 40k + 400
9.
(m − 7)(m − 7)
6.
3.
(x + 10)(x + 10)
x2
− 14m + 49
(c + 12)(c + 12)
7.
(6 + x)(6 + x)
r2 − 30r + 225
10. (5 − d)(5 − d)
11. (9 + m)(9 + m)
8.
(y + 25)(y + 25)
y 2 + 50y + 625
12. (4 − p)(4 − p)
36 + 12x + x2
25 − 10d + d2
13. (11 − h)(11 − h)
14. (30 + y)(30 + y)
15. (13 − a)(13 − a)
16. (16 + k)(16 + k)
121 − 22h + h2
900 + 60y + y 2
169 − 26a + a2
256 + 32k + k 2
17. (5x + y)(5x + y)
25x2 + 10xy + y 2
21. (2a − 7b)(2a − 7b)
4a2 − 28ab + 49b2
25. (m − 8)2
18. (2c − d)(2c − d)
19. (m − 3n)(m − 3n)
4c2 − 4cd + d2
m2 − 6mn + 9n2
22. (8x + 3y)(8x + 3y)
1 − 2x + x2
26. (x + 1)2
h2 + 22h + 121
x2 + 2x + 1
30. (10 − w)2
100 − 20w
33. (h + 11)2
23. (5c + 4d)(5c + 4d)
64x2 + 48xy + 9y 2
m2 − 16m + 64
29. (1 − x)2
81 + 18m + m2
(a − 1)(a − 1)
a2 − 2a + 1
+ 20x + 100
(r − 15)(r − 15)
c2 + 24c + 144
4.
25c2 + 40cd + 16d2
16 − 8p + p2
20. (p + 4r)(p + 4r)
p2 + 8pr + 16r2
24. (11h − 2k)(11h − 2k)
121h2 − 44hk + 4k2
27. (r − 5)2
r2 − 10r + 25
28. (p + 9)2
p2 + 18p + 81
31. (y + 3)2
y 2 + 6y + 9
32. (a + 7)2
a2 + 14a + 49
+ w2
34. (k + 14)2
k2 + 28k + 196
35. (m − 30)2
m2
37. (15 + d)2
225 + 30d + d2
38. (12 − p)2
41. (3w + 2)2
9w2 + 12w + 4
42. (9h − 5)2
144 − 24p + p2
36. (c − 16)2
c2 − 32c + 256
− 60m + 900
39. (18 − r)2
324 − 36r + r2
40. (20 + x)2
400 + 40x + x2
43. (2u + 7)2
4u2 + 28u + 49
44. (6y − 1)2
36y 2 − 12y + 1
47. (1 + 9k)2
1 + 18k + 81k 2
48. (10 − 3c)2
81h2 − 90h + 25
45. (7 − 4m)2
46. (2 + 15a)2
4 + 60a + 225
49 − 56m + 16m2
49. (4a + b)2
100 − 60c + 9c2
16a2 + 8ab + b2
50. (10x − y)2
100x2
53. (3w − 10x)2
9w2
− 60wx + 100x2
57. (h + 6k)2
h2
+ 12hk
− 20xy
51. (k − 6m)2
+ y2
k2
54. (6a + 5y)2
55. (12c + 2d)2
36a2
144c2
+ 60ay
+ 25y 2
58. (x − 2y)2
x2 − 4xy + 4y 2
+ 48cd + 4d2
59. (8a − b)2
+ 36k 2
64a2
p4 − 2p2 + 1
62. (1 + x2 )2
1 + 2x2 + x4
63. (c2 + 5)2
65. (6 + r3 )2
36 + 12r3 + r6
66. (y 4 − 7)2
y 8 − 14y 4 + 49
67. (10 − x4 )2
100 − 20x4
25k 4
− 20k2
70. (3 + 4w3 )2
9 + 24w3
+4
73. (10c3 − 7ad2 )2
100c6
− 140ac3 d2
+ 49a2 d4
74. (8rt2 + 3s3 )2
64r2 t4
c4 + 10c2 + 25
36p6
+ 12p3
+1
75. (3b3 c4 + 11x)2
+ 48rs3 t + 9s6
ALG catalog ver. 2.6 – page 82 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
9b6 c8
56. (9c − 7w)2
81c2 − 126cw + 49w2
121w2 + 22wy + y 2
64. (4 − k 2 )2
16 − 8k 2 + k 4
68. (a3 + 2)2
a6 + 4a3 + 4
+ x8
71. (6p3 + 1)2
+ 16w6
n2 + 16np + 64p2
60. (11w + y)2
− 16ab + b2
61. (p2 − 1)2
69. (5k 2 − 2)2
52. (n + 8p)2
− 12km + 36m2
+ 66b3 c4 x + 121x2
72. (2 − 9y 4 )2
4 − 36y 4 + 81y 8
76. (5ax2 − 8y 4 )2
25a2 x4 − 80ax2 y 4 + 64y 8
EE
Fractions and decimals
2
1
x+y
77.
2
78.
1 2
x + xy + y 2
4
81.
3
8m − r
4
85.
2
a+b
5
82.
9 2
r
16
2
3
2
c−
5
5
2
9 2 12
4
c −
c+
25
25
25
93. (m2 + 1.1)2
m4
+ 2.2m + 1.21
97. (2.1 − 3rs)2
4.41 − 12.6rs + 9r2 s2
101. (3b + 2.5c)2
9b2
+ 15bc + 6.25c2
105. (3.6r2 − 1.1p2 )2
12.96r4 − 7.92r2 p2 + 1.21p4
79.
1 2
a
16
4
k − 10
5
2
86.
2
y− z
3
y2 −
90.
2
3
c − 2p
4
2
2 2
6b − n
80.
3
9 2
c − 3cp + 4p2
16
36b2 − 8bn +
5 2 2
83.
12w + x
6
16 2
k − 16k + 100
25
4 2 4
a + ab + b2
25
5
89.
1 2
2+ a
4
4+a+
2
64m2 − 12mr +
144w2 + 20wx2 +
87.
1
+ 3r
2
2
84.
2
4n2 −
91.
1 2 1
1
p + pr + r2
16
6
9
94. (0.9 − wx)2
2 2 3
w −
3
4
2
4 4
9
w − w2 +
9
16
95. (y + 0.5)2
y 2 + y + 0.25
0.81 − 1.8wx + w2 x2
98. (1.2d2 + 5)2
99. (8kn + 0.3)2
64k 2 n2 + 4.8kn + 0.09
103. (0.4p − 1.1)2
0.49 + 0.42a + 0.09a2
106. (2.4x3 − 1.5y)2
92.
16
16 2
nk +
k
3
9
1 3 1
a +
2
5
2
1 6 1 3
1
a + a +
4
5
25
96. (0.4r3 + 1)2
0.16r6 + 0.8r3 + 1
1.44d4 + 12d2 + 25
102. (0.7 + 0.3a)2
2
4 2
2n − k
88.
3
4
4
yz + z 2
3
9
1
1
p+ r
4
3
1
+ 9p
3
1
+ 6p + 81p2
9
25 4
x
36
1
+ 3r + 9r2
4
4 2
n
9
0.16p2
− 0.88p + 1.21
107. (0.2km3 + 0.3)2
5.76x6 − 7.2x3 y + 2.25y 2
0.04k 2 m6 + 0.12km3 + 0.09
100. (0.7 + 6y)2
0.49 + 8.4y + 36y 2
104. (6r − 0.25s)2
36r2 − 3rs + 0.625s2
108. (0.9a2 + 0.4x3 )2
0.81a4 + 0.72a2 x3 + 0.16x6
Variable exponents
109. (x2a − y 3a )2
x4a − 2x2a y 3a + y 6a
113. (an + bn )2
a2n + 2an bn + b2n
110. (a3x + bx )2
111. (ma − n2a )2
a6x + 2a3x bx + b2x
114. (xn+1 − x)2
m2a − 2ma n2a + n4a
115. (ax+1 + ax )2
x2n+2 − 2xn+2 + x2
ALG catalog ver. 2.6 – page 83 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
a2x+2 + 2a2x+1 + a2x
112. (x5n + y 2n )2
x10n + 2x5n y 2n + y 4n
116. (m2a − ma−1 )2
m4a − 2m3a−1 + m2a−2
EF
Topic:
Differences of squares. For other multiplication problems, see categories ED (binomials),
EE (binomial squares), and EG (polynomials).
Directions:
3—Multiply.
1.
(c + 1)(c − 1)
c2 − 1
2.
(x − 5)(x + 5)
x2 − 25
3.
(m + 6)(m − 6)
m2
5.
(p − 10)(p + 10)
p2
9.
6.
(r + 14)(r − 14)
r2
− 100
(8 + k)(8 − k)
64 − k 2
7.
− 196
10. (2 − y)(2 + y)
4 − y2
(d − 3)(d + 3)
8.
(p + 20)(p − 20)
d2 − 9
− 36
(a − 11)(a + 11)
a2
4.
p2 − 400
− 121
11. (9 + m)(9 − m)
12. (1 − x)(1 + x)
1 − x2
81 − m2
13. (12 − u)(12 + u)
14. (16 + p)(16 − p)
144 − u2
17. (6c − 5)(6c + 5)
w2
33. (4x + 7y)(4x − 7y)
h2
a2 − 100c2
38. (9a − 8y)(9a + 8y)
49r2 − 81u2
39. (4x + 15z)(4x − 15z)
81a2 − 64y 2
16x2 − 225z 2
41. (x2 + 1)(x2 − 1)
42. (a2 − 5)(a2 + 5)
43. (3 − p2 )(3 + p2 )
36x4
50. (b − c3 )(b + c3 )
b2 − c6
54. (x2 + 8y)(x2 − 8y)
− 81m6
x4
57. (11ax3 − 2by)(11ax3 + 2by)
59. (10k + 3m5 x2 )(10k − 3m5 x2 )
Fractions and decimals
1
1
61.
a+
a−
2
2
c2
16
62.
1
4
−7
− 49
16k 4 − 25
− 49
52. (a3 + d4 )(a3 − d4 )
r10 − p4
53. (2k − 9m3 )(2k + 9m3 )
4
44. (8 + y 2 )(8 − y 2 )
48. (4k 2 − 5)(4k 2 + 5)
51. (r5 − p2 )(r5 + p2 )
x4 − y 2
65.
9 − p4
47. (6x2 + 7)(6x2 − 7)
1 − 81r4
49. (x2 + y)(x2 − y)
c
144c2 − 25k 2
64 − y 4
46. (1 + 9r2 )(1 − 9r2 )
16 − 9m4
a2 −
40. (12c + 5k)(12c − 5k)
a4 − 25
45. (4 − 3m2 )(4 + 3m2 )
4k2
36. (7r + 9u)(7r − 9u)
9m2 − 64p2
121h2 − 9k 2
x4 − 1
32. (a − 10c)(a + 10c)
− 49m2
35. (3m − 8p)(3m + 8p)
25c2 − 4d2
37. (11h − 3k)(11h + 3k)
36k 2 − n2
31. (h + 7m)(h − 7m)
− 16y 2
34. (5c − 2d)(5c + 2d)
16x2 − 49y 2
28. (6k + n)(6k − n)
169a2 − x2
30. (w + 4y)(w − 4y)
− 225d2
16 − 25p2
27. (13a − x)(13a + x)
81p2 − r2
29. (c − 15d)(c + 15d)
24. (4 − 5p)(4 + 5p)
1 − 64a2
26. (9p − r)(9p + r)
4x2 − y 2
49h2 − 121
23. (1 − 8a)(1 + 8a)
25 − 144j 2
25. (2x + y)(2x − y)
20. (7h + 11)(7h − 11)
4x2 − 81
22. (5 + 12j)(5 − 12j)
100 − 9m2
225 − r2
19. (2x + 9)(2x − 9)
9k 2 − 1
21. (10 + 3m)(10 − 3m)
16. (15 + r)(15 − r)
900 − a2
18. (3k − 1)(3k + 1)
36c2 − 25
c2
15. (30 − a)(30 + a)
256 − p2
55. (7a4 + 4c)(7a4 − 4c)
− 64y 2
49a8
2
k+1
3
63.
4 2
k −1
9
c
4
+7
66.
n n
12 +
12 −
6
6
144 −
36w2 − z 4
60. (2h2 − 6c3 x)(2h2 + 6c3 x)
100k 2 − 9m10 x4
2
k−1
3
56. (6w − z 2 )(6w + z 2 )
− 16c2
58. (8r2 x5 + 3y 6 )(8r2 x5 − 3y 6 )
121a2 x6 − 4b2 y 2
a6 − d8
1
y − a2
5
y2 −
67.
n2
36
ALG catalog ver. 2.6 – page 84 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1
y + a2
5
1 4
a
25
2
h − 3k
3
4 2
h − 9k 2
9
64r4 x10 − 9y 12
4h4 − 36c6 x2
64.
4
w − z2
3
4
w + z2
3
16 2
w − z4
9
2
h + 3k
3
1
1
8pr +
8pr −
68.
4
4
64p2 r2 −
1
16
EF
69.
2
3
rx −
3
5
2
3
rx +
3
5
70.
4 2 2
9
r x −
9
25
4 2 1
c + d
5
2
4 2 1
c − d
5
2
71.
x
74. (h3 + 0.6k)(h3 − 0.6k)
− y2
h6
77. (0.4x2 + 0.3)(0.4x2 − 0.3)
79. (1.1x2 + 0.7z)(1.1x2 − 0.7z)
1.21x4 − 0.49z 2
72.
a
b
−
6 2
83. (xn−1 − xn )(xn−1 + xn )
85. (x2a − y a )(x2a + y a )
87. (wa + y b )(wa − y b )
9y 2a+4 − y 2
6.25p2 − w2
78. (0.1 − 1.2ay)(0.1 + 1.2ay)
0.01 − 1.44a2 y 2
80. (0.2mn + 0.5y 3 )(0.2mn − 0.5y 3 )
x2n−2 − x2n
x4a − y 2a
w2a − y 2b
82. (xa − 4x)(xa + 4x)
0.04m2 n2 − 0.25y 6
x2a − 16x2
84. (4xa + 7x2a )(4xa − 7x2a )
86. (a3x + b5x )(a3x − b5x )
88. (an+1 − bn )(an+1 + bn )
ALG catalog ver. 2.6 – page 85 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
a
b
+
6 2
76. (2.5p − w)(2.5p + w)
0.0081 − a2
Variable exponents
81. (3y a+2 − y)(3y a+2 + y)
b2
a2
−
36
4
75. (0.09 + a)(0.09 − a)
− 0.36k 2
0.16x4 − 0.09
y x y +
−
4
9 4
9
x2
y2
−
16
81
16
1
− d2
25c4
4
73. (1.5x − y)(1.5x + y)
2.25x2
16x2a − 49x4a
a6x − b10x
a2n+2 − b2n
EG
Topic:
Multiplying polynomials.
Directions:
3—Multiply.
1.
(a + 3)(b − 7)
(x + w)(y + z)
2.
ab − 7a + 3b − 21
5.
(u + 7)(u2 − u)
u3
9.
+ 6u2
− y3
(m2 − m)(m − 4)
6.
m3
− 7u
(y 2 − y)(y 2 + 2)
y4
+ 2y 2
− 5m2
(d2 − 4)(d − 3)
25t3
− 2y
7.
x2
+ p2 r2
− pr2
k 4 + 9k 3 − k 2 − 9k
− 4d + 12
p4 + 3p3 + 2p2
−x
+ x2 y
− xy
12. (6a − 3)(a − 2c)
− xy 2
6a2 − 12ac − 3a + 6c
a2 b4
− ab3
+ ab4
16. (5c2 d − 3)(cd − d2 )
− b3
5c3 d2 − 5c2 d3 − 3cd + 3d2
17. (y + 3)(y 2 + 5y − 4)
18. (a − 2)(a2 + 3a − 8)
19. (w2 − 2w − 7)(w + 5)
y 3 + 8y 2 + 11y − 12
a3 + a2 − 14a + 16
w3 + 3w2 − 17w − 35
21. (c2 + 2c + 8)(3c − 2)
3c3
+ 4c2
22. (n2 − 4n − 8)(7n + 1)
7n3
+ 20c − 16
25. (w + z)(w + x − z)
− 2y 2
30. (n − r)(n2 + nr − r2 )
n3
−x+y
33. (3n − p)(5n2 + 3np − 2p2 )
35. (5c − 2d)(5c + 2d − 2)
37. (a + b + c)(a + b)
39. (m − r − 2)(m − r)
6y 3
− 2nr2
25c2 − 4d2 − 10c + 4d
a2 + b2 + 2ab + ac + bc
m2 − 2mr − 2m + r2 + 2r
41. (y 2 − 4y)(5y 2 − 2y − 8)
5y 4 − 22y 3 + 32y
43. (6x3 − 2x2 − x)(x3 − 7x2 )
45. (2w − 9)(3w + 5 − 2w2 )
47. (5x − x2 + 6)(3 − 7x)
6x6 − 44x5 + 13x4 + 7x3
−4w3 + 24w2 − 17w − 45
7x3 − 38x2 − 27x + 18
− 25y 2
24. (4k + 3)(k 2 + k − 9)
4k 3 + 7k 2 − 33k − 27
− 3y + 28
28. (m − 4)(m − r + 4)
y 2 + xy + 3x − 9
m2 − mr + 4r − 16
31. (a + 2b)(a2 − ab + 2b2 )
+ r3
15n3 + 4n2 p − 9np2 + 2p3
r3 − 10r2 + 29r − 30
27. (y + 3)(x + y − 3)
a2 − k 2 − ap + kp
29. (x − y)(2x + 2y − 1)
20. (r2 − 4r + 5)(r − 6)
23. (6y − 7)(y 2 − 3y − 4)
− 60n − 8
26. (a − k)(a + k − p)
w2 − z 2 + wx + xz
2x2
− 27n2
a3
32. (p + r2 )(p2 − pr2 − r4 )
+ a2 b + 4b3
p3 − 2pr4 − r6
34. (2a + 3b)(4a − 6b + 1)
8a2 − 18b2 + 2a + 3b
36. (x + 2y)(2x2 − xy − y 2 )
38. (x − y)(x − y + z)
40. (c + d)(c + d − 1)
2x3 − 2y 3 + 3x2 y − 3xy 2
x2 + y 2 − 2xy + xz − yz
c2 + d2 + 2cd − c − d
42. (2x4 + 3x2 + 4)(x2 + 11)
2x6 + 25x4 + 37x2 + 44
44. (r + 1)(3r3 + 9r2 − 5r)
3r4 + 12r3 + 4r2 − 5r
46. (4 − 2a + 3a2 )(5a + 4)
15a3 + 2a2 + 12a + 16
48. (2 − 5c)(c − 4 + 3c2 )
−15c3 + c2 + 22c − 8
49. (a + 3)(a3 − 2a2 + 5a − 2)
a4 + x3 − a2 + 13a − 6
50. (w3 + 3w2 − 7w − 5)(w + 8)
51. (y − 5)(y 3 − 4y 2 − 6y + 3)
y 4 − 9y 3 + 14y 2 + 33y − 15
52. (p3 + 3p2 + 8p − 2)(p − 3)
53. (d3 + 5d2 − 3d + 7)(2d − 9)
55. (y 3 + 9y 2 − y + 4)(5y + 6)
2d4 + d3 − 51d2 + 41d − 63
5y 4 + 51y 3 + 49y 2 + 14y + 24
57. (a + b)(2a2 + 3a − 4ab − 4b)
(p2 + 2p)(p2 + p)
8.
15. (ab + b)(ab3 − b2 )
−r
(k + 9)(k 3 − k)
4.
11. (x + xy)(x − y)
−t
p3 r3
− 3d2
(x3 − x)(x2 + 1)
+ 4m
14. (p2 r − 1)(pr2 + r)
+ t5
d3
x5
10. (5t + 1)(5t2 − t)
13. (st − t3 )(s − t2 )
s2 t − 2st3
3.
xy + xz + wy + wz
w4 + 11w3 + 17w2 − 61w − 40
p4 − p2 − 26p + 6
54. (3k − 8)(k 3 − 2k 2 − 7k + 10)
56. (2n + 7)(n3 + n2 + n + 1)
3k 4 − 14k 3 − 5k 2 + 86k − 80
2n4 + 9n3 + 9n2 + 9n + 7
58. (x − y)(3x2 + 2xy − 5y + 2y)
3x3 − x2 y − 2xy 2 + 3y 2 − 3xy
2a3 − 2a2 b − 4ab2 + 3a2 − 4b2 − ab
59. (2a − 3b)(5a + 3b + ab − 4b2 )
60. (4x + y)(2x + y − 3xy − 8y 2 )
12b3 + 2a2 b − 11ab2 + 10a2 − 9b2 − 9ab
61. (x2 + 5x − 1)(x2 + 3x + 2)
x4 + 8x3 + 16x2 + 7x − 2
−8y 3 − 12x2 y − 35xy 2 + 8x2 + y 2 + 6xy
62. (m2 − 8m − 4)(m2 + 3m − 5)
m4 − 5m3 − 33m2 + 28m + 20
63. (u2 − 7u + 3)(u2 + 5u + 6)
u4 − 2u3 − 26u2 − 27u + 18
64. (w2 + w − 9)(w2 − 4w − 5)
ALG catalog ver. 2.6 – page 86 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
w4 − 3w3 − 18w2 + 31w + 45
EG
65. (5c2 + 2c − 1)(c2 + 3c + 7)
5c4 + 17c3 + 40c2 + 11c − 7
67. (y 2 + 8y + 11)(11y 2 − 8y − 1)
11y 4
+ 80y3 + 56y 2
66. (3a2 − a − 9)(a2 + 2a − 7)
68. (x2 − 3x + 7)(4x2 − 9x + 2)
w4 + 3w − 2
71. (r2 − 4r + 12)(r2 + 4r + 4)
73. (a + b + c)(a − b + c)
77. (x2 + 3x + 2)2
r4 + 32r + 48
a2 − b2 + c2 + 2ac
75. (m + p − 3)(m + p + 3)
m2 + p2 + 2mp − 9
s4 + 6x3 + 13x2 + 12x + 4
79. (4m2 − m + 7)2
16m4 − 8m3 + 57m2 − 14m + 49
81. (8w + 2w2 − 3)(−6w − 2 + 5w2 )
70. (a2 + 2a + 1)(a2 − 2a + 3)
2p4 − 7p3 − 13p2 + 29p − 12
85. (a2 + 2ab + b2 )(3a2 − 4ab − b2 )
74. (a2 − 2a − 1)(a2 + 2a + 4)
a4 − a2 − 10a − 4
76. (x2 − 3x + 2)(x2 + 3x − 1)
x4 − 8x2 + 9x − 2
78. (2y 2 + 5y − 3)2
4y 4 + 20y 3 + 13y 2 − 30y + 9
80. (5h2 − 3h − 1)2
25h4 − 30h3 − h2 + 6h + 1
82. (7 + 4x2 − 2x)(4x + 3x2 − 6)
84. (2y − 1 + 3y 2 )(−9 + 2y 2 − y)
6x4 + x3 y − 20x2 y 2 + 17xy 3 − 4y 4
87. (x2 − 8xy − 3y 2 )(5x2 + 2xy + 3y 2 )
− 30xy 3
6y 4 + y 3 − 31y 2 − 17y + 9
86. (2x2 − 3xy + y 2 )(3x2 + 5xy − 4y 2 )
3a4 + 2a3 b − 6a2 b2 − 6ab3 − b4
89. (2c − 1)(c + 3)(c − 2)
n4 + 3n + 20
12x4 + 10x3 − 11x2 + 40x − 42
83. (3 − 5p + p2 )(3p − 4 + 2p2 )
− 28x2 y 2
a4 + 4a + 3
72. (n2 + 3n + 4)(n2 − 3n + 5)
10w4 + 28w3 − 67w2 + 2w + 6
− 38x3 y
4x4 − 21x3 + 57x2 − 69x + 14
− 96y − 11
69. (w2 − w + 2)(w2 + w − 1)
5x4
3a4 + 5a3 − 32a − 11a + 63
88. (6a2 + 4ab − b2 )(2a2 + 3ab + 5b2 )
− 9y 4
2c3 + c2 − 13c + 6
91. (3y − 5)(y − 1)(y + 3)
3y 3 + y 2 − 19y + 15
93. (r + 5)(2r − 7)(3r + 2)
6r3 + 13r2 − 99r − 70
95. (n − 6)(4n − 9)(2n − 3)
8n3 − 78n2 + 207n − 162
12a4 + 26a3 b + 40a2 b2 + 17ab3 − 5b4
90. (x + 4)(2x − 3)(x + 2)
2x3 + 9x2 − 2x − 24
92. (a + 6)(a − 4)(2a + 3)
2a3 + 7a2 − 42a − 72
94. (3a + 8)(3a + 1)(a + 5)
9a3 + 72a2 + 143a + 40
96. (4x − 1)(5x − 2)(x + 4)
20x3 + 67x2 − 50x + 8
97. (3y − 2)2 (y + 4)
9y 3 + 24y 2 − 44y + 16
98. (2k − 3)(k + 6)2
2k 3 + 21k 2 + 36k − 108
99. (x − 4)2 (3x − 1)
3x3 − 25x2 + 56x − 16
100. (r + 2)(2r + 5)2
4r3 + 28r2 + 65r + 50
Binomial cubes
101. (x + 3)(x + 3)(x + 3)
x3 + 9x2 + 27x + 27
102. (y − 1)(y − 1)(y − 1)
y 3 − 3y 2 + 3y − 1
103. (5 + a)(5 + a)(5 + a)
125 + 75a + 15a2 + a3
104. (4 − n)(4 − n)(4 − n)
64 − 48n + 12n2 − n3
105. (a + b)(a + b)(a + b)
a3 + 3a2 b + 3ab2 + b3
106. (x + 2y)(x + 2y)(x + 2y)
x3 + 6x2 y + 12xy 2 + 8y 3
108. (c + 5d)(c + 5d)(c + 5d)
c3 + 15c2 d + 75cd2 + 125d3
107. (3p + r)(3p + r)(3p + r)
27p3 + 27p2 r + 9pr2 + r3
109. (2 − k)3
8 − 12k + 6k 2 − k3
110. (x + 1)3
x3 + 3x2 + 3x + 1
111. (c − 4)3
c3 − 12c2 + 48c − 64
112. (3 + y)3
27 + 27y + 9y 2 + y 3
113. (1 + w)3
1 + 3w + 3w2 + w3
114. (4 − h)3
64 − 48h + 12h2 − h3
115. (x + 5)3
x3 + 15x2 + 75x + 125
116. (a − 2)3
a3 − 6a2 + 12a − 8
117. (3x + y)3
27x3 + 27x2 y + 9xy 2 + y 3
118. (m − 4r)3
119. (2h − k)3
8h3 − 12h2 k + 6hk 2 − k 3
120. (x + y)3
ALG catalog ver. 2.6 – page 87 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
m3 − 12m2 r + 48mr2 − 64r3
x3 + 3x2 y + 3xy 2 + y 3
EG
121. (r − 2w)3
r3 − 6r2 w + 12rw2 − 8w3
122. (3c − d)3
123. (x + 4y)3
x3 + 12x2 y + 48xy 2 + 64y 3
124. (5m + n)3
125. (y 2 + 1)3
y 6 + 3y 4 + 3y 2 + 1
126. (2 + x2 )3
8 + 12x2 + 6x4 + x6
127. (1 − c2 )3
1 − 3c2 + 3c4 − c6
128. (n2 − 3)3
n6 − 9n4 + 27n2 − 27
27c3 − 27c2 d + 9cd2 − d3
125m3 + 75m2 n + 15mn2 + n3
129. (3w − 5)3
27w3 − 135w2 + 225w − 125
130. (5a + 2)3
125a3 + 150a2 + 60a + 8
131. (4 + 3h)3
64 + 144h + 14h2 + 27h3
132. (2 − 3y)3
8 − 36y + 54y 2 − 27y 3
Sums and differences of cubes
133. (a + 1)(a2 − a + 1)
135. (w + 7)(w2 − 7w + 49)
137. (2 − y)(4 + 2y + y 2 )
139. (1 − x)(1 + x + x2 )
134. (x − 3)(x2 + 3x + 9)
a3 − 1
w3 + 343
8 − y2
1 − x3
141. (x + y)(x2 − xy + y 2 )
8c3 − d3
145. (a2 − 5)(a4 + 5a2 + 25)
147. (y 3 − 3)(y 6 + 3y 3 + 9)
a6 − 125
y 9 − 27
149. (a + 4b2 )(a2 − 4ab2 + 16b4 )
151. (w2 − x)(w4 + w2 x + x2 )
a3 + 64b6
w6 − x3
n3 − 125
138. (6 + c)(36 − 6c + c2 )
216 + c3
140. (4 + k)(16 + 4k + k 2 )
64 + k 3
142. (a − b)(a2 + ab + b2 )
x3 + y 3
143. (2c − d)(4c2 + 2cd + d2 )
136. (n − 5)(n2 + 5n + 25)
x3 − 27
a3 − b3
144. (m + 3r)(m2 − 3mr + 9r2 )
146. (1 + x2 )(1 + x2 + x4 )
1 + x6
148. (r3 − 2)(r6 + 2r3 + 4)
r9 − 8
150. (3x − y 4 )(9x2 + 3xy 4 + y 8 )
152. (c2 + d2 )(c4 − c2 d2 + d4 )
ALG catalog ver. 2.6 – page 88 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
m3 + 27r3
27x3 − y 12
c6 + d6
EH
Topic:
Dividing monomials and polynomials.
Directions:
6—Simplify. 9—Simplify (assume no denominator equals zero).
76—Find the quotient. 52—Write as a mixed expression.
1.
3a − 12
3
5.
30k 2 + 15k
15
9.
14m − 28mr + 7r
−7
a−4
2k2 + k
2.
20x + 5
5
6.
42y 2 − 56
−14
10.
−8c + 24d + 72
8
13.
n2 + n
n
17.
−ay 3 − by 2
−y 2
21.
48m2 − 60m
12m
25.
5ab2 + a2 b2
ab2
29.
24n2 p3 − 8n2 p
−4n2 p
n+1
ay + b
4m − 5
5+a
cx − cy
−c
18.
4k 4 + k 2
k2
22.
32x2 − 40xy
−8x
26.
9w2 y 2 − 18w2
3w2
30.
−2a4 b2 + 5a2 b3
−ab2
7.
−20xy + 40x
−20
11.
33x2 − 11x + 11
11
5ac + 7a2 c − a3 c
ac
34.
38.
5 + 2b − 6x
42.
−3r + 6p + 1
45. (14c − 42) ÷ (7)
6m + 32
2
8.
55a − 11b
11
12.
−18y 3 + 6y 2 − 24y
−6
−x2 − x
20.
6a2 c − a2
−a2
3z + 1
24.
8x6 + 12x4
4x3
28.
−3cd + 42cd2
−3cd
32.
2r3 w4 − 4r3 w3
2r2 w3
23.
15z 3 + 5z 2
5z 2
3y 2 − 6
27.
k 2 m2 − k 3 m
−km
31.
26xy 3 + 65x2 y 2
13xy 2
6u6 − 24u4 − 10u2
−2u2
8x2 y + 24x2 y 2 + 8xy 2
8xy
46. (10w + 28) ÷ (2)
−3b + 2
−km + k 2
−4a2 + 12ab − 4ab2
−4a
36.
a − 3b + b2
39.
15b2 y 2 − 20by 2 − 5by
5by
2x2 + 3x
1 − 14d
2xy + x2 y − xy 2
xy
40.
8cd − 4cd2 − 4d2
−4d
−2c + cd + d
27x3 z − 9x2 z 2 + 45x2 z 3
9x2 z
44.
3x − z + 5z 2
47. (12x − 9) ÷ (3)
−6c + 1
2+x−y
3by − 4y − 1
43.
2u + 3w
rw − 2r
2y + 5x
35.
5a − b
2uw + 3w2
w
−4x + 5y
14n2 − 7n + 28nx
7n
3m + 16
16.
−x5 − x4
x3
x + 3xy + y
2c − 6
4.
3y 3 − y 2 + 4y
19.
4k 2 + 1
−3u4 + 12u2 + 5
18pr3 − 36p2 r2 − 6pr2
−6pr2
xy − 2
3ab − 2a
−a
2n − 1 + 4x
35bx + 14b2 x − 42bx2
7bx
3y − 2
15.
−x + y
2a3 − 5ab
5 + 7a − a2
41.
24y − 16
8
3x2 − x + 1
14.
−6p2 + 2
37.
−3y + 4
3.
−c + 3d + 9
−2m + 4mr − r
33.
4x + 1
4—Divide.
n4 p3 + 2n3 p3 − 3n2 p4
n 2 p2
n2 p + 2np − 3p2
4x − 3
48. (25a + 5) ÷ (5)
5a + 1
5w + 14
49. (20yz − 80y) ÷ (20)
50. (−11b + 88c) ÷ (−11)
yz − 4y
b − 8c
53. (72a2 − 12a + 12) ÷ (12)
6a2 − a + 1
55. (14w − 35wz + 7z) ÷ (−7)
57. (2xy − 5x) ÷ (−x)
−2y + 5
61. (−a5 − a4 ) ÷ (a3 )
−a2 − a
65. (45n3 + 15n2 ) ÷ (15n2 )
3n + 1
51. (48n2 + 16n) ÷ (16)
−2w + 5wz − z
58. (7mr + 3r2 ) ÷ (r)
3n2
−3x + 4
+n
54. (−24k 3 + 6k 2 − 18k) ÷ (−6)
56. (−9c + 36d + 72) ÷ (9)
59. (k 2 + k) ÷ (k)
k+1
63. (−uw3 − xw2 ) ÷ (−w2 )
−2c + 1
66. (4a6 + 16a4 ) ÷ (4a3 )
4k3 − k 2 + 3k
−c + 4d + 8
60. (ab − ac) ÷ (−a)
−b + c
7m + 3r
62. (2x2 y − x2 ) ÷ (−x2 )
52. (45x2 − 60) ÷ (−15)
uw + x
67. (42w2 − 98w) ÷ (14w)
a3 + 4a
ALG catalog ver. 2.6 – page 89 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
3w − 7
64. (5e5 + e2 ) ÷ (e2 )
5e3 + 1
68. (72a2 − 54ab) ÷ (−6a)
−12a + 9b
EH
69. (u2 w2 − u3 w) ÷ (−uw)
71. (6xy 2 + x2 y 2 ) ÷ (xy 2 )
−uw + u2
6+x
70. (−7cd + 49cd2 ) ÷ (−7cd)
1 − 7d
72. (9m2 p2 − 18m2 ) ÷ (3m2 )
3p2 − 6
73. (22ab3 + 66a2 b2 ) ÷ (11ab2 )
2b + 6a
74. (3k 3 m4 − 9k 3 m3 ) ÷ (3k 2 m3 )
km − 3k
75. (28p2 r3 − 8p2 r) ÷ (−4p2 r)
−7r2 + 2
76. (−2x4 y 2 + 5x2 y 3 ) ÷ (−xy 2 )
2x3 − 5xy
77. (−5x2 + 15xy − 5xy 2 ) ÷ (−5x)
79. (2cx + 7c2 x − c3 x) ÷ (cx)
78. (4ab + a2 b − ab2 ) ÷ (ab)
x − 3y + y 2
80. (63k 2 − 7k + 21kn) ÷ (7k)
2 + 7c − c2
81. (15p2 s2 − 21ps2 − 3ps) ÷ (3ps)
5ps − 7s − 1
83. (32mr + 16m2 r − 48mr2 ) ÷ (8mr)
85. (9a3 c − 36a2 c2 + 81a2 c3 ) ÷ (9a2 c)
4+a−b
4 + 2m − 6r
82. (5xy − 45xy 2 − 5y 2 ) ÷ (−5y)
−x + 9xy + y
84. (8s6 − 20s4 − 14s2 ) ÷ (−2s2 )
−4s4 + 10s2 + 7
86. (4u4 w3 + u3 w3 − 2u2 w4 ) ÷ (u2 w2 )
a − 4c + 9c2
87. (24w3 z 2 − 44w2 z 2 − 4wz 2 ) ÷ (−4wz 2 )
9k − 1 + 3n
−6w2 + 11w + 1
88. (8p2 r + 56p2 r2 + 8pr2 ) ÷ (8pr)
1
− c+1
2
91.
15r2 − 21
15
4u2 w + uw − 2uw2
p + 7pr + r
Mixed expressions
90.
4c − 8
−8
94.
3x2 + 35
15
1 2 7
x +
5
3
95.
−16a + 33
−12
98.
−ab2 + b
−b3
a
1
− 2
b
b
99.
y 3 − 3y
−y 3
102.
15y 3 + 12
6y
5y 2
2
+
2
y
103.
km + 2m
km
106.
cx2 − c2 x
c2 x2
1
1
−
c
x
107.
5
−1
x4
110.
−3a5 + 8a2
−4a7
1
1
+
14a
7b
114.
8p2 s4 + 4p4 s2
12p2 s4
89.
12x + 4y
6
93.
7w − 20
−14
1
10
− w+
2
7
97.
5x2 − x
x2
5−
101.
−8a2 b − 10b
−4a
105.
4s3 + 9s
6s3
109.
25x2 + 5x6
−5x6
113.
a3 b2 + 2a4 b
14a4 b2
2x +
2
y
3
1
x
2ab +
5b
2a
2
3
+ 2
3
2s
−
3
2
− 5
4a2
a
2
p2
+ 2
3
3s
12u3 − 15u2 − 3u
6u2
2u −
118.
x2 y + xy 2 − xy 3
x2 y 2
5
1
−
2
2u
121.
−5c2 d2 − 2c2 d + 10cd2
−10c2 d
123.
92.
14m + 16
−4
7
− m−4
2
11
4
a−
3
4
96.
8y 3 − 15y
10
4 3 3
y − y
5
2
−1 +
3
y2
100.
4c3 + 9c
c2
4c +
1+
2
k
104.
2p − pr2
−pr
−
−ab2 − a3
−a2 b
a
b
+
a
b
108.
20uw + 4w2
8w2
111.
4c2 − 6c5
12c5
1
1
−
3c3
2
112.
7y 4 + 2y 7
3y 6
115.
10x5 y 2 − x2 y 6
5x5 y 2
116.
h2 k 5 − 5h4 k 3
−5h4 k 3
2−
117.
7
5
r2 −
119.
y4
5x3
−16c4 + 12c3 − 3c
−12c
120.
1
4c3
− c2 +
3
4
1
1
y
+ −
y
x
x
d
1
d
+ −
2
5
c
122.
12m2 r2 − 16mr + 32r
−16mr
12a3 b2 − 9a4 b − 15ab
9ab
4a2 b
5
− a3 −
3
3
124.
14x4 y 2 + 7x2 y 4 − 21xy 3
14xy 2
125.
20w4 − 10w2 z 2 + 8z 4
−4w2 z 2
−
126.
14c3 d + 35c2 d3 − 7c2 d4
21c3 d3
127.
x3 y 2 − xyz 2 + x2 y 3 z
x2 y 2 z
x
z
+y
−
z
xy
128.
k 3 mp2 + k 2 p3 + kmp2
k 2 mp2
5w2
5
2z 2
+ − 2
2
z
2
w
−
ALG catalog ver. 2.6 – page 90 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
p
t
+1+
t
p
xy 2
3y
−
2
2
2
5
d
+
−
3d2
3c
3c
k+
5u
1
+
2w
2
2y
7
+
3y 2
3
k2
+1
5h2
3mr
2
+1−
4
m
x3 +
2
+r
r
p3 t2 − p2 t3 − pt4
−p2 t3
−
−
9
c
p
1
+
m
k
EH
129. (18n + 16) ÷ (−8)
130. (12a + 4b) ÷ (6)
9
− n−2
4
2
2a + b
3
10
2
x−
3
7
5e +
2
m
−
145. (−xy 2 − x3 ) ÷ (−x2 y)
2
+s
s
4xy +
−
140. (−xy 2 + y) ÷ (−y 3 )
144. (15a3 + 12) ÷ (9a)
5y
x
5a2
4
+
3
3a
147. (9c3 + 4c) ÷ (6c3 )
148. (by 2 − b2 y) ÷ (b2 y 2 )
1
1
−
b
y
150. (2d4 + 5d7 ) ÷ (3d6 )
7
−1
a4
2
5d
+
3d2
3
3
4
− 5
5x2
x
152. (−3x5 + 20x2 ) ÷ (−5x7 )
b4
5a3
153. (15a5 b2 − a2 b6 ) ÷ (5a5 b2 )
3−
155. (x3 y 2 + 2x4 y) ÷ (16x4 y 2 )
1
1
+
16x
8y
157. (−12h4 + 16h3 − 3h) ÷ (−12h)
159. (12m3 − 15m2 − 3m) ÷ (6m2 )
1
a
3
2
+ 2
2
3c
1
3
−
4n3
8
151. (35a2 + 5a6 ) ÷ (−5a6 )
7−
143. (−8x2 y − 10y) ÷ (−2x)
1
7k
+
2n
2
149. (4n2 − 6n5 ) ÷ (16n5 )
1 2 11
a +
3
5
x
1
− 2
y
y
146. (28kn + 4n2 ) ÷ (8n2 )
y
x
+
x
y
139. (7a2 − a) ÷ (a2 )
154. (u2 w5 − 5u4 w3 ) ÷ (−5u4 w3 )
h3 −
2m −
161. (12x3 y 2 − 9x4 y − 15xy) ÷ (9xy)
156. (4p2 r4 + 6p4 r2 ) ÷ (12p2 r4 )
5
1
−
2
2m
160. (a2 b + ab2 − ab3 ) ÷ (a2 b2 )
1
r
2r
+ −
2
5
m
a
z
−
+ b2
z
ab
−
7u2
7
2w2
+ − 2
w2
2
u
w2
+1
5u2
1
p2
+ 2
3
2r
158. (c3 d2 − c2 d3 − cd4 ) ÷ (−c2 d3 )
4x2 y
5
− x3 −
3
3
167. (28u4 − 14u2 w2 + 8w4 ) ÷ (−4u2 w2 )
−
4h2
1
+
3
4
163. (−5m2 r2 − 2m2 r + 20mr2 ) ÷ (−10m2 r)
165. (a3 b2 − abz 2 + a2 b4 z) ÷ (a2 b2 z)
w
4
+
2
5
136. (5a2 + 33) ÷ (15)
3
e
142. (2p − ps2 ) ÷ (−ps)
141. (mr + 2r) ÷ (mr)
−
1
3
− w+
4
2
138. (5e3 + 3e) ÷ (e2 )
3
c2
132. (5w − 8) ÷ (−10)
7
−
3
135. (4u − 24) ÷ (−16)
4 3 5
b − b
9
6
137. (c3 − 3c) ÷ (−c3 )
1+
h2
134. (8b3 − 15b) ÷ (18)
133. (−14x + 30) ÷ (−21)
−1 +
131. (12h2 − 28) ÷ (12)
−
d
c
+1+
d
c
b
1
1
+ −
b
a
a
162. (14a4 b2 + 7a2 b4 − 21ab3 ) ÷ (14ab2 )
164. (32y 2 z 2 − 16yz + 12z) ÷ (−16yz)
166. (c3 dh2 + c2 h3 + cdh2 ) ÷ (c2 dh2 )
a3 +
−2yz + 1 −
c+
168. (14b3 y + 56b2 y 3 − 7b2 y 4 ) ÷ (21b3 y 3 )
Variable exponents
169.
x3n − x2n + xn+1
xn
171.
a4x+1 + a3x+1 − a2x+3
a2x+1
173.
a5x + a3x+1 − a2x+3
a2x+2
175.
y 4k − y 3k+3 − y k+1
y 3k+1
170.
y a+3 + y 2a + y a
ya
a2x + ax − a2
172.
mk+5 + m2k+1 + mk+2
mk+1
a3x−2 + ax−1 − a
174.
k 3n−1 + k 2n + k n+4
k n+2
176.
x3n+5 + x4n + x2n+1
x3n+1
x2n − xn + x
y k−1 − y 2 −
1
k2
ALG catalog ver. 2.6 – page 91 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
y3 + ya + 1
m4 + mk + m
k 2n−3 + k n−2 + k 2
x4 + xn−1 +
ab2
3b
−
2
2
1
xn
3
4y
h
1
+
d
c
2
8
y
+
−
3y 2
3b
3b
EI
Topic:
Dividing polynomials.
Directions:
6—Simplify. 9—Simplify (assume no denominator equals zero).
76—Find the quotient. 52—Write as a mixed expression.
4—Divide.
1.
(5a + 20) ÷ (a + 4)
5
2.
(9x − 6) ÷ (3x − 2)
3.
(6y − 42) ÷ (y − 7)
6
4.
(21n + 7) ÷ (3n + 1)
5.
(7w2 − 8w) ÷ (7w − 8)
w
6.
(2a3 + a2 ) ÷ (2a + 1)
7.
(5c4 − 3c2 ) ÷ (5c2 − 3)
c2
8.
(9x3 − 7x) ÷ (9x2 − 7)
9.
(3y 3 + 15y 2 ) ÷ (y + 5)
3y 2
10. (8m3 − 8m2 ) ÷ (2m2 − m)
3
7
a2
x
4m
11. (20k 4 + 5k 3 ) ÷ (4k + 1)
5k 3
12. (4u3 + 12u2 ) ÷ (2u + 6)
2u2
13. (x2 − 3x − 4) ÷ (x + 1)
x−4
14. (a2 + 9a − 22) ÷ (a − 2)
a + 11
15. (p2 − 17p + 60) ÷ (p − 5)
p − 12
16. (w2 + 12w + 27) ÷ (w + 3)
17. (2y 2 − y − 21) ÷ (2y − 7)
y+3
18. (5x2 − 59x + 44) ÷ (5x − 4)
19. (7c2 − 26c − 45) ÷ (7c + 9)
20. (8h2 + 19h + 6) ÷ (8h + 3)
c−5
w+9
x − 11
h+2
21. (3m2 + 7m − 20) ÷ (m + 4)
3m − 5
22. (4x2 + 47x + 120) ÷ (x + 8)
4x + 15
23. (5w2 − 38w + 21) ÷ (w − 7)
5w − 3
24. (12a2 − 67a − 30) ÷ (a − 6)
12a + 5
25. (8p2 + 10p + 3) ÷ (2p + 1)
4p + 3
26. (20m2 + 3m − 2) ÷ (5m + 2)
27. (24y 2 − 7y − 6) ÷ (3y − 2)
8y + 3
28. (20x2 − 64x + 35) ÷ (10x − 7)
29. (x3 + 5x2 + 5x − 3) ÷ (x + 3)
30. (a3 + 4a2 − a + 20) ÷ (a + 5)
x2 + 2x − 1
31. (k 3 − 5k 2 − 2k + 16) ÷ (k − 2)
32. (y 3 − 2y 2 − 25y + 6) ÷ (y − 6)
k 2 − 3k − 8
33. (2w3 + 11w2 + 19w + 10) ÷ (2w + 5)
w2 + 3w + 2
4m − 1
2x − 5
a2 − a + 4
y 2 + 4y − 1
34. (3m3 − 5m2 − 26m − 8) ÷ (3m + 1)
35. (6p3 + p2 + 25p − 25) ÷ (6p − 5)
p2 + p + 5
36. (4x3 − 17x2 + 30x − 27) ÷ (4x − 9)
37. (5a3 + 19a2 + 5a + 36) ÷ (a + 4)
5a2 − a + 9
38. (2y 3 + 17y 2 + 15y − 42) ÷ (y + 7)
39. (8x3 − 50x2 + 17x − 30) ÷ (x − 6)
8x2 − 2x + 5
40. (7w3 − 17w2 − 10w − 6) ÷ (w − 3)
m2 − 2m − 8
x2 − 2x + 3
2y 2 + 3y − 6
7w2 + 4w + 2
41. (6x3 + 11x2 − 14x − 10) ÷ (2x + 5)
3x2 − 2x − 2
42. (15m3 + 53m2 + 13m + 10) ÷ (3m + 10)
43. (32a3 + 4a2 − 62a + 21) ÷ (8a − 3)
4a2 + 2a − 7
44. (15y 3 − 7y 2 − 26y − 18) ÷ (5y − 9)
45. (p4 + 5p3 + p2 + 20p − 12) ÷ (p2 + 4)
p2 + 5p − 3
47. (3k 4 + 2k 3 + 4k 2 + 6k − 15) ÷ (k 2 + 3)
3k 2 + 2k − 5
49. (25a4 − 5a3 − 5a2 − 3a − 12) ÷ (5a2 + 3)
5a2 − a − 4
46. (x4 + x3 + 6x2 − 2x − 16) ÷ (x2 − 2)
5m2 + m + 1
3y 2 + 4y + 2
x2 + x + 8
48. (2c4 − 6c3 − 25c2 + 48c + 72) ÷ (c2 − 8)
2c2 − 6c − 9
50. (24w4 + 12w3 − 22w2 − 21w − 35) ÷ (4w2 − 7)
6w2 + 3w + 5
51. (6y 4 − 4y 3 + 31y 2 − 18y + 18) ÷ (2y 2 + 9)
3y 2 − 2y + 2
52. (15m4 − 6m3 − 29m2 + 2m + 8) ÷ (3m2 − 1)
5m2 − 2m − 8
53. (x3 − 7x2 + 10x + 8) ÷ (x2 − 3x − 2)
x−4
55. (2w4 − 7w3 − 13w2 + 29w − 12) ÷ (w2 − 5w + 3)
54. (p3 − 4p2 − p + 10) ÷ (p2 − 2p − 5)
p−2
56. (6a4 + a3 − 31a2 − 17a + 9) ÷ (3a2 + 2a − 1)
2w2 + 3w − 4
ALG catalog ver. 2.6 – page 92 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
2a2 − a − 9
EI
Mixed order
57. (5c − 52 + 2c2 ) ÷ (c − 4)
2c + 13
59. (3x2 − 40 − 19x) ÷ (3x + 5)
x−8
58. (43y − 10 + 9y 2 ) ÷ (y + 5)
9y − 2
60. (6k 2 − 3 + 17k) ÷ (6k − 1)
k+3
61. (28 + 71x + 18x2 ) ÷ (2x + 7)
9x + 4
62. (−90 + 7m + 6m2 ) ÷ (3m − 10)
63. (35 + 32y 2 − 76y) ÷ (8y − 5)
4y − 7
64. (−7 + 30w2 − 29w) ÷ (5w + 1)
65. (2a2 − 17a + a3 + 6) ÷ (a − 3)
p2 + 3p − 3
69. (3x3 − 32 − 14x2 − 48x) ÷ (3x + 4)
x2 − 6x − 8
71. (9m − 5m2 + 3m3 − 24 + m4 ) ÷ (m2 + 3)
6w − 7
66. (−2x2 − 24 + x3 − 14x) ÷ (x + 4)
a2 + 5a − 2
67. (9p + 2p3 − 15 + 11p2 ) ÷ (2p + 5)
2m + 9
m2 + 3m − 8
68. (−2 − 3y + 4y 3 + y 2 ) ÷ (y − 1)
x2 − 2x − 6
4y 2 + 5y − 2
70. (17w2 + 2 + 13w + 10w3 ) ÷ (5w + 1)
2w2 + 3w + 2
72. (18a − 21a2 + 4a4 − 8a3 + 27) ÷ (4a2 − 9)
a2 − 2a − 3
Missing terms
73. (y 2 − 4) ÷ (y − 2)
75. (9h2 − 16) ÷ (3h + 4)
77. (x3 − 27) ÷ (x − 3)
74. (x2 − 25) ÷ (x + 5)
y+2
76. (4d2 − 121) ÷ (2d − 11)
3h − 4
78. (m3 + 125) ÷ (m + 5)
x2 + 3x + 9
79. (8w3 − 343) ÷ (2w − 7)
80. (27y 3 + 64) ÷ (3y + 4)
4w2 + 14w + 49
81. (10a4 + 3a3 + 4a − 3) ÷ (5a2 − a + 3)
83. (3d3 − 5d2 + 16) ÷ (3d + 4)
2a2 + a − 1
d2 − 3d + 4
85. (2w3 − 100w + 14) ÷ (w − 7)
87. (y 3 + 5y 2 − 18) ÷ (y + 3)
x−5
2w2 + 14w − 2
2d + 11
m2 − 5m + 25
9y 2 − 12y + 16
82. (2p3 − 30p − 8) ÷ (2p − 8)
84. (5x3 + 3x − 8) ÷ (x − 1)
86. (a3 + 7a2 − 50) ÷ (a + 5)
p2 + 4p + 1
5x2 + 5x + 8
a2 + 2a − 10
88. (4m3 − 7m + 3) ÷ (2m − 1)
y 2 + 2y − 6
2m2 + m − 3
Remainders
89. (x2 + x − 2) ÷ (x + 3)
90. (w2 + 6w + 2) ÷ (w + 5)
x − 2, r: 4
91. (c2 + 4c − 37) ÷ (c − 4)
c + 8, r: −5
93. (2y 2 − y + 1) ÷ (2y + 3)
y − 1, r: 4
92. (u2 − 5u − 3) ÷ (u − 7)
94. (4a2 + 3a − 7) ÷ (4a − 5)
w + 1, r: −3
u + 2, r: 11
a + 2, r: 3
95. (6p2 + 25p + 13) ÷ (6p + 7)
p + 3, r: −8
96. (5m2 − 29m + 26) ÷ (5m − 9)
m − 4, r: −10
97. (18x2 + 43x − 2) ÷ (9x − 1)
2x + 5, r: 3
98. (12w2 − 13w − 47) ÷ (3w + 5)
4w − 11, r: 8
99. (40y 2 − 17y − 6) ÷ (5y + 1)
8y − 5, r: −1
100. (42a2 − 17a − 11) ÷ (7a + 3)
101. (c3 − 2c2 − 9c + 9) ÷ (c − 4)
c2 + 2c − 1, r: 5
103. (x3 + 9x2 + 20x + 1) ÷ (x + 6)
x2 + 3x + 2, r: −11
105. (2w3 − 2w2 − 17w + 21) ÷ (w − 3)
107. (3a3 + 9a2 − 38a − 42) ÷ (a + 5)
2w2 + 4w − 5, r: 6
3a2 − 6a − 8, r: −2
109. (6x3 + 13x2 − 38x − 27) ÷ (2x + 7)
111. (9y 3 + 36y 2 + 35y + 2) ÷ (3y + 8)
6a − 5, r: 4
102. (k 3 + 3k 2 − 33k − 29) ÷ (k + 7)
104. (y 3 − 8y 2 + 7y + 30) ÷ (y − 5)
k 2 − 4k − 5, r: 6
y 2 − 3y − 8, r: −10
106. (5m3 − 13m2 + 15m − 14) ÷ (m − 2)
108. (4p3 + 7p2 + 2p + 10) ÷ (p + 1)
5m2 − 3m + 9, r: 4
4p2 + 3p − 1, r: 11
3x2 − 4x − 5, r: 8
110. (8n3 − 2n2 + 29n − 20) ÷ (4n − 3)
2n2 + n + 8, r: 4
3y 2 + 4y + 1, r: −6
112. (18a3 − 40a2 + 53a − 7) ÷ (9a − 2)
2a2 − 4a + 5, r: 3
ALG catalog ver. 2.6 – page 93 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
EI
113. (w4 + 5w3 − 2w2 + 10w + 7) ÷ (w2 + 2)
w2
114. (x4 + 8x3 − 10x2 − 48x − 17) ÷ (x2 − 6)
x2 + 8x − 4, r: −7
+ 5w − 4, r: 15
115. (2h4 − 14h3 + 6h2 − 77h − 21) ÷ (2h2 + 11)
h2
116. (6m4 + 12m3 − 13m2 − 20m + 10) ÷ (3m2 − 5)
2m2 + 4m − 1, r: 5
− 7h − 3, r: 12
117. (7y + 6y 2 − 14) ÷ (2y + 5)
118. (20p − 8 + 32p2 ) ÷ (4p − 1)
3y − 4, r: 6
119. (29w2 − 4 + 8w3 − 14w) ÷ (w + 4)
8w2 − 3w − 2, r: 4
121. (5x2 + 2 − 16x + 2x3 ) ÷ (4x − 2 + x2 )
2x − 3, r: −4
123. (10m − 7m2 + 9 + m3 ) ÷ (m2 − 3m − 2)
125. (y 3 − 9) ÷ (y − 2)
m − 4, r: 1
127. (3w3 − 5w2 + 20) ÷ (w2 − 3w + 4)
129. (9a3 − 5a + 3) ÷ (3a2 + a − 1)
3w + 4, r: 4
3a − 1, r: −a + 2
131. (4w3 − 51w − 48) ÷ (2w2 − 6w − 8)
120. (18a + 2a3 − 2 + 13a2 ) ÷ (2a + 5)
2w + 6, r: w
a2 + 4a − 1, r: 3
122. (10b − 5b2 − 5 + 12b3 ) ÷ (3b2 − 2b + 3)
124. (10c − 9c2 + 5 + 2c3 ) ÷ (c2 + 3 − 4c)
126. (a3 + 65) ÷ (a + 4)
y 2 + 2y + 4, r: −1
8p + 7, r: −1
2c − 1, r: 8
a2 − 4a + 16, r: 1
128. (6p3 − 22p + 2) ÷ (2p − 4)
3p2 + 6p + 1, r: 6
130. (4x3 − 8x − 1) ÷ (2x2 + x − 3)
132. (16y 3 + y) ÷ (4y 2 − 2y + 1)
ALG catalog ver. 2.6 – page 94 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
4b + 1, r: −8
2x − 1, r: −x − 4
4y + 2, r: y − 2
EJ
Topic:
Combined methods (polynomials).
Directions:
6—Simplify.
98—Perform the indicated operation(s).
1.
24 − (6 + 7c) + 5c
3.
2x − (−8x + 12) − 18
5.
−x − (5 − x) + 17 − (−x)
7.
−13 + 5a − (7 − a) − (−6a)
9.
(4k − 7) − (k + 3) + (3k + 8)
−2c + 18
10x − 30
x + 12
12a − 20
−6r + 9 − (1 − 10r)
4.
−15 + 4p − (8 + 11p)
6.
4c − (−20) − (−3c + 2) − 8c
8.
2y − (−10y) + 17 − (3y − 12)
4r + 8
−7p − 23
−c + 18
9y + 29
10. −(−14 + 2y) + (3 − 8y) − (1 + 6y)
6k − 2
11. −(11p − 20) + (−2p − 1) − (9p + 4)
2.
−22p + 15
12. (7 − a) − (−12a + 3) + (6 − 5a)
−16y + 16
6a + 10
14. (2x2 − 8) + (−x − 10) − (−x2 ) − (4x2 − 5x)
13. −13ab − (14a + 4b) + (−2b + 6a) − (−6ab)
−7ab − 8a − 6b
−x2 + 4x − 18
15. (y 3 − 2y) − (−5y 2 ) + (−2y 3 + y 2 ) − (−8y)
16. −(9m − r) + (4m + 8) − (−2r) − (−6r + 3)
−5m + 9r + 5
−y 3 + 6y 2 + 6y
17. (3xy − 4xy 2 ) − (2x2 y − xy) + (5x2 y + xy 2 )
18. (−2a + 3b) − (−4ab + 2a − 12b) + (21ab − 15a)
−19a + 15b + 25ab
3x2 y − 3xy 2 + 4xy
19. (14c2 − 2cd) + (8cd + 14) − (−7c2 + cd − 17)
20. (x4 − 5x2 ) − (2x3 − x2 ) + (−x4 − 2x3 + x2 )
21c2 + 5cd + 31
−4x3 − 3x2
21. − [3 − 6n − (−n + 2)]
22. − −7 + 6p2 − (8p2 − 3)
5n − 1
23. − [5ax − (−3 + 2ax) + 12] −3ax − 15
25. 2r2 − −8r2 − (−5r + r2 ) + 3r 11r2 − 8r
27. 11m3 − 10m3 + 8m2 − (m2 − m3 ) − 2m
2
24. − [−3c − (5 − 10c) − 8]
2p2 + 4
−7c + 13
26. 8xy − [9y − (8xy − 6y) − 2y + xy ]
−5m2
29. [−(2cd − d) − (−5d)] − [5cd + (−3d) − (3cd + 7d)]
−4cd + 16d
15xy − 13y
28. 18 − 14 − 6y 2 − (−y 2 − 5) + 6 − 2y 2
30.
7y 2 − 7
−r3 + 2r − (−2r3 ) + 3r − −4r − (4r3 + r)
5r3 + 10r
31. − [−6y − (y + 15) − (−22)] + [6 − (−10 + 3y) + 4y ]
32. − [4k + 3p − (8k − 2p)] + [−(3k − 4p) + 7k − (−p)]
8y + 9
8k
33. −(x2 + 12) − −3x − (−4x2 + 2) + (7x2 − 4x)
34. 8km + 16k − [3m + 24k − (9km − 8k)] − (−2km + 7m)
35. 4c − (−5d + 9) + 2d − [−3d − (7c + 6) − 2d] + 11
36. −2a3 − −(−a2 + 3a) + 5a + 4a2 − (3a3 − 2a2 )
19km − 16k − 10m
2x2 − x − 10
11c + 12d + 8
37. −2(3y − 5) + 12
39. 9a − 7(4 − 7a)
−5a3 − 5a2 − 2a
38. −6(−x + 9) − 6x
−6y + 22
40. 6w − 5(1 + 2w)
58a − 28
41. 8(a − b) − 4(a − b)
43. −2(h − k) − (h + k)
4a − 4b
−3h + k
−54
−4w − 5
42. −7(c + 4) + 2(c − 18)
44. 5(x + 3) − 8(x − 2)
−5c − 64
−3x + 31
45. 5(c2 − 4c) − (3c2 − 10c)
2c2 − 10c
46. 12(2a + b) − 8(3a − b)
47. −3(5y 3 + 4) + 9(y 3 − 2)
−6y 3 − 30
48. −(6kn − n) − 2(3kn + n)
−12kn − n
50. 4y 2 − 14y + 5(−3y 2 + y)
−11y 2 − 9y
49. 14p + 2(r − 7p) − 6r
−4r
ALG catalog ver. 2.6 – page 95 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
20b
EJ
51. 8(3w − 4wz) + 25wz − 12w
52. 2(6a + 15) − 11a + 33
12w − 7wz
53. −7(k 2 m − km + 2) + 2(3k 2 m − km + 7)
55. 2(7a − 3b + 2) − (10a − 6b + 5)
57. 6(p − 4) − (p − 4) − 7(p − 4)
−k 2 m + 5km
58. −(a + 4) − 7(a − 2) + 3(a − 5)
−2p + 8
61. 29 − (−5y 3 ) − 2(3y 3 − 8) − 35
6c + 10k
−19x2 + 3x
65. −3ab − 5(a + 3b) − 7a + 2(6b − ab)
67. 14r − 2(r + 2s) + 6rs + 5(rs + 3r)
−5ab − 12a − 3b
27r − 4s + 11rs
69. 12(2xy 2 − x + 2y 2 ) − 9(xy 2 + 2x − 3y 2 )
−5a − 5
23u + x
5d
3r − 6q − 4
66. −8x2 − 7x + 4(4x − 3) − (x2 − 14)
−9x2 + 9x + 2
68. 2x2 y + 11(x2 − y) − (x2 y + 8y) − 9x2
x2 y + 2x2 − 19y
70. −7(3mr − 8m + 4) + 4(10mr + 2m − 5)
19mr + 64m − 48
71. −3(k 2 + 11k − 10) + 5(2k 2 + 5k − 6)
73. −6(x2 − 2) − 2x(3x + 5)
7k 2 − 8k
76. −3p(5p − 4) + 7(p2 − 2)
y 2 − 18y + 5
78. c3 (4c − 1) − 4c2 (c2 − 6c)
3y 3 + y 2 + 4y
79. −x2 (x3 − 3x) + x3 (4x − 3)
81. −p(4p2 − 2p − 5) + 4(p3 + 3p2 − p + 1)
83. −7(x4 − 3x2 + x) + 4x(2x3 + 3x − 3)
14p2 + p + 4
x4 + 33x2 − 19x
87. −5y(4y 2 + y) + 2y 2 (2y 2 + 10y − 3)
−8p2 + 12p − 14
−25c3
4y 4 − 11y 2
89. 5a2 (a + 4) − 2a(6a2 − 3a + 7) + 7(a3 − 2)
2m4 + 7m3 + 6m
82. 2y(2y 2 − 7y + 3) − 6(y 3 + 4y 2 − 2y)
84. 2(a3 + 6a + 5) − 4a(2a2 − a + 5)
86. 2p2 (2p − 4) − 4p(p2 − 2p + 3)
3x3 + 9x
−8c2 + 50c − 22
10a2 − 21a + 6
80. 6m(2m2 + 1) + m2 (2m2 − 5m)
−x5 + 4x4
85. 3x(x2 + 2x − 5) − 6x(x − 4)
72. 10(c2 + 2c − 1) − 6(3c2 − 5c + 2)
74. 3(a2 − 4a + 2) + a(7a − 9)
−12x2 − 10x + 12
75. y(6y − 8) − 5(y 2 + 2y − 1)
26a2
−14y 2 + y + 13
64. −13 − (−5r) − 16q − 2(r − 5q) + 9
15xy 2 − 30x + 51y 2
77. y 2 (3y − 1) + 2y(y + 2)
14rs + 21r + 38s
60. 10(u − x) − (−u + x) + 12(u + x)
62. 32d − 4(c + 8d) − (−4c) + 5d
−y 3 + 10
63. −3x2 − 3(6x2 + x) + 6x − (−2x2 )
54. 9(2rs + r + 2s) − 4(rs − 3r − 5s)
56. −5(2y 2 − y − 1) − 4(y 2 + y − 2)
4a − 1
59. −2(c − k) + 5(c + k) − 3(−c − k)
a + 63
−2y 3 − 38y 2 + 18y
−6a3 + 4a2 − 8a + 10
−12p
88. −8a(a2 − a + 3) + 6(−3a2 + 5a)
−8a3 − 10a2 + 6a
90. 6x2 (3x − 4) + 5x(3x2 + 6x − 5) − 6(x2 − 4x)
33x3 − x
− 14a − 14
91. 3m2 (m − 4) + 6m(3m2 − m + 2) − 4(3m − 7)
92. 8s2 (s + 5) − 2s(4s2 − 11) − 9(3s2 − 2s + 1)
21m3 − 18m2 + 28
13s2 + 40s − 9
93. a2 b2 (2a2 − 3ab + b) + ab(a3 b + 3a2 b2 − 2ab2 )
3a4 b2
94. 2c2 d(cd − 3d + d2 ) − cd2 (c2 − 6c + 2cd)
c3 d2
− a2 b3
95. xy 3 (x2 y + 2xy 2 + 2y) + 2x2 y(xy 3 − y 4 + xy)
96. mr2 (3m3 r + 6r3 − 2mr4 ) − 3mr(mr3 + 2r4 − mr5 )
3x3 y 4 + 2xy 4 + 2x3 y 2
3m4 r3 + m2 r6
97. a(b − 3) − b(a + 3) + 3(a + b)
98. −w(w + 4) + 2(w2 − 3) − w(w3 − w)
0
−w4 + 2w2 − 4w − 6
99. c2 (c2 − c) − 5(c3 − c2 ) + c(c2 + c)
2
3
101. 5(2r − ) − 8(r − )
5
4
c4 − 5c3 + 6c2
1
1
102. 3(4a2 + ) − 2(5a2 − )
6
4
2r + 4
5
2
103. −9( y 2 − y) + 10(y 2 − y)
3
2
100. −x(xy + y) + y(x − y) − xy(x + 2)
4y 2 − 16y
3
1
104. −4(3k + ) + 6(2k + )
8
12
ALG catalog ver. 2.6 – page 96 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
2a2 + 1
−1
−2x2 y − 2xy − y 2
EJ
1
5
105. − (y + 18) + (y − 12)
6
6
2
3y
107.
2
7
(cd + 20) − (cd + 5)
5
5
109.
1
1
(9m3 − 6r) + 10( m3 − r)
3
5
111.
1
1
(18r2 − 3) + 4(r2 + )
6
8
3
1
106. − (x − 6y) + (x + 18y)
2
2
− 13
108.
−cd + 1
5m3 − 12r
7r2
2
4
113. − (9ab − 21a − 3b) − a(10b + 30)
3
5
115.
1
3
1
(2k 2 + ) + (50k 2 + 10k − 5)
2
3
10
117. 0.25(16x − 48) − 0.1(10x − 20)
−14ab − 10a + 2b
8k2 + k
3a + 4
121. 2.5(m2 − 6) + 0.25(2m2 + 4m + 50)
3m2 + m − 2.5
123. 0.5(a2 b − 2ab + 8b) − 0.2a(7.5ab − 5b)
125. (4a + 1)(3a − 2) + (a − 2)(a + 2)
127. (c + 4)2 + (2c + 1)(c − 5)
129. (2p − 5)2 − (p + 4)2
133. 3x(x − y) − (x + y)2
−a2 b + 4b
13a2 − 5a − 6
3c2 − c + 11
3p2 − 28p + 9
131. (2x − 1)(2x + 1) − (2x + 3)2
−12x − 10
2x2 − 5xy − y 2
135. (4a − b)(a + 4b) − (2a + b)2
137.
5y(2y − 4) + (y − 2)(4y + 7)
7
139.
3(5b2 + 7ab) − 5b(7ab + 3b)
7b
11ab − 5b2
2y 2 − 3y − 2
3a − 5ab
2 2
4
(n − 15n) + (n2 + 9n)
3
3
2n2 + 2n
2
5
110. 5(2p − ) − (12p − 4)
5
4
−5p + 3
1
3
112. 6( n + 2) − (6n + 10)
2
2
−6n − 3
114.
1
3
3
(6x2 − 20x + 1) − (3x2 + )
4
2
2
116.
5
1
(x + 6y) + (x2 − 18xy + 24y)
6
6
118. −0.5(7y − 2z) + 1.5(3y − 5z)
3x − 10
119. 0.2(30a2 + 50a) − 0.3(10a2 + 20a)
−x + 18y
3x2 − 15x
x2 + 2xy + 4y
y − 6.5z
120. −1.5(12wx − 3) + 0.25(8wx − 2)
16wx + 3
122. −1.1c(2d + 20) + 0.4(10c2 − 2cd + 15c)
124. 1.6(5r2 + 50r − 3) − 2.1(5r2 − 8)
−1.5r2 + 80r + 12
126. (w − 6)(w + 6) + (5w − 1)(2w + 3)
128. (y + 8)(3y − 2) + (y − 4)2
130. (a − 10)2 − (4a + 5)2
11w2 + 13w − 39
4y 2 + 14y
−15a2 − 60a + 75
132. (3r + 4)2 − (4r + 5)(4r − 5)
134. (c + d)(c − d) + (c + 2d)2
−7r2 + 24r + 41
2c2 + 4cd + 3d2
136. 4s(p + s) + (p − 2s)(p + 2s)
p2 + 4ps
138.
(2x − 3)(x + 4) + 3x(6x − 7)
4
140.
6r(4m − 5r) − 3m(8r − 11mr)
3r
ALG catalog ver. 2.6 – page 97 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−3cd + 4c2 − 16c
5x2 − 4x − 3
−10r + 11m2
EK
Topic:
Applications of polynomials. See also category DN (monomials).
Directions:
0—(No explicit directions.)
49—Write each answer as a polynomial in simplest form.
1
4 x − 5.
1.
The side of a square is 3n − 2r. What is the
perimeter? 12n − 8r
2.
The side of a square is
perimeter? x − 20
3.
The length and width of a rectangle are 3k + 1
and k + 12 . What is the perimeter? 8k + 3
4.
The length and width of a rectangle are
and 32 c + d. What is the perimeter? 4c
5.
The side of a square is 7y − 1. What is the area?
6.
The side of a square is
49y 2 − 14y + 1
1 2
4x
+ xy
1
2 x + y.
What is the
4
3c − d
What is the area?
+ y2
7.
Find the area of a rectangle whose dimensions are
k − 2 and k − 3. k2 − 5k + 6
8.
Find the area of a rectangle whose dimensions are
3x + 5 and 2x − 1. 6x2 + 7x − 5
9.
The length and width of a rectangle are a + b
and a − b. What is the area and perimeter?
10. The length and width of a rectangle are 2n − 3
and 2n + 3. What is the area and perimeter?
a2 − b2 ; 4a
4n2 − 9; 16n
11. The length and width of a rectangle are 3a − b
and a − 3b. What is the area and perimeter?
12. The length and width of a rectangle are x + 2y
and x + y. What is the area and perimeter?
3a2 − 10ab + 3b2 ; 8a − 8b
x2 + 3xy + 2y 2 ; 4x + 6y
13. The perimeter of a rectangle is 4c + 10d and the
width is 5d. What is the length? 2c
14. The perimeter of a rectangle is 8k + 12 and the
length is 4k − 1. What is the width? 7
15. The perimeter of a rectangle is 3x + 3y and the
width is x + 12 y. What is the length? 1 x + y
16. The perimeter of a rectangle is 12w + 8 and the
length is 4w − 5. What is the width? 2w + 9
17. The area of a rectangle is 3y 2 + 6y and the length
is 3y. What is the width? y + 2
18. The area of a rectangle is 8a2 + 4a and the width
is 4a. What is the length? 2a + 1
19. The area of a rectangle is c2 − 3c − 18 and the length
is c + 3. What is the width? c − 6
20. The area of a rectangle is n2 + 5n − 14 and the
width is n − 2. What is the length? n + 7
21. The area of a rectangle is 3x2 + 13x + 4 and the
length is 3x + 1. What is the width? x + 4
22. The area of a rectangle is 2y 2 − 13y + 6 and the
width is y − 6. What is the length? 2y + 1
23. The area of a rectangle is 2a2 + 11a − 21 and the
length is 2a − 3. What is the width? a + 7
24. The area of a rectangle is 5n2 + 34n − 7 and the
width is n + 7. What is the length? 5n − 1
25. The sides of a triangle are 5, 2k − 3 and 4k. What
is the perimeter? 6k + 2
26. The sides of a triangle are 2a, a − 2 and a − 1.
What is the perimeter? 4a − 3
27. The sides of a triangle are 2x + 5, x − 4 and x − 1.
What is the perimeter? 4x
28. The sides of a triangle are y + 3, 3y + 1 and y − 6.
What is the perimeter? 5y − 3
29. The perimeter of a triangle is 6a + 1. Two sides
are 3a and a + 5. What is the other side? 2a − 4
30. The perimeter of a triangle is 7y − 3z. Two sides
are 4y + 2z and 7z. What is the other side? 3y − 12z
31. The perimeter of a triangle is 6x + 5. Two sides
are 2x − 1 and 4x. What is the other side? 6
32. The perimeter of a triangle is 5k + 4m. Two sides
are k and 3k + 4m. What is the other side? k
33. The perimeter of an isosceles triangle is 9y − 15, and
the base is 5y + 1. What is the length of each side?
34. The perimeter of an isosceles triangle is 12a + 8b,
and the base is 2a − 4b. What is the length of each
side? 5a + 6b
2
2y − 8
ALG catalog ver. 2.6 – page 98 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
EK
35. The perimeter of an isosceles triangle is 7w − 20, and
each side is 2w − 10. What is the length of the base?
36. The perimeter of an isosceles triangle is 10x + 9, and
each side is 5x − 3. What is the length of the base?
3w
12
37. The first side of a triangle is 8 less than the second.
The third side is 9 more than the second side. What
is the perimeter? 3x + 1, where x is second side
38. The second side of a triangle is 15 more than the
first. The third side is twice the first side. What is
the perimeter? 4x + 15, where x is first side
39. The first side of a triangle is twice the second. The
second is 1 less than the third side. What is the
perimeter? 4x − 3, where x is third side
40. The second side of a triangle is 10 less than the first.
The first is 6 more than the third side. What is the
perimeter? 3x + 2, where x is third side
41. The third side of a triangle is 5 less than twice the
second. The second is 3 more than the first side.
What is the perimeter? 4x + 4, where x is first side
42. The third side of a triangle is four times the second.
The second is 1 less than twice the first side. What
is the perimeter? 11x − 5, where x is first side
43. The third side of a triangle is 12 more than the first,
and five times longer than the second side. What is
the perimeter? 11x − 12, where x is second side
44. The third side of a triangle is half of the second, and
6 more than the first side. What is the perimeter?
45. Find the volume of a rectangular solid whose
dimensions are y − 4, 3y, and y + 4. 3y3 − 48y
46. Find the volume of a rectangular solid whose
dimensions are 2x, 5x, and 2x + 1. 20x3 + 10x
47. Find the volume of a rectangular solid whose
dimensions are r − 2, r + 1, and r + 2. r3 + r2 − 4r − 4
48. Find the volume of a rectangular solid whose
dimensions are 3k, k + 6, and k − 2. 3k3 + 12k2 − 36k
49. In a rectangle, the width is half of the length.
A 4 mm wide strip is cut from all sides. What is the
new perimeter? 3` − 32
50. The width of a rectangle is 8 cm less than the length.
A strip that is 3 cm wide is cut from two adjacent
sides. What is the new perimeter? 4` − 28
51. In a rectangle, the length is three times the width.
A 2 cm wide strip is cut from all sides. What is the
new area? 3w2 − 16w + 16
52. The length of a rectangle is 10 mm more than the
width. A strip that is 5 mm wide is cut from two
adjacent sides. What is the new area? w2 − 25
53. The side of a square is 8. A strip of uniform width
is cut from opposite sides. What is the new area and
perimeter? 64 − 8w; 32 − 2w
54. The side of a square is 12. A strip of uniform width
is added to three sides. What is the new area and
perimeter? 144 + 36w + 2w2 ; 48 + 6w
55. A rectangular piece of carpet is 8 m × 5 m. A strip of
uniform width is cut from two adjacent sides. What
is the new area and perimeter? 40 − 13w + w2 ; 26 − 4w
56. A rectangular photograph is 10 cm × 8 cm. A border
of uniform width is added to all sides. What is the
new area and perimeter? 80 + 36w + 4w2 ; 36 + 8w
57. The length of a rectangle is 5 more than twice the
width. It is made into a box by cutting out four
squares from the corners. The side of each square
is 2. What is the volume of the box? 4w2 − 14w − 8
58. The length of a rectangle is 3 less than twice the
width. It is made into a box by cutting out four
squares from the corners. The side of each square
is 1. What is the volume of the box? 4w2 − 18w + 20
59. A rectangular piece of cardboard has dimensions
12 × 5 cm. The cardboard is folded into box by
cutting squares from the corners. The side of each
square is p. What is the volume of the box?
60. A rectangular piece of cardboard has dimensions
10 × 8 cm. The cardboard is folded into box by
cutting squares from the corners. The side of each
square is r. What is the volume of the box?
2x − 6, where x is second side
60p − 34p2 + 4p3
ALG catalog ver. 2.6 – page 99 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
80r − 36r2 + 4r3
EL
Topic:
Mixed practice with polynomials.
Directions:
0—(No explicit directions.)
The expression 2x3 − 5x2 + x − 1 has how many
terms? 4
2.
3.
Tell how many terms are in the expression
4a(bc)2 − abc. 2
4.
Tell how many terms are in the expression
8n5 − 4n3
. 1
2
5.
Give the degree of the polynomial 5x3 y − xy 5 .
6.
Give the degree of the polynomial 3r8 − 2r4 + 1.
7.
The polynomial (3x2 y 3 − 7x5 y 2 + 8y 6 ) is what
degree? 6
8.
The polynomial (2a − 3ab − 5b + 6) is what degree?
Find the sum of 11x + y − 3z and 8y + 4z − 10x.
10. Find the sum of 3c3 − c + 4 and −2c2 + c − 10.
1.
9.
5
The expression 5xy + x + 2 has how many terms?
3
8
1
3c3 − 2c2 − 6
x + 9y + z
11. Find the sum of 8x2 − 5x4 − 3 and −x2 − x + 4x4 + 7.
12. Find the sum of 2r3 + r + 9 and r3 + 4r2 − r.
−x4 + 7x2 − x + 4
3r3 + 4r2 + 9
13. Add: (2 − a3 − 3a) + (4a2 + a − 3)
−a3 + 4a2 − 2a − 1
15. Add: (−n2 + 7n3 − 3) + (6n4 + 3n2 − 7)
14. Add: (−yz − 4z) + (4y − 3yz + 5z)
−4yz + 4y + z
16. Add: (xyz − yz + 2xz) + (xy − xyz + 3yz)
2yz + 3xz
6n4 + 7n3 + 2n2 − 10
17. Subtract 2a2 − 3 from a2 − 3.
18. Subtract −n + 3p from 4n + p.
−a2
19. Subtract x2 + 2x − 3 from −3x2 − x + 5.
−4x2
5n − 2p
20. Subtract 4y 3 + 2y 2 + y from 5y 3 − y.
y 3 − 2y 2 − 2y
− 3x + 8
21. Subtract: (−4wz + 6w − z) − (4w + 6wz + z)
22. Subtract: (4a − b + c) − (a + b + 3c)
3a − 2b − 2c
−10wz + 2w − 2z
23. Subtract: (3k 3 + 2k + 1) − (2k 3 + 2k − 1)
k3 + 2
25. Find the product of f + 1 and f 2 + 4f − 2.
f3
+ 5f 2
24. Subtract: (n3 + 3n2 − 5n) − (3n2 − 5)
n3
26. Find the product of m − 3 and m2 − 3m + 4.
m3 − 6m2 + 13m − 12
+ 2f − 2
27. Find the product of a + b and a2 − ab + b2 .
a3 + b3
28. Find the product of 6h − k and 6h + k + 3.
36h2 − k2 + 18h − 3k
29. Multiply: (4p − 3r)(4p + 3r)
31. Multiply: (x + y)(4x + 1)
30. Multiply: (2k 2 + 5)(2k 2 − 5)
16p2 − 9r2
32. Multiply: (2x + 5)(3x − 7)
4x2 + 4xy + x + y
33. Divide: (m3 − 6m2 + 13m − 12) ÷ (m − 3)
m2
4k 4 − 25
6x2 + x − 35
34. Divide: (y 5 + 1) ÷ (y + 1)
y4 − y3 + y2 − y + 1
36. Divide: 4x − 7 16x2 − 49
4x + 7
− 3m + 4
35. Divide: 3n − 2 3n2 − 8n + 4
n−2
37. Find the remainder of (2n3 − 5n2 − n) ÷ (n − 4).
38. Find the remainder of (r2 − 5r − 7) ÷ (r + 1).
−1
44
39. Find the remainder of (8y 3 − 27) ÷ (2y − 3).
0
40. Find the remainder of (2x2 + 11x − 18) ÷ (2x − 3).
3
ALG catalog ver. 2.6 – page 100 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
EM
Topic:
3 dimensional figures (see also category HI).
Directions:
171—Write a polynomial for the volume of the figure.
172—Write a polynomial for the surface area of the figure.
173—Write polynomials for the volume and surface area of the figure.
1.
2.
4.
3DFIG02.PCX
3DFIG01.PCX
7.
6.
3DFIG04.PCX
3DFIG03.PCX
A = 22x + 144, V = 28x + 112
A = 36x + 18, V = 36x − 18
A = 24x + 70, V = 35x
A = 48x + 24, V = 36x
5.
3.
8.
3DFIG07.PCX
3DFIG06.PCX
3DFIG05.PCX
A = 4x2 + 10x − 12,
V = 6x2 − 12x
A = 14x2 + 16x, V = 3x3 + 6x2
10.
9.
3DFIG08.PCX
A = 6x2 + 8x − 6,
V = x3 + 2x2 − 3x
11.
A = 2x2 + 24x + 6,
V = 5x2 + 10x + 5
12.
3DFIG10.PCX
3DFIG09.PCX
3DFIG11.PCX
A = 6x + 52, V = 2x + 24
A = 36x + 60, V = 40x
13.
14.
15.
3DFIG16.PCX
3DFIG15.PCX
A = 26x + 168, V = 24x + 32
18.
A = 40x + 94, V = 55x + 35
16.
3DFIG14.PCX
3DFIG13.PCX
A = 20x + 94, V = 25x + 60
17.
3DFIG12.PCX
A = 24x + 160, V = 32x + 112
A = 26x + 156, V = 30x + 80
A = 26x + 204, V = 36x + 108
19.
20.
3DFIG20.PCX
3DFIG17.PCX
A = 42x + 36, V = 36x
3DFIG19.PCX
A = 32x + 82, V = 30x + 30
3DFIG18.PCX
A = 24x + 12, V = 12x
A = 22x + 216, V = 24x + 160
21.
22.
24.
23.
3DFIG24.PCX
3DFIG21.PCX
A = 2x2 + 40x + 96,
V = 6x2 + 33x + 45
A = 2x2 + 30x + 42,
V = 7x2 + 7x
3DFIG22.PCX
A = 2x2 + 36x + 120,
V = 6x2 + 36x + 72
3DFIG23.PCX
A = 4x2 + 36x + 30,
V = 10x2 + 15x
ALG catalog ver. 2.6 – page 101 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FA
Topic:
Averages.
Directions:
67—Find the average.
1.
−8, 22
5.
−10, −25, 6
9.
−8, −31, 0, −3
7
−9 23
2.
0, −9
6.
−12, 21, −9
−4 12
0
10. −6, 6, 12, −20
−10 12
13. −17, 17, −15, 9, −9
−21, −13
7.
−4, 28, 0
−17
8
11. −34, −16, 25, 25
−2
14. 34, −13, 6, −7, 0
3.
3
−3
17. 8 35 , 5 52
8.
−8, −12, −15
0
11 23
1
4
16. 11, −25, 9, −13, −12
−6
7
3
21. − 19
5 , 10
1 43
25. 4, − 12 , −6 12
29.
16, −16
12. 19, −5, 8, −21
0
15. −50, 17, 13, 15, 20
4
4.
1 12
19. − 43 , − 10
3
22. − 34 , − 58
− 11
16
23. 3 12 , −8
26.
−1
3
1 7
4 , − 2 , 4 , −2
18. −4 12 , 7 21
4
1
1
3, −5, −3
27. − 92 , 0, 3
4
− 15
30. −3 12 , − 23 , 3 12 , 16
0
20. −3 14 , 1 14
−2 13
24. 0, −6 32
2 14
−3 13
28. − 23 , −6, −5 13
− 12
31. 5 13 , −8 13 , −2 21 , − 12
1
8
−1
32. 5, 35 , − 12 , −5
−4
1
40
−1 12
33. −6.4, 4.0
34. −2.9, 8.1
−1.2
37. 0.02, −0.8, −0.42
−0.4
35. −3.6, −7.4
2.6
38. 4.1, −7.6, −3.1
−2.2
−5.5
39. −0.05, −0.15, −0.4
−0.2
41. −3.5, −0.5, 1.9, 2.1
0
42. −0.06, 0.1, 0.9, −0.34
0.15
45. a2 , 5a2
49. −3y, 8y
46. −x, 3x
3a2
50. −4c3 , −7c3
5
2y
53. −9c, 2c, −5c
−4c
43. 3.5, −0.11, −2.5, −0.09
51. x, −10x
54. 4w, 18w, −4w
−6u2
44. −3.2, −8, 3.6, −2.4
48. 3p, −11p
52. 5ab, 2ab
− 92 x
55. 2k, −11k, −15k, 4k
6w
40. −2.01, 2.02, −0.1
−2.5
47. −2u2 , −10u2
3
− 11
2 c
1.9
−0.03
0.2
x
36. 4.3, −0.5
−4p
7
2 ab
56. −10r, −14r, 25r, −r
0
−5k
57. 4p − 10, 6p
5p − 5
61. −4n, 2n + 1, 2n − 7
−2
58. −3y + 5z, 9y − 5z
62. 8, p2 − 5, 2p2 − 6
p2 − 1
3y
59. −7a, −a + 4
−4a + 2
63. m + 3n, −4m − n, −8n
−m − 2n
ALG catalog ver. 2.6 – page 102 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
60. −5x2 + 8, 5x2 − 2
3
64. x − y, x + y, −5x
−x
FB
Topic:
Percents I.
Directions:
0—(No explicit directions.)
1.
What is 50% of 24?
2.
12
5.
Find 10% of 10.
What is 50% of 90?
3.
Find 25% of 204.
7.
What is 75% of 56?
51
6.
1
Find 10% of 500.
50
42
9.
What is 33 31 % of 48?
16
10. What is 33 13 % of 195?
11. Find 66 23 % of 78.
52
32.5
17. What is 75% of 94?
70.5
14. Find 25% of 70.
17.5
18. What is 75% of 250?
22. Find 66 23 % of 8.
5 13
13 13
15. What is 50% of 5?
30
26.5
19. Find 10% of 145.
14.5
23. What is 33 13 % of 52?
450
33. What is 12 21 % of 160?
20
26. What is 20% of 115?
27. Find 60% of 45.
27
30. Find 70% of 30.
21
31. What is 30% of 330?
24. What is 33 13 % of 74?
28. Find 80% of 205.
164
34. What is 62 12 % of 72?
45
3
38. Find 55% of 240.
41. What is 400% of 8?
32
32. What is 30% of 90?
27
35. Find 37 12 % of 800.
36. Find 87 12 % of 104.
91
300
132
39. What is 45% of 120?
54
42. What is 1000% of 14?
40. What is 85% of 60?
51
43. Find 500% of 19.
95
44. Find 300% of 6.
18
140
45. Find 150% of 102.
46. Find 125% of 44.
55
153
47. What is 110% of 60?
66
49. What is 4% of 700?
28
50. What is 1% of 1200?
48. What is 250% of 300?
750
51. Find 2% of 3000.
60
52. Find 8% of 650.
52
12
47
54. Find 18% of 150.
27
55. What is 72% of 75?
54
57. What is 37.5% of 88?
33
58. What is 87.5% of 16?
14
61. What is 40% of 2.5?
1
62. What is 500% of 1.6?
8
4.5
66. Find 25% of 3.24.
0.81
3
70. What is 10.5% of 400?
42
73. What is 0.05% of 600?
26.84
0.8
24 23
99
77. Find 88% of 30.5.
20. Find 10% of 8.
23
29. Find 90% of 500.
0.3
12. Find 66 23 % of 150.
16. What is 50% of 53?
17 13
25. What is 40% of 75?
69. Find 7.5% of 40.
What is 75% of 16?
187.5
21. Find 66 23 % of 20.
65. Find 150% of 3.
8.
8
100
2.5
53. Find 94% of 50.
Find 25% of 32.
12
65
13. Find 25% of 130.
37. Find 15% of 20.
4.
45
74. What is 6.5% of 900?
56. What is 56% of 125?
70
59. Find 12.5% of 120.
60. Find 62.5% of 48.
63. What is 800% of 4.5?
36
64. What is 80% of 12.5?
10
67. Find 75% of 50.8.
68. Find 125% of 6.
13.64
7.5
38.1
71. What is 0.3% of 2000?
6
75. Find 2.5% of 90.
72. Find 0.04% of 2500.
1
2.25
76. Find 0.6% of 60.
58.5
78. Find 110% of 12.4.
30
15
79. What is 88% of 4.75?
4.18
ALG catalog ver. 2.6 – page 103 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
80. What is 124%
of 19.25? 23.87
0.36
FC
Topic:
Percents II.
Directions:
0—(No explicit directions.)
1.
50% of what number
is 26? 52
2.
50% of what number
is 85? 170
3.
25% of what number
is 14? 56
4.
25% of what number
is 110? 440
5.
10% of what number
is 12? 120
6.
10% of what number
is 101? 1010
7.
75% of what number
is 33? 44
8.
75% of what number
is 90? 120
9.
33 31 % of what number
is 240? 720
10. 33 13 % of what number
is 15? 45
11. 66 23 % of what number
is 82? 123
12. 66 32 % of what number
is 48? 72
13. 40% of what number
is 60? 150
14. 20% of what number
is 24? 120
15. 60% of what number
is 207? 345
16. 80% of what number
is 300? 375
17. 90% of what number
is 81? 90
18. 70% of what number
is 350? 500
19. 30% of what number
is 9? 30
20. 30% of what number
is 21? 70
21. 12 21 % of what number
is 11? 88
22. 37 12 % of what number
is 30? 80
23. 67 23 % of what number
is 76? 114
24. 87 21 % of what number
is 105? 120
25. 400% of what number
is 20? 5
26. 1000% of what number
is 140? 14
27. 500% of what number
is 95? 19
28. 300% of what number
is 63? 21
29. 15% of what number
is 6? 40
30. 55% of what number
is 22? 40
31. 45% of what number
is 27? 60
32. 85% of what number
is 102? 120
33. 150% of what number
is 210? 140
34. 125% of what number
is 190? 152
35. 110% of what number
is 121? 110
36. 250% of what number
is 80? 32
37. 4% of what number
is 8? 200
38. 1% of what number
is 16? 1600
39. 2% of what number
is 1? 50
40. 8% of what number
is 22? 275
41. 25% of what number
is 6.5? 26
42. 50% of what number
is 20.5? 41
43. 75% of what number
is 22.5? 30
44. 75% of what number
is 16.5? 22
45. 75% of what number
is 6.3? 8.4
46. 25% of what number
is 0.01? 0.04
47. 50% of what number
is 1.8? 3.6
48. 10% of what number
is 0.55? 5.5
49. 66 32 % of what number
is 5? 7.5
50. 66 23 % of what number
is 0.2? 0.3
51. 33 13 % of what number
is 0.05? 0.15
52. 33 31 % of what number
is 2.1? 6.3
53. 3% of what number
is 5.4? 180
54. 6% of what number
is 0.36? 6
55. 5% of what number
is 0.6? 12
56. 9% of what number
is 1.71? 19
57. 800% of what number
is 1000? 125
58. 200% of what number
is 146? 73
59. 250% of what number
is 50? 20
60. 125% of what number
is 7.5? 6
61. 12 12 % of what number
is 7.2? 57.6
62. 37 12 % of what number
is 13.2? 35.2
63. 62 12 % of what number
is 0.4? 0.64
64. 87 12 % of what number
is 1.4? 1.6
65. 7.5% of what number
is 4.5? 60
66. 2.5% of what number
is 18? 720
67. 10.5% of what number
is 21? 200
68. 6.5% of what number
is 0.78? 12
69. 0.3% of what number
is 9? 3000
70. 0.05% of what number
is 0.01? 20
71. 0.6% of what number
is 0.3? 50
72. 0.04% of what number
is 3.2? 8000
ALG catalog ver. 2.6 – page 104 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FD
Topic:
Percents III.
Directions:
0—(No explicit directions.)
1.
What percent of 140
is 70? 50%
2.
What percent of 90
is 45? 50%
3.
What percent of 50
is 35? 70%
4.
What percent of 200
is 120? 60%
5.
What percent of 60
is 45? 75%
6.
What percent of 80
is 60? 75%
7.
What percent of 56
is 14? 25%
8.
What percent of 252
is 63? 25%
9.
What percent of 90
is 27? 30%
10. What percent of 80
is 72? 90%
11. What percent of 80
is 52? 65%
12. What percent of 600
is 330? 55%
13. What percent of 84
is 56? 66 2 %
14. What percent of 66
is 44? 66 2 %
15. What percent of 246
is 82? 33 1 %
16. What percent of 501
is 167? 33 1 %
17. What percent of 56
is 21? 37.5%
18. What percent of 96
is 60? 62.5%
19. What percent of 210
is 134.4? 64%
20. What percent of 140
is 134.4? 96%
21. What percent of 125
is 60? 48%
22. What percent of 75
is 33? 44%
23. What percent of 12
is 30? 250%
24. What percent of 16
is 20? 125%
25. What percent of 11
is 110? 1000%
26. What percent of 5
is 100? 2000%
27. What percent of 150
is 6? 4%
28. What percent of 150
is 9? 6%
29. What percent of 12
is 7.2? 60%
30. What percent of 18
is 13.5? 75%
31. What percent of 40
is 1.2? 3%
32. What percent of 30
is 0.3? 1%
33. What percent of 1000
is 2? 0.2%
34. What percent of 200
is 5? 2.5%
35. What percent of 20
is 0.7? 3.5%
36. What percent of 24
is 0.06? 0.25%
37. 30 is what percent
of 120? 25%
38. 55 is what percent
of 220? 25%
39. 6 is what percent
of 30? 20%
40. 11 is what percent
of 110? 10%
41. 18 is what percent
of 36? 50%
42. 62 is what percent
of 124? 50%
43. 144 is what percent
of 192? 75%
44. 66 is what percent
of 88? 75%
45. 45 is what percent
of 300? 15%
46. 133 is what percent
of 140? 95%
47. 32 is what percent
of 40? 80%
48. 48 is what percent
of 120? 40%
49. 24 is what percent
of 72? 33 1 %
50. 36 is what percent
of 108? 33 1 %
51. 168 is what percent
of 252? 66 2 %
52. 274 is what percent
of 411? 66 2 %
53. 49.6 is what percent
of 310? 16%
54. 46.8 is what percent
of 130? 36%
55. 13 is what percent
of 104? 12.5%
56. 77 is what percent
of 88? 87.5%
57. 25 is what percent
of 5? 500%
58. 30 is what percent
of 10? 300%
59. 198 is what percent
of 275? 72%
60. 132 is what percent
of 550? 24%
61. 1 is what percent
of 50? 2%
62. 4 is what percent
of 80? 5%
63. 95 is what percent
of 20? 475%
64. 60 is what percent
of 8? 750%
65. 1.4 is what percent
of 35? 4%
66. 1.8 is what percent
of 30? 6%
67. 1.2 is what percent
of 6? 20%
68. 6.3 is what percent
of 42? 15%
69. 0.09 is what percent
of 75? 0.12%
70. 0.4 is what percent
of 25? 1.6%
71. 8 is what percent
of 500? 1.6%
72. 12 is what percent
of 3000? 0.4%
3
3
3
3
3
3
ALG catalog ver. 2.6 – page 105 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
3
3
FE
Topic:
Applications of averages.
Directions:
0—(No explicit directions.)
1.
In the games so far this season, a soccer team has
scored 3, 0, 5, 3, 2, 4, 1 and 2 goals. What is the
average number of goals per game? 2.5
2.
At a regional basketball tournament, the team scored
48, 59, 54, 43 and 48 points in its games. On
average, what were the team’s points per game?
50.4
3.
The members of the varsity wrestling team weigh
75.3, 81, 58.2, 64.4, and 85.6 kg. What is the average
weight of the wrestlers? 72.9 kg
4.
There are several bike paths at Balboa Park. Their
distances are 24, 21.9, 14.5 and 8 kilometers. What
is the average distance? 17.1 km
5.
On the girl’s basketball team, the starting players
are 6 ft-1 in., 6 ft, 5 ft-8 in., 5 ft-7 in., and 5 ft-5 in.
What is their average height? 5 ft-9 in.
6.
When shopping, Martin weighed some 5-pound bags
of potatoes, and found their actual weights to be
5 lb-4 oz, 5 lb-9 oz, 6 lb, and 5 lb-3 oz. What was the
average weight? 5.5 lb or 5 lb-8 oz
7.
A pair of blue jeans costs $23.70 at Men’s Fashion
Store, $29.45 at Pants Emporium and $34.99 at
Clothes-n-More. What is the average price of jeans
at those stores? $29.38
8.
One liter of soda costs $1.15 at John’s Grocery, $1.19
at Midtown Market, $1.39 at the convenience store,
and $0.99 at the Warehouse Supermarket. What is
the average price of the soda? $1.18
9.
In a football game, the fullback gained 8 yards,
22 yards, 2 yards, 6 yards, and 10 yards on various
running plays. He also had 0 yards (no gain) on a
play, and losses of 3 yards and 1 yard. What was his
average per play? 5.5 yds
10. The stock market can be highly volatile. In
one week, the market closed up 22 points, down
3 12 points, down 18 14 points, up 11 points, and up
5 43 points. What was the average close during that
week? up 3.4 points
11. U.S. Foreign Assistance is measured in millions of
dollars of new credits. These were the changes in
recent years: up 302 in 1986, down 2,657 in 1987,
up 195 in 1988, and down 296 in 1989. What was
the average change during those years? −614
12. New home construction tends to go in cycles.
Construction once fell by 298,000 (1970–1975), rose
by 142,000 (1975–1980), rose by 432,000 (1980–1985),
and fell by 136,000 (1985–1990). What was the
average change? +35, 000
13. During the week in Nome, Alaska, the low
temperatures were 2, −16, −28, −21, −10, 3, and
7 degrees Fahrenheit. What was the average low
temperature? −9
14. During the week in Goose Bay, Newfoundland, the
high temperatures were 1, 3, −4, 3, −7, −11, and
−6 degrees. What was the average high temperature?
15. A saleswoman drove her car 562 km one week.
During the next two weeks, she drove a total of
1220 km. The week after that she drove 316 km. On
average, how much did she drive per week? 524.5 km
16. 485 people went to the theater on opening night, and
441 the second night. Then a total of 730 people
went on the next two nights. What was the average
attendance per night? 414
17. When selling newspaper subscriptions, Jaime earned
$45 the first week, $80 the next two weeks, nothing
the following week, and then $55 the last week of
the subscription drive. On average, how much did he
earn per week? $36
18. The Serv-U-Rite Computer Store, in its opening
month, sold $35,000 worth of computers. It sold
$58,000 in the next two months, and $78,000 in the
three months after that. On average, how much did
the store sell per month? $28,500
19. In Mr. Wilkes’ history class, eight students scored a
92 on the last test, ten students scored an 86, four
students scored an 82, and three students scored
a 72. What was the average score in the class? 85.6
20. According to company records, one employee missed
6 days of work in April, eight missed 2 days,
twelve missed 1 day, and four missed no days. What
was the average number of days missed by an
employee during April? 1.36
−3
ALG catalog ver. 2.6 – page 106 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FF
Topic:
Applications of percent. See also category FJ (interest and investment).
Directions:
0—(No explicit directions.)
1.
In a survey of 1240 people, 12 21 % of them expressed
no opinion. How many people expressed no opinion?
In an election, the winner received 62% of the 850
votes that were cast. How many votes did the
winner receive? 527
2.
3.
There are 26,000 employees at a large corporation,
and 0.75% of them deal with quality control. How
many employees deal with quality control? 195
4.
Of the 45,360 commuters who go across a bridge
each day, only 2 21 % are in car pools. Find out how
many commuters are in the car pools. 1134
5.
In one month, a stereo store sold $17,500 worth of
VCR’s, which was 28% of its total sales. Find the
total sales for the month. $62,500
6.
A movie theater makes 65% of its profit from the
snack bar. What was the total profit during a month
when $3510 was made from the snack bar? $5400
7.
In a recent survey, 63% of the people said they
support the new governor. Find out how many
people were surveyed, if 252 of them expressed
support. 400
8.
In a class election, the second place finisher got 129,
or 32%, of the votes. Find out how many people
voted. 403 (rounded)
9.
Last week Tony spent $41.15 on compact discs. That
was 55% of what he earned on his part-time job.
What did he earn? $74.82 (rounded)
10. The Torres’ food budget each month is $540, which
is 15% of their household income. What is their
household income? $3,600/month
11. A bowl of Mega-brand Cereal provides 2.6 g of
protein, which is 4% of the recommended daily
allowance (RDA). What is the RDA of protein?
65 g
155
12. An 8-oz. serving of Flavortime Ice Cream contains
176 mg, or 22%, of the recommended daily allowance
(RDA) of calcium. What is the RDA of calcium?
800 mg
13. During the softball season, Jeane reached first base
45 times. If she had 144 “at bats,” what percent of
the time did she reach first base? 31.25%
14. During the baseketball season, Guy Johnson made
119 free throws out of 136 opportunities. What
percent of his free throws were made? 87.5%
15. On 51 of the last 75 business days, the stock market
either went down or did not change. What percent
of the time did the market go up? 32%
16. In the last 90 days, the high temperature was normal
or above normal 63 times. What percent of the time
was the temperature below normal? 30
17. 32 men and 50 women are enrolled in the afternoon
swimming class. What percent of the total are
women? ≈ 61%
18. The police department in small town employs 45 men
and 19 women. What percent of the total are men?
19. A gumball machine contains 93 blue, 51 green and
128 red gum balls. Find the percent of the gum balls
that are green. 18.75%
20. An airplane has 18 first class, 52 business class, and
170 economy class seats. What percent of the seats
are first class? 7.5%
21. The speedometer in a car may be inaccurate by as
much as 5%. If a speedometer reads 70 km/hr, by
how many kilometers per hour could it be wrong?
22. A 2 × 4 piece of wood may be off by as much as
0.5% from its specified length. If a 2 × 4 is labeled
as being 10 feet long, by how many inches could it
be off? 0.6 in.
3.5 km/hr
≈ 70%
23. Carl made 325 tee shirts for the baseball world
series. The first week he sold 48% of them. The
next week he sold 25% of the remainder. How many
did he sell in the two-week period? 198
24. Darlene’s Donut Shop made 850 glazed donuts on
Monday. 80% of them were sold that day. 60% of
the remainder were sold on Tuesday. How many
donuts were left after Tuesday? 68
25. Last year Francis spent 32% of his income on rent.
What was his rent per month if his yearly income
was $28,500? $760
26. Last year Julia used 8% of her income to make car
payments. What was her car payment each month if
her yearly income was $35,850? $239
ALG catalog ver. 2.6 – page 107 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FF
27. “Ice-man” Dan scored 35% of the points on his
hockey team. What was his average per game if the
team scored 110 times in a 24-game season? 1.75
28. “Super” Mario scored 25% of the points on his water
polo team. What was his average per game if the
team scored 243 points in a 30-game season? 2.025
29. In one week, Ms. Sanders drove her car 416 miles,
and 75% of those miles came from her daily commute
(between her home and office). Find the approximate
distance between her home and office, assuming she
went back and forth 5 times during the week.
30. In the 30 shopping days before Christmas, a toy
store does about 60% of its yearly business. What
is the average amount of business the store does on
each of those days, if its total sales for the year is
$472,000? $9440
31.2 mi
Sales and commission
31. The regular price of a tool kit is $49.99. If the tool
kit is marked “15% off,” what is the selling price?
32. A car stereo that normally goes for $199 is on sale
for 25% off. What is the sale price? $149.25
$42.50 (rounded)
33. Mr. Koch spent $6.99 on a can of paint that was on
sale for 30% off. What would the paint have cost if
it were not on sale? $9.98 (rounded)
34. Consuelo paid $16.72 for an item which was
discounted 24%. What was the price of the item
before it was discounted? $22
35. Melanie bought a coat that was marked down 20%.
What was the original cost of the coat, if she
saved $8.00? $40
36. Mr. Sheridan bought a new suit that was on sale for
35% off. As a result, he saved $79. What was the
original price of the suit? $220
37. One night at Mel’s Diner, Alice received $31.50
in tips. If her customers’ bills totaled $226, what
percent did she receive in tips? ≈ 14%
38. A certain microwave oven sells for $405. The store
will charge an additional $25.30 for sales tax. What
percent of the selling price is the sales tax? ≈ 6.25%
39. A furniture set that was originally priced at $850 is
now selling for $748. By what percent was the price
reduced? 12%
40. A carton of eggs costs $1.09 without a coupon and
79/
c with a coupon. By what percent does the
coupon lower the price? ≈ 27.5%
41. At the appliance store, a saleswoman gets 5%
commission on everything she sells. How much would
she have to sell to get $750 in commission? $15,000
42. At the shoe store, the employees earn 4.5%
commission on their sales. What amount of sales
would be needed for an employee to earn $180 in
commission? $4,000
43. The publisher of Maria’s book is giving her $18,000,
plus 0.5% of sales. How much will have to be made
in sales, in order for Maria to get a total of $25,000?
44. Mrs. Kim is a real estate broker. Her salary is $1650
per month, plus 0.25% of sales. To earn $4000 in a
month, how much does she need to sell? $940,000
$1.4 million
45. Assume you are the manager of a convenience store,
and the local sales tax is 5.5%. At the end of the
day, you find receipts totalling $2310. What amount
of that is sales tax? $120.43 (rounded)
46. At a local bank, the service charge on traveler’s
checks is 0.75%. On a very busy day, sales of
traveler’s checks totalled $34,758.75. What amount
of that was service charges? $258.75
“Facts and figures”
47. Alaska has 31.7 million acres of national park land.
The total national park land in the United States is
47.4 million acres. What percent is in Alaska?
≈ 67%
49. The land area of the the U.S. is 3.6 million sq mi, or
6.2% of the total land area on earth. What is the
total land area on earth? ≈ 58 million sq mi
48. New Hampshire has 13 miles of coastline, the least
amount of any state that borders an ocean. If the
U.S. has 12,383 miles of coastline, what percent is in
New Hampshire? ≈ 0.1%
50. 15.4 million sq km, or 41%, of the moon’s surface can
be never seen from earth. What is the total surface
area of the moon? ≈ 37.6 million sq km
ALG catalog ver. 2.6 – page 108 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FF
51. In the U.S., 118,000 women and 534,000 men are
employed as lawyers and judges. What percent of
the total are women? ≈ 18.1%
52. Each day, the average American throws away 0.35 lbs
of garbage that are eventually recycled and 2.93 lbs
that are not. What percent of the total is recycled?
≈ 10.7%
53. North America has 5.2 percent of the world’s
population. If there are 5,320 million people in the
world, how many live in North America? ≈ 277 million
54. Of the 10.6 million cars sold in the U.S. in 1988,
29.2% were imported. What was the number of
imported cars sold that year? ≈ 3.1 million
55. In 1988, airplanes handled only 0.3% of all domestic
freight in the U.S. If the total amount of freight
(measured in ton-miles) was 3,100 billion, how much
was handled by airplanes? 9.3 billion ton-miles
56. In 1989, women in executive and managerial
positions earned 70.4% as much as men in similar
positions. If the average weekly pay for men was
$488, what was the average pay for women?
$343.55 (rounded)
57. New York City is about 44% further from Honolulu,
Hawaii, than it is from London, England. If the air
distance between New York and London is 3450 mi,
what is it between New York and Honolulu?
≈ 4968 mi
58. O’Hare Airport in Chicago is the world’s
busiest. In 1989, it handled 59.1 million
passengers—24% more than the second busiest
airport, Dallas/Ft. Worth. How many passengers
were handled at the second busiest airport?
≈ 47.7 million
59. Between the years 1850 and 1950, the world’s
population grew by 130 percent. If the population
was 1.1 billion in 1850, what was it a hundred years
later? 2.53 billion
60. In the U.S., total passenger car mileage increased
30% between 1980 and 1988. If 1,110 billion miles
were driven in 1980, what was the total mileage in
1988? 1,433 billion
61. World oil production increased by 5.5 percent
between 1989 and 1990. If 22.6 billion barrels were
produced in 1990, how much oil was produced the
previous year? ≈ 21.4 bb
62. Monthly automobile production increased by 40%
in the U.S. between 1982 and 1988. 592,000
automobiles were made each month in 1988. How
many were made each month in 1982? ≈ 425,000
63. In 1950, the gross national product of the U.S. was
288.3 billion dollars. In 1989, it was 5,234 billion
dollars. Ignoring inflation, what was the percent
increase? ≈ 1715%
64. When measured in constant dollars, the gross
national product of the U.S. was 1,204 billion in 1950
and 4,144 billion in 1989. What was the percent
increase? ≈ 244%
65. 3.43 billion barrels of oil were produced in the U.S.
and Canada in 1989, and 3.24 billion barrels in 1990.
What was the percent decrease in oil production?
66. Per capita energy consumption in the U.S. was
equivalent to 10,386 kg of coal in 1980 and 9,542 kg
in 1987. What was the percent decrease in energy
consumption? ≈ 8.1%
≈ 5.5%
67. The average person watched television for 6 34 hours
per day in 1980 and 7 hours in 1990. What percent
change does that represent? ≈ 3.7%
68. World population increased from about 4.48 billion
in 1980 to 5.32 billion in 1990. What percent change
does that represent? ≈ 18.75%
ALG catalog ver. 2.6 – page 109 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FG
Topic:
Explicit formulas. See also categories MK (square roots) and NH (quadratic formula).
Directions:
0—(No explicit directions.)
1.
Use the formula P = 2(` + w) to find the perimeter
of a rectangle whose length ` = 9 14 ft and width
w = 5 12 ft. 29 1 ft
2.
Use the formula P = 2(` + w) to find the perimeter
of a rectangle whose length ` = 20.2 mm and width
w = 13.6 mm. 66.6 mm2
The perimeter of a rectangle is given by the formula
P = 2` + 2w. Find the perimeter if the length
` = 32 in. and the width w = 49 in. 2 2 sq in.
4.
The perimeter of a rectangle is given by the formula
P = 2` + 2w. Find the perimeter if the length
` = 1.4 m and the width w = 0.35 m. 3.5 m2
The formula S = 6d 2 gives the surface area of a
cube (d is the length of an edge). Find S when
d = 4.5 cm. 121.5 cm2
6.
The formula S = 6d 2 gives the surface area of a
cube (d is the length of an edge). Find S when
d = 10 12 inches. 661 1 sq in.
The formula A = πr2 gives the area of a circle (r is
radius and π ≈ 3.14). Find the area if r = 0.5 km.
8.
2
3.
9
5.
7.
0.785 sq km
9.
The area of a triangle is given by the formula
A = 21 bh. Find the area if the base b = 12.4 and the
height h = 7.5. 46.5
2
The formula A = πr2 gives the area of a circle (r is
4
radius and π ≈ 22
7 ). Find the area if r = 2 5 mi.
431 15 sq mi
10. The area of a triangle is given by the formula
A = 21 bh. Find the area if the base b = 5 34 and the
height h = 3 32 . 7 2
3
11. The area of a trapezoid is given by the formula
A = 21 (a + b)h. Find the area if the height h = 16,
base a = 13.5, and base b = 22.5. 288
12. The area of a trapezoid is given by the formula
A = 21 (a + b)h. Find the area if the height h = 5,
base a = 6.4, and base b = 8. 36
13. The area of a pentagon can be approximated by the
formula A = 1.720s2 , where s is the length of a side.
Find the area of a pentagon whose side is 15 cm.
14. The area of a pentagon can be approximated by the
formula A = 1.720s2 , where s is the length of a side.
Find the area of a pentagon whose side is 8 inches.
387 cm2
110.08 sq in.
15. The formula A = 2.598s2 gives the approximate area
of a hexagon (s is the length of a side). If s = 2 ft,
what is the area? 10.392 sq ft
16. The formula A = 2.598s2 gives the approximate area
of a hexagon (s is the length of a side). If s = 5 m,
what is the area? 64.95 m2
17. The formula A = 4.828s2 gives the approximate area
of an octagon (s is the length of a side). Find the
area of an octagon whose side is 20 cm. 1931.2 cm2
18. The formula A = 4.828s2 gives the approximate area
of an octagon (s is the length of a side). Find the
area of an octagon whose side is 100 inches.
48280 sq in.
19. In a regular n-sided polygon, the measure of each
angle (in degrees) is given by the formula:
180(n − 2)
.
n
What is the measure of each angle in an octagon?
d=
135 ◦
20. In a regular n-sided polygon, the measure of each
angle (in degrees) is given by the formula:
180(n − 2)
.
n
What is the measure of each angle in a 12-sided
polygon? 150 ◦
d=
21. The surface area of a rectangular solid is given by
the formula S = 2(`w + d` + wd). Find the surface
area if the length ` = 6, the width w = 2, and the
depth d = 3. 72
22. The surface area of a rectangular solid is given by
the formula S = 2(`w + d` + wd). Find the surface
area if the length ` = 5, the width w = 10, and the
depth d = 3.2. 196
23. The formula V = πr2 ` is used to find the volume of
a cylinder. What is the volume, if the length ` = 1 m
and the radius r = 7 cm? ≈ 15386 cm3
24. The formula V = πr2 ` is used to find the volume of a
cylinder. What is the volume, if the length ` = 2 feet
and the radius r = 4 inches? ≈ 1206 cu in.
ALG catalog ver. 2.6 – page 110 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FG
25. The volume of a cone is given by the formula
V = 13 πr2 h (r is radius and h is height). What is the
volume of a cone whose radius is 15 cm and height
is 18 cm? ≈ 4239 cm3
26. The volume of a cone is given by the formula
V = 13 πr2 h (r is radius and h is height). What is
the volume of a cone whose radius is 4 inches and
height is 6 inches? ≈ 100.5 cu in.
27. Use the formula V = 34 πr3 to find the volume of
sphere whose radius r = 10 inches. ≈ 4187 cu in.
28. Use the formula V = 34 πr3 to find the volume of
sphere whose radius r = 2.6 m. ≈ 73.6 m3
Interest rates
29. The formula I = prt is used to calculate simple
interest (p is the principal amount deposited, r is the
annual rate, and t is the time in years). Find the
interest on $1000 deposited for 3 years at an annual
rate of 5.5%. $165
30. The formula I = prt is used to calculate simple
interest (p is the principal amount deposited, r is
the annual rate, and t is the time in years). Find
the interest on $15,000 deposited for 1 21 years at an
annual rate of 8%. $1880
31. The formula A = p (1 + rt) gives the total amount
of an investment (or loan) with simple interest.
Find the amount A if the principal p is $12,500, the
annual rate r is 7.5%, and the time t is 4 years.
32. The formula A = p (1 + rt) gives the total amount of
an investment (or loan) with simple interest. Find
the amount A if the principal p is $800, the annual
rate r is 4.75%, and the time t is 10 years. $1180
$16250
33. The amount of an investment or loan after one year
is given by the formula:
34. The amount of an investment or loan after one year
is given by the formula:
A = p (1 + rt )t ,
A = p (1 + rt )t ,
where p is the original amount of money, r is the
annual interest rate, and t is the number of times
interest is compounded during the year. Find A if
p = 2000, r = 0.055 and t = 4. $2112.29
where p is the original amount of money, r is the
annual interest rate, and t is the number of times
interest is compounded during the year. Find A if
p = 20000, r = 0.18 and t = 12. $23912.36
35. When interest is compounded quarterly, the total
amount of an investment (or loan) is given by the
formula:
36. When interest is compounded weekly, the total
amount of an investment (or loan) is given by the
formula:
A = p (1 + 4r )4t .
Find the total amount A if the principal p is
1200 dollars, the annual interest rate r is 10 percent,
and the time t is 3 years. $1613.87
37. Assume the interest on a certificate of deposit is
compounded monthly. Use the formula
A = p (1 +
A = p (1 +
Find the total amount A if the principal p is
500 dollars, the annual interest rate r is 12 percent,
and the time t is 12 year. $530.88
38. Assume the interest on a credit card is compounded
daily. Use the formula
r t
12 )
to find the amount that the certificate is worth after
two years, if the principal p is $10,000, the annual
interest rate r is 8.5%, and the time t is 24 (the
number of months the certificate is held). $11,845.95
r 52t
52 ) .
A = p (1 +
r t
365 )
to find the total amount owed after ten weeks, if
the principal p is $2000, the annual interest rate r
is 18%, and the time t is 70 (the number of days the
money is owed). $2070.23
Science and engineering
39. The formula C = 95 (F − 32) is used to convert
temperatures from Fahrenheit to Celsius. How many
degrees Celsius is 167 ◦ F ? 75 ◦ C
40. The formula C = 95 (F − 32) is used to convert
temperatures from Celsius to Fahrenheit. How many
degrees Fahrenheit is −25 ◦ C ? −13 ◦ F
41. Use the formula C = 95 (F − 32) to convert −4 degrees
Fahrenheit to Celsius. −20 ◦ C
42. Use the formula F = 95 C + 32 to convert 70 degrees
Celsius to Fahrenheit. 158 ◦ F
ALG catalog ver. 2.6 – page 111 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FG
43. Air temperature affects the speed of sound.
The relationship is shown in the formula
S = 331.5 + 0.61t , where t is measured in degrees
Celsius, and S in meters/sec. Find the speed of
sound when t = 10 ◦, then convert your answer
to km/hr (1 m/s ≈ 3.6 km/hr). 337.6 m/s; ≈ 1216 km/hr
44. Air temperature affects the speed of sound.
The relationship is shown in the formula
S = 1053.5 + 1.14t , where t is measured in degrees
Fahrenheit, and S in ft/sec. Find the speed of sound
when t = 60 ◦, then convert your answer to miles per
hour (1 ft/sec ≈ 0.68 mph). 1121.9 ft/sec; ≈ 763 mph
45. In an electrical circuit, the total resistance of two
separate, parallel resistors can be calculated using
the formula:
46. In an electrical circuit, the total resistance of two
separate, parallel resistors can be calculated using
the formula:
RT =
R1 · R2
.
R1 + R2
Find RT , if R1 = 1.5 ohms and R2 = 4.5 ohms.
1.125 ohms
47. The current in an electrical circuit is given by the
formula:
V
,
R + 2r
where I is current (amperes), V is potential energy
(volts), R is circuit resistance (ohms), and r is cell
or battery resistance. Find I if V = 22, R = 1.3 and
r = 0.05. ≈ 15.7 amps
I=
49. Air friction is related to an object’s speed as well as
its shape. For a reasonably aerodynamic object, such
as a baseball, air friction can be calculated with the
formula F = 0.0064cv 2 , where F is the amount of
friction in pounds, v is velocity in mph, and c is the
“drag” coefficient (about 1 for a baseball). Use the
formula to find the air friction on a tennis ball, if its
velocity is 110 mph and its drag coefficient is 1.2.
≈ 93 lbs
RT =
R1 · R2
.
R1 + R2
Find RT , if R1 = 11 ohms and R2 = 5.5 ohms.
3.67 ohms (rounded)
48. The current in an electrical circuit is given by the
formula:
V
,
R + 2r
where I is current (amperes), V is potential energy
(volts), R is circuit resistance (ohms), and r is cell
or battery resistance. Find I if V = 12.6, R = 1.05
and r = 0.35. 7.2 amps
I=
50. Air friction is related to an object’s speed as well as
its shape. For a reasonably aerodynamic object, such
as a baseball, air friction can be calculated with the
formula F = 0.0064cv 2 , where F is the amount of
friction in pounds, v is velocity in mph, and c is the
“drag” coefficient (about 1 for a baseball). Use the
formula to find the air friction on a golf ball, if its
velocity is 80 mph and its drag coefficient is 1.15.
≈ 47 lbs
51. Air friction is related to an object’s speed as well
as its shape. For an object whose front end is
somewhat flat, air friction can be calculated with
the formula F = 0.1008Av 2 , where F is the amount
of friction in pounds, A is the cross-sectional area
in sq ft, and v is velocity in mph. Use the formula to
find the air friction on a car, if its velocity is 50 mph
and its front area is 20 sq ft. 5040 lbs
52. Air friction is related to an object’s speed as well as
its shape. For an object whose front end is somewhat
flat, air friction can be calculated with the formula
F = 0.1008Av 2 , where F is the amount of friction
in pounds, A is the cross-sectional area in sq ft, and
v is velocity in mph. Use the formula to find the air
friction on a school bus, if its velocity is 45 mph and
its front area is 40 sq ft. ≈ 8165 lbs
53. The formula h = v 2 /2g can be used to find the
maximum height of an object that is shot upward
(if air friction is ignored). In the formula, h is the
maximum height in feet, v is the initial velocity in
ft/sec, and g is the “deceleration” due to gravity
(g = 32 ft/sec2 on earth). Find the maximum height
of a bullet shot upward at 800 ft/sec. 10,000 ft
54. The formula h = v 2 /2g can be used to find the
maximum height of an object that is shot upward
(if air friction is ignored). In the formula, h is the
maximum height in meters, v is the initial velocity
in m/sec, and g is the “deceleration” due to gravity
(g = 9.8 m/sec2 on earth). Find the maximum height
of an arrow shot upward at 65 m/sec. ≈ 216 m
55. The resistance of a wire (measured in ohms) can be
calculated with the formula R = ρ`/A, where ` is the
length (cm), A is the cross-sectional area (cm2 ), and
ρ is a constant based on the type of metal. Find the
resistance of 5 m of copper wire, if A = 0.0007 sq cm
and ρ = 1.7 × 10−6 . ≈ 1.2 ohms
56. The resistance of a wire (measured in ohms) can be
calculated with the formula R = ρ`/A, where ` is
the length (cm), A is the cross-sectional area (cm2 ),
and ρ is a constant based on the type of metal.
Find the resistance of 1 m of aluminum wire, if
A = 0.0013 sq cm and ρ = 2.1 × 10−5 . ≈ 1.6 ohms
ALG catalog ver. 2.6 – page 112 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FG
57. The horse power of an automobile can be
approximated by the formula P = d 2 N/f , where
N is the number of cylinders, d is the diameter of
each (inches), and f is an efficiency rating, usually
between 0.35 and 0.65. Find the horse power when
N = 6, d = 3.5 and f = 0.505. ≈ 145 h.p.
58. The horse power of an automobile can be
approximated by the formula P = d 2 N/f , where
N is the number of cylinders, d is the diameter of
each (inches), and f is an efficiency rating, usually
between 0.35 and 0.65. Find the horse power when
N = 4, d = 2.5 and f = 0.48. ≈ 52 h.p.
59. When a heavy object is suspended by a metal
wire, the wire stretches a little bit. The amount of
stretching can be approximated by the formula:
60. When a heavy object is suspended by a metal
wire, the wire stretches a little bit. The amount of
stretching can be approximated by the formula:
4w`
,
πd 2 E
where w is the weight of the object, ` is the original
length of wire, d is the diameter of the wire, and
E is a coefficient which depends on the type of
metal. Find out how much 10 inches of steel wire
will stretch if it is holding up 75 pounds, its diameter
is 0.03 in., and E = 1.5 × 107 . ≈ 0.07 in.
4w`
,
πd 2 E
where w is the weight of the object, ` is the original
length of wire, d is the diameter of the wire, and
E is a coefficient which depends on the type of
metal. Find out how much 60 cm. of copper wire
will stretch if it is holding up 100 kg, its diameter is
0.04 cm, and E = 2.5 × 107 . ≈ 0.2 cm
61. The efficiency of an automobile drops rapidly
at higher speeds. According to the formula
F = N/e0.15k , an automobile with normal efficiency N
at 50 mph will have efficiency F at higher speeds.
In the formula, e is 2.72 and k is the number of
multiples of 10 mph above 50 (for example, when a
car is going 70 mph, k = 2). Find the efficiency of an
automobile going 80 mph, if N = 24 miles/gal.
62. The efficiency of an automobile drops rapidly
at higher speeds. According to the formula
F = N/e0.094k , an automobile with normal
efficiency N at 80 km/hr will have efficiency F at
higher speeds. In the formula, e is 2.72 and k is
the number of multiples of 10 km/hr above 80 (for
example, when a car is going 90 km/hr, k = 1). Find
the efficiency of an automobile going 120 km/hr, if
N = 11 km/liter. ≈ 7.6 km/`
s=
≈ 15.3 mpg
63. Heat transfer, which is the flow of heat from a
warmer object to a cooler object, is represented by
the formula:
k(t2 − t1 )
.
d
In the formula, Q is the rate of transfer (measured
in calories per second), d is the thickness of the wall
or space separating the objects, t1 and t2 are the two
temperatures, and k, which is called “specific heat,”
depends on the materials involved. Find the heat
transfer between two liquids whose temperatures
are 25 ◦ and 15 ◦, if k = 0.65 and d = 0.1 cm.
Q=
65 cal./sec
s=
64. Heat transfer, which is the flow of heat from a
warmer object to a cooler object, is represented by
the formula:
k(t2 − t1 )
.
d
In the formula, Q is the rate of transfer (measured
in calories per second), d is the thickness of the wall
or space separating the objects, t1 and t2 are the
two temperatures, and k, which is called “specific
heat,” depends on the materials involved. Find
the heat transfer between two metal plates whose
temperatures are 102 ◦ and 77 ◦, if k = 0.32 and
d = 0.05 cm. 160 cal./sec
ALG catalog ver. 2.6 – page 113 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
Q=
FH
Topic:
Area, perimeter and volume. See also categories DN and EK (applications of monomials and polynomials),
and HI (word problems for first degree equations).
Directions:
0—(No explicit directions.)
1.
The side of a square is 14 12 m. What is the
perimeter? 58 m
2.
The side of a square is 7.5 inches. What is the
perimeter? 30 in.
3.
What is the perimeter of a square building whose
sides are 81.6 feet? 326.2 ft
4.
What is the perimeter of a square table whose sides
are 1 34 meters? 7 m
5.
The side of a square is 1.8 inches. What is the area?
6.
The side of a square is 5 12 feet. What is the area?
3.24 sq in.
30 14 sq ft
7.
What is the area of a city block (assuming it is
square), if one side is 210 m? 44100 m2
9.
The side of a square is
and area? 3 1 ft; 25 sq ft
3
5
6
foot. What is the perimeter
36
11. Mr. Ramirez is going to re-seed his lawn and put a
fence around it. The lawn is square and one side is
22 feet. What is the perimeter and area of the lawn?
88 ft; 484 sq ft
8.
What is the area of a square picture frame, if one
side measures 7.5 cm? 56.25 cm2
10. The side of a square is 1.5 km. What is the perimeter
and area? 6 km; 2.25 km2
12. A woman needs to carpet the floor and put trim
around the walls of her office. The office is square
and one side is 8.5 m. What is the perimeter and
floor area? 34 m; 72.25 m2
13. Find the perimeter of a rectangle whose dimensions
are 5.7 × 9.3 cm. 30 cm
14. Find the perimeter of a rectangle whose dimensions
are 3 14 00 × 2 14 00 . 11 in.
15. A rectangular horse stable is 10 12 by 12 meters.
What is the perimeter? 45 m
16. A rectangular piece of property is 108 by 128 feet.
What is the perimeter? 472 ft
17. What is the area of a rectangle which is 3 13 by
2 12 inches? 8 1 sq in.
18. What is the area of a rectangle which is 7.5 by
5.4 centimeters? 40.5 cm2
19. Maureen is going to hang up a sign inside the school
auditorium. The sign is rectangular and measures
4 12 × 12 ft. What is its area? 54 sq ft
20. Mrs. Flecher wants more ceiling insulation in her
house. What is the area of her attic, if it is
rectangular and measures 18.5 × 24.4 m? 451.4 m2
21. The length and width of a rectangle are 1.4 m
and 0.8 m. What is its perimeter and area?
22. The length and width of a rectangle are 6 ft
and 9 34 ft. What is its perimeter and area?
3
4.4 m; 1.12 m2
31 12 ft; 58 21 sq ft
23. The health club is getting a new swimming pool,
with colored tiles along the edges. The pool is
rectangular and measures 35 0 × 75 0 . What is the
area of the pool, and what is the distance around
the edges? 2625 sq ft; 220 ft
24. Katie’s grandmother is going to make a quilt
bedspread with lace trim around the edges. The quilt
will be rectangular and measure 255 cm by 300 cm.
What is the area of the quilt, and how much trim is
required? 1110 cm; 67500 cm2
25. The perimeter of a rectangle is 22 m and the width
is 3.5 m. What is the length? 7.5 m
26. The perimeter of a rectangle is 155 inches and the
length is 50 inches. What is the width? 27.5 in.
27. The perimeter of a rectangle is 0.72 cm. Find the
width if the length is 0.3 cm. 0.06 cm
28. The perimeter of a rectangle is 32 ft. Find the length
if the width is 7.9 ft. 8.1 ft
29. The area of a rectangle is 4.2 cm2 . Find the length if
the width is 1.5 cm. 2.8 cm
30. The area of a rectangle is 97.5 sq. inches. Find the
width if the length is 15 inches. 6.5 in.
31. What is the length of a rectangle, if the area is
306 sq. feet and the width is 15 feet? 20.4 ft
32. What is the width of a rectangle, if the area is 0.4 m2
and the length is 0.8 m? 0.5 m
ALG catalog ver. 2.6 – page 114 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FH
33. Ms. Giraudo wants to plant new grass in the front
and back of her house. The grassy areas are
rectangular, and measure 14 × 12 m and 8 × 8 m.
The grass sod is $2.35/sq. m. What is the total cost
of the sod? $545.20
34. Two walls in a building need to be re-plastered and
painted. The walls are 130 × 95 ft and 75 × 95 ft
(there are no doors or windows). The materials cost
$0.40/sq. ft. What is the total cost of the materials?
35. Robert is going to put new carpet in two rooms of
his house. The dimensions of the rooms are 12 0 × 18 0
and 21 0 × 15 0 , and the carpet costs $6.70 per square
yard. What is Robert going to have to spend on the
carpet? $395.30
36. Mr. and Mrs. Boyd have decided to make new
curtains for their living room. They need two pieces
of fabric, 260 × 180 cm and 220 × 160 cm. The fabric
costs $3.95 per square meter. What will the Boyds
spend on fabric? $32.39
37. Find the volume of a sugar cube whose edge is
1
2 inch. 1 cu in.
38. Find the volume of a milk crate whose edge is 0.3 m
(assume it is a cube). 0.027 m3 or 27000 cm3
39. Find the surface area of a cube whose edge is 1 13 ft.
40. Find the surface area of a cube whose edge is 0.5 m.
8
$7790
1.5 m2
32 sq ft
41. The edge of a cube is 0.6 m. What is the volume and
surface area? 0.216 m3 ; 2.16 m2
42. The edge of a cube is 41 ft. What is the volume and
surface area? 1 cu ft or 27 cu in.; 3 sq ft
43. The edge of a block of ice is 10 cm (assume it is a
cube). What is the surface area and volume of ice?
44. The edge of an ice cube is 1 41 inches. What is the
surface area and volume of ice? 9 3 sq in.; 1 61 cu in.
64
8
8
600 cm2 ; 1000 cm3
64
45. The base of a triangle is 220 mm and the height
is 180 mm. Find the area. 19800 mm2
46. The base of a triangle is 75 inches and the height is
162 inches. Find the area. 6075 sq in.
47. What is the area of a triangle whose base is 6.8 cm
and height is 10.5 cm? 35.7 cm2
48. What is the area of a triangle whose base is
10 12 inches and height is 6 32 inches? 35 sq in.
49. Find the circumference of a bicycle wheel whose
diameter is 700 mm. ≈ 2198 mm
50. Find the circumference of a circle whose diameter is
3 12 inches. ≈ 11 in.
51. The diameter of a large truck wheel is 1.2 m. What
is the circumference? ≈ 3.768 m
52. The diameter of a circle is 35 ft. What is the
circumference? ≈ 110 ft
53. What is the circumference of a circle whose radius
is 2 13 ft? ≈ 14 2 ft
54. What is the circumference of a “hula-hoop” whose
radius is 0.5 m? ≈ 1.57 m
55. Find the circumference of a circle whose radius is
210 inches. ≈ 1320 in.
56. Find the circumference of a “frisbee” whose radius
is 20 cm. ≈ 62.8 cm
57. Find the area of a circle whose radius is 0.1 km.
58. Find the area of a circle whose radius is 1.4 ft.
3
≈ 0.0314 km2
≈ 616 sq ft
59. The radius of a paper plate is 12 cm (assume it is
flat). What is the area? ≈ 452.16 cm2
60. The radius of a small speaker is 2 45 inches (assume it
is flat). What is the area? ≈ 24 16 sq in.
61. The diameter of a circle is 14 inches. What is the
area? ≈ 154 sq in.
62. The diameter of a circle is 400 cm. What is the area?
63. Find the area of a circle whose diameter is 1 ft.
64. Find the area of a circle whose diameter is 0.14 m.
≈
11
14
sq ft
65. The circumference of a circle is 96π. What is its
radius? 48
25
≈ 125600 cm2
≈ 0.061544 m2
66. The circumference of a circle is 50π. What is its
radius? 25
ALG catalog ver. 2.6 – page 115 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FH
67. Find the radius of a circle whose circumference
is 12
5 π. 6
68. Find the radius of a circle whose circumference is
69. Find the volume of a rectangular solid whose
dimensions are 2.5 × 4 × 6.5 cm. 65 cm3
70. Find the volume of a rectangular solid whose
dimensions are 3 × 4 12 × 8 ft. 108 cu ft
5
71. The dimensions of a rectangular solid are 1 ×
What is the surface area? 16 sq in.
1
2
× 5 in.
73. The dimensions of a rectangular solid are 4 × 5 × 6 in.
What is the volume and surface area?
120 cu in.; 148 sq in.
2
3 π.
1
3
72. The dimensions of a rectangular solid are
1.5 × 3 × 0.4 m. What is the surface area?
12.6 m2
74. The dimensions of a rectangular solid are
5 × 10 × 8 mm. What is the volume and surface
area? 400 mm3 ; 340 mm2
75. A rectangular solid is 1 × 3 × 12 cm. What is the
volume and surface area? 36 cm3 ; 102 cm2
76. A rectangular solid is 2 × 2 × 1 21 ft. What is the
volume and surface area? 6 cu ft; 20 sq ft
77. The dimensions of a milk carton are 4 × 4 × 8 in.
What is its volume (ignore the triangular part at the
top)? 128 cu in.
78. The dimensions of a shoe box are 14 × 28 × 9.5 cm.
What is the volume of the shoe box? 3724 cm3
79. A rectangular swimming pool is 8 × 12 × 3 m.
What is its volume (assume its depth is the same
everywhere)? 288 m3
80. A rectangular closet is 1.5 × 2 × 2.5 m. What is the
volume of the closet? 7.5 m3
81. A can of orange juice concentrate is 12 cm in length
and 4.8 cm in diameter. What is the volume of juice
concentrate? ≈ 217 cm3
82. A can of soup can is 4 inches tall and 3 inches in
diameter. What is the volume of soup in the can?
83. The length of a drinking straw is 20 cm and the
diameter is 4 mm. How much liquid can the entire
straw hold? ≈ 2.512 cm3
84. The inner diameter of a lead pipe is 1 inch. How
much water could you hold in 2 feet of the pipe?
≈ 28.26 cu in.
≈ 18.84 cu in.
ALG catalog ver. 2.6 – page 116 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FI
Topic:
Time, distance, rate. See also category HE (word problems for first degree equations).
Directions:
0—(No explicit directions.)
1.
If Melissa walks at a rate of 5 12 km/hr, how far will
she walk in 4 hours? 22 km
2.
An airplane flies for 2 13 hours at an average speed of
348 mph. How far does it go? 812 mi
3.
If a woman walks at a rate of 3 12 mph, how far will
she go in 1 21 hours? 5 1 mi
4.
Gerard runs at a rate of 10.8 km/hr. How far can he
run in three-quarters of an hour? 8.1 km
5.
A boat travels at an average speed of 40 km/hr. How
far will it go in 5 hours and 24 minutes? 216 km
6.
If a race car travels at a speed of 150 mph, how far
will it go in 1 hour and 18 minutes? 195 mi
7.
A bicyclist rides for 40 minutes at 22 12 mph. How far
does she ride? 15 mi
8.
Mr. Sornees runs for 36 minutes at 13.75 km/hr.
How far does he run? 8.25 km
9.
One day Morris went on a hike 7:00 am to 4:30 pm.
His average speed during the hike was 3.2 km/hr.
What was his total distance? 30.4 km
10. Jean left on a bicycle trip at 1:00 pm and returned at
3:45 pm. Her average speed on the trip was 16 mph.
What was her total distance? 44 mi
4
11. An airplane flies non-stop from San Francisco to
Montreal at a rate of 765 km/hr. If the airplane
leaves at 3:25 pm and arrives at 8: 45 pm, what is
the distance between the cities? 4080 km
12. A freight train left the station at 8:30 am and arrived
at its destination at 2:00 pm on the same day. If the
train’s average speed was 120 km/hr, how far did it
go? 660 km
13. An airplane travels 4110 km in 5 hours. What is the
airplane’s speed? 822 km/hr
14. What is the average speed of a bicyclist who goes
54 miles in 3 hours? 18 mph
15. In 10 21 hours of hiking, Mrs. Sornees covered a
distance of 28 km. How fast did she walk? 2 2 km/hr
16. In a half-hour workout, Michael swam three-quarters
of a mile. What was his speed in the water? 1.5 mph
17. Find the speed of a truck (in mph) that goes
28 miles in 35 minutes. 48 mph
18. Find the speed of a jogger (in km/hr) who goes 8 km
in 36 minutes. 13 1 km/hr
19. What is the speed of a race horse (in mph) that goes
around a 1 12 mile track in 2 12 minutes? 36 mph
20. A satellite orbits the earth every 112 minutes, and
follows a path of 84,000 km. How fast is it moving
(in km/hr)? 45000 km/hr
21. A race car went 328 miles in 2 hours and 8 minutes.
What was the car’s average speed? 153.75 mph
22. A fishing boat travels 144 km in 6 hours and
45 minutes. What is the boat’s average speed?
3
3
21 13 km/hr
23. An airplane left Chicago at 10:30 am and arrived in
Cleveland at 11:20 am. If the cities are 340 miles
apart, what was the average speed of the plane?
24. A train left Calgary at 9:45 am and arrived in Regina
at 5:45 pm on the same day. If the cities are 760 km
apart, what was the train’s average speed? 95 km/hr
408 mph
25. What is a bullet’s speed in miles per hour, if it
travels 1760 feet in 1.5 seconds? 800 mph
26. What is a rocket’s speed in km/hr, if it goes 9500
meters in 10 seconds? 3420 km/hr
27. In the 1988 Olympics, Florence Griffith-Joyner won
the 200 meter dash in 21.34 seconds. What was her
speed in km/hr? ≈ 33.7 km/hr
28. In the 1980 Winter Olympics, Eric Heiden won the
1000 meter speed skating event in 75.18 seconds.
What was his speed in km/hr? ≈ 47.9 km/hr
29. A man drove his car 180 km at an average speed of
75 km/hr. How much time did he spend driving?
30. A woman ran a marathon at an average speed
of 8.8 mph. The marathon course is 26.4 miles. How
long did it take her to finish? 3 hr
2.4 hr
31. How long (in minutes) does it take someone to run a
7.5 mile race whose average speed is 9 mph? 50 min
32. How long (in minutes) does it take someone to run a
10 km race whose average speed is 15 km/hr? 40 min
ALG catalog ver. 2.6 – page 117 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FI
33. In 1990, the Lockheed SR-71 set a new record for
transcontinental flight. It went 2404 miles at an
average speed of 2124 mph. How long did the flight
last? (Round your answer to the nearest minute.)
34. At the Kentucky Derby, a horse ran the 1 41 mile race
at an average speed of 36 mph. How long did it
take the horse to finish? (Round your answer to the
nearest second.) 2 min 5 sec
1 hr 8 min
35. The planet Mars is approximately 78.3 million miles
from Earth. If you could fly straight to Mars at a
speed of 10,000 mph, how long would it take to get
there? (Express your answer in days and hours.)
326 days, 6 hours
37. How long (in seconds) does it take a bullet traveling
at 600 mph to reach a target that is 220 feet away?
0.25 sec
36. In 1986, the Voyager aircraft circled the Earth
without stopping or refueling. It went about
24,940 mi at an average speed of 116 mph. How long
did the flight last? (Express your answer in days and
hours.) 8 days, 23 hours
38. How long (in seconds) does it take a baseball
pitched at 90 mph to reach home plate (a distance of
60 feet)? 2 sec
3
39. A truck needs to go back and forth between Chicago
and Peoria. The distance between the cities is
168 miles, and the truck can go no faster than
48 mph. What is the least amount of time required
for the round-trip? 7 hr
40. A sailboat can travel at a maximum rate of 18 km/hr.
It is entering a race from the harbor to a lighthouse
31.5 km away, and then back again. What is the
least amount of time required for the round-trip?
41. In the first 2 hours of a trip, a backpacker walked
4 12 miles uphill, and in the last 3 hours, he walked
8 miles downhill. What was his average speed for
the trip? 2.5 mph
42. A truck went 55 km in the first 30 minutes of a trip,
and 62 km in the last 1 hour and 45 minutes. What
was the truck’s average speed for the trip? 52 km/hr
43. Two cars, going in opposite directions, pass each
other at 2:30 pm. Their speeds are 65 and 80 km/hr.
How far apart are they at 5:15 that same afternoon?
44. Two trains, going in opposite directions, pass each
other at 11:15 am. Their speeds are 45 and 33 mph.
How far apart are they at 2:45 in the afternoon?
385 km
3.5 hr
273 mi
45. An automobile, traveling at 55 mph, passes a truck
going in the same direction at 40 mph. How far
ahead will the car be 40 minutes later? 10 mi
46. A motorcycle, traveling at 90 km/hr, passes a
tractor-trailer going in the same direction at
48 km/hr. How far ahead will the motorcyle be
50 minutes later? 35 km
47. It takes a boat 1 21 hours to go 7.5 miles up river. If
the boat’s stillwater speed is 9 mph, what must be
the speed of the river current? 4 mph
48. It takes an airplane 2 14 hours to go 414 km against
the wind. What is the wind’s speed, if the airplane’s
speed in still air is 200 km/hr? 16 km/hr
ALG catalog ver. 2.6 – page 118 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FJ
Topic:
Interest and investment. See also category HG (word problems for first degree equations).
Directions:
0—(No explicit directions.)
1.
Suppose a savings account pays 5 12 % annual interest.
How much would you earn in interest if you put $650
into the account for one year? $35.75
2.
Suppose the annual interest on savings bonds is 8%.
If you bought $1500 worth of bonds, how much
interest would you get the first year? $120
3.
Maurice took out an auto loan of $3200 for 3 years.
The simple annual rate was 18%. How much interest
did he pay on his loan? $1728
4.
Maria got a home improvement loan of $10,000 for
4 years. The simple annual rate was 14%. How
much interest did she pay on her loan? $5600
5.
Find the simple interest on a business loan of
$26,000, if the annual rate is 12% and the money is
borrowed for 3 21 years. $10,920
6.
Find the simple interest on a home equity loan of
$1500, if the annual rate is 17% and the money is
borrowed for 2 12 years. $637.50
7.
Rex borrowed $4200 at an annual rate of 13.8%.
If he repaid the money after 3 months, how much
interest did he owe? $144.90
8.
Mr. Ellis repaid a loan of $2800 after 6 months.
How much interest did he owe if the annual rate was
12.75%? $178.50
9.
Mrs. Sinclair invested $5000 at a simple annual rate
of 7%. How much was her investment worth after
10 years? $8500
10. Mr. Guerrero invested $800 at a simple annual rate
of 6 12 %. How much was his investment worth after
8 years? $1216
11. Sandy receives a $3000 student loan for five years.
He is charged 6% simple annual interest on the
entire loan, and he has to repay it in equal monthly
installments. What will be the amount of each
installment? $65
12. To buy a sailboard, Nina borrowed $1200 for two
years. She was charged 11% simple annual interest
on the entire loan, and had to repay it in equal
monthly installments. What was the amount of each
installment? $61
13. Ms. De’angelo bought a new truck for $7800. The
dealer gave her $2000 on a trade-in, and she financed
the rest over a four year period at a simple annual
rate of 11%. What were her monthly payments, if
the total amount she owed was divided up equally?
14. Manfred bought a synthesizer for $1300. He paid
$550 in cash and financed the rest over a two year
period at a simple annual rate of 14%. What did he
have to repay each month, if the total amount he
owed was divided up equally? $40
$174
15. A municipal bond pays 8.5% annual interest. Find
the amount invested if the interest after one year is
$74.80. $880
16. A saving bond pays 7.2% annual interest. Find the
amount invested if the interest after one year is
$91.80. $1275
17. Richard’s savings certificate was worth $1602 after
one year. If the interest rate was 6.8% per year, how
much did he originally invest? $1500
18. Jennifer’s certificate of deposit was worth $3723 after
one year. If the interest rate was 9.5% per year, how
much did she originally invest? $3400
19. After 6 years, the total amount to be paid back on a
loan is $13,125. The simple annual rate was 12 12 %.
How much was originally borrowed? $7500
20. After 8 years, the total amount to be paid back on a
loan is $10,800. The simple annual rate was 14 21 %.
How much was originally borrowed? $5000
21. Alfred bought some silver coins as an investment. He
kept them for one year and then sold them for $1239.
His profit on the original investment was 18%. What
was the original investment? $1050
22. Mr. Phelan bought a painting as an investment. He
kept it for one year and then sold it for $6270. His
profit on the original investment was 4.5%. What
was the original investment? $6000
23. Roberto put $2500 into a savings account for one
year. He earned $220 in interest. What was the rate
of return? 8.8%
24. Ms. Garin invested $2000 in savings bonds for one
year. She earned $140 in interest. What was the
rate of return? 7%
25. The interest charged on a 5-year, $3600 loan is
$2475. What is the simple annual rate for the loan?
26. The interest charged on 6-year, $12,000 loan is
$10,512. What is the simple annual rate for the
loan? 14.6%
13.75%
ALG catalog ver. 2.6 – page 119 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
FJ
27. At what simple annual rate can you earn $900 on a
2-year, $6000 investment? 7.5%
28. At what simple annual rate can you earn $396 on a
4-year, $1200 investment? 8.25%
29. Find the simple annual rate on a loan of $16,000, if
the total amount paid back after 3 years is $24,160.
30. Find the simple annual rate on a loan of $1400, if
the total amount paid back after 4 years is $2324.
17%
16.5%
31. Mr. Barnes got a loan of $2800, and repaid it after
6 months. If he was charged $220.50 in interest,
what was the annual rate? 15.75%
32. Claire invested $5000 in a 6 month certificate of
deposit. If she earned $255 in interest, what was the
annual rate? 10.2%
33. A credit union will lend you $2000 for 3 months, but
charge you $65 in interest. What is the annual rate
for the loan? 13%
34. If you owe your credit card company $600 for one
month, you will be charged $8.75 in interest. What
is the annual interest rate? 17.5%
35. Find the value of an investment of $8500 after
5 years, if the interest rate is 8% per year. Assume
the interest is compounded annually. $12,489.29
36. Find the value of an investment of $1200 after
5 years, if the interest rate is 9% per year. Assume
the interest is compounded annually. $1846.35
37. Find the value of an investment of $2000 after one
year, if the annual rate is 12% and the interest is
compounded quarterly. $2251.02
38. Find the value of an investment of $2000 after one
year, if the annual rate is 12% and the interest is
compounded monthly. $2253.65
39. $1000 is invested for 4 years at an annual rate
of 8%. What is the total interest if it is compounded
(a) yearly, (b) quarterly, and (c) monthly?
40. $10,000 is invested for 5 years at an annual rate
of 9%. What is the total interest if it is compounded
(a) yearly, (b) quarterly, and (c) monthly?
$360.49; 372.79; 375.67
$5386.24; 5605.09; 5656.81
41. Suppose you invest $5000 for one year at an annual
rate of 10%. How much more would you earn if
interest were compounded quarterly instead of
yearly? $19.06
42. Suppose you invest $5000 for one year at an annual
rate of 10%. How much more would you earn if
interest were compounded monthly instead of yearly?
43. At the beginning of the month, Silvia receives a bill
from her credit card company for $800. The interest
rate is 1.5% per month on the balance due. If she
pays $150 each month for three months, how much
will she owe on the next bill? $372.91
44. At the beginning of the month, Jon receives a bill
from a department store for $1150. The interest rate
is 1.5% per month on the balance due. If he pays
$250 each month for three months, what will be the
amount of the next bill? $428.80
45. How long will it take an investment of $650 to
increase to $1000 at a simple annual rate of 6.25%?
(Round answer to nearest 0.1 year). 8.6 years
46. How long will it take an investment of $7000 to
increase to $11,000 at a simple annual rate of 8.5%?
(Round answer to nearest 0.1 year). 6.7 years
47. At a simple annual rate of 10 12 %, how long will it
take any investment to triple in value? (Round
answer to the nearest 0.1 year.) 19 years
48. At a simple annual rate of 7 12 %, how long will it
take any investment to double in value? (Round
answer to the nearest 0.1 year.) 13.3 years
$23.57
ALG catalog ver. 2.6 – page 120 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
GA
Topic:
Equations involving addition and subtraction.
Directions:
15—Solve.
1.
9 − 17 = r
5.
d = −11 − (−20)
9.
y+5=9
16—Solve and check.
−8
9
2.
c = −12 + (−18)
6.
−14 + 7 = k
13. 22 = k + 14
8
14. 18 = x + 2
17. 28 + r = 39
11
18. n + 51 = 74
21. 150 = 85 + a
25. p + 17 = 9
−8
−45
33. c + 16 = −14
44
41. −250 + k = 53
45. x + 13 = 0
303
3
w = −5 + 23
7.
34 − (−6) = a
23
100
18
40
4.
−26 − (−14) = p
8.
r = −30 − 46
12. 5 + m = 11
3
15. 27 = 13 + d
14
16. 10 = 8 + z
19. 35 = 25 + w
10
20. 79 = f + 44
6
2
35
23. h + 45 = 178
133
24. 238 + x = 296
27. 21 + m = 16
−5
28. 11 = 15 + k
−4
30. 45 + m = 6
−39
31. 25 = 40 + s
−15
32. a + 59 = 27
−32
−44
35. z + 57 = −7
−64
36. −20 = p + 25
38. −56 + r = 55
101
39. 9 = −49 + n
58
40. k − 97 = 23
42. 166 = w − 25
191
43. u − 184 = 39
223
44. 157 = −192 + h
47. y − 25 = 0
25
8
50. −11 = n − 12
1
51. −1 = k − 9
48. w − 17 = 0
52. m − 15 = −4
11
−15
54. k − 26 = −37
−11
55. −35 = b − 34
−1
56. −23 = y − 8
57. −18 = a − 32
14
58. −16 = x − 25
9
59. y − 45 = −23
22
60. r − 15 = −12
61. r − 24 = −24
0
62. c − 18 = −18
0
63. 17 + w = 17
8
20
67. −68 = −81 + m
69. −58 = y − 16
−42
70. −126 = b − 118
−8
71. x − 21 = −56
73. n − (−12) = 8
−4
74. 16 = w − (−22)
−6
75. b − (−11) = 10
77. 32 = x − (−45)
81. b + (−24) = 1
−13
25
85. 105 = x + (−105)
89. x − (−5) = 12
210
7
93. 50 = −(−35) + a
97. y + (−6) = −12
15
−6
101. −200 = k + (−99)
78. −(−48) + y = 21
−27
37
83. 10 = m + (−6)
86. 72 = y + (−48)
120
87. m + (−93) = 57
90. 17 = −(−8) + n
9
91. 14 = k − (−12)
94. x − (−10) = 15
98. m + (−9) = −10
−1
5
106. −14 = k + (−21)
109. −24 = x + (−58)
34
110. n + (−105) = −33
−12
117. −34 = m − (−16)
114. −13 = n − (−6)
118. k − (−28) = −28
7
72
−19
−56
−13
−8
80. m − (−79) = 78
−1
84. 12 = n + (−10)
16
150
2
25
99. −22 = a + (−15)
−7
22
88. b + (−61) = 129
190
92. −(−14) + y = 20
6
96. 90 = b − (−45)
45
100. −17 = x + (−12)
−5
104. w + (−210) = −550
−125
105. w + (−17) = −12
113. x − (−5) = −7
−32
103. m + (−20) = −145
−225
56
76. 8 = w − (−16)
95. −(−40) + m = 65
5
0
72. k − 49 = −62
−1
79. 52 = −(−84) + c
3
68. −22 = −78 + n
13
−35
82. x + (−16) = 21
102. −400 = h + (−175)
−101
64. 31 = b + 31
66. −65 + y = −45
349
17
−6
0
−45
120
53. w − 14 = −20
65. −107 + k = −99
58
−7
−37
−12
−76
26. 13 = y + 20
46. h + 37 = 0
−13
49. b − 10 = −7
16
34. −36 = u + 8
−30
3.
11. 4 + x = 7
3
22. 412 = p + 312
65
29. 21 = 66 + x
37. 36 = w − 8
−7
10. a + 12 = 15
4
−30
−340
107. −11 = r + (−19)
8
108. h + (−26) = −22
4
111. c + (−76) = −47
29
112. −36 = z + (−99)
63
115. m − (−8) = −12
−20
116. −1 = k − (−14)
119. −49 = w − (−32)
−50
ALG catalog ver. 2.6 – page 121 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−81
120. n − (−19) = −27
−15
−46
GA
2
121. m = − 1
3
7
122. − + 2 = x
6
− 13
4
6
=s
125. − + −
5
5
1
1
129. y = 3 − 2
2
4
133. n +
137. −
141. 3
1
5
=
2
2
153.
11
7
=−
9
9
161. −5
169.
7
2
+f =
6
3
173. −
5
2
−4
155. t + 2
− 12
170. −
178. −
4
2
181. r + −
=
3
3
171. a +
4
3
175. −
7
7
=h−
8
12
7
− 24
179. x +
5
1
182. y − −
=
6
6
2
5
1
+d
=− −
4
4
− 20
7
−1
2
1
189. w − −
=−
5
10
186. −4
1
1
=k+ −
3
3
−4
11
5
156.
− 94
− 14
4
3
=− +y
5
10
5
3
=−
4
18
2
2
= −3 + a
5
5
3
1
7
=x+
6
6
−1
− 35
5
164. − + w = −6
2
− 72
7
4
=v−
4
7
65
28
9
8
=−
15
5
− 73
1
5
176. − + r = −
6
3
− 12
180. −
− 37
36
7
11
=n−
6
8
− 32
5
24
3
7
184. = z + −
8
8
0
7
5
=t+ −
6
12
− 12
− 23
172. x +
3
4
1
3
3
9
=−
10
10
168.
17
6
3
2
7
1
=−
12
12
160. y −
−6
1
1
187. h + −5
=1
2
2
191. −
152. −
5
1
1
183. = c − −
4
4
− 23
3
7
=
190. r + −
2
10
1
11
=
6
12
−5
1
1
7
=
4
12
5
=3+d
2
148. k +
− 34
5
1
167. − + k =
2
3
11
1
=−
6
2
144.
1
13
=−
2
2
3
4
2
3
=r−
5
5
140. c +
− 56
7
5
=
8
8
163. −1 = z −
− 75
23
15
5
5
=w+
14
2
− 17
10
159. x −
− 13
136.
1
1
=w−7
2
2
− 12
1
2
132. −1 + 2 = r
6
3
− 43
3
2
1
6
9
124. h = − −(−4)
2
3
1
128. − − 4 = a
4
4
3
1
5
+a=−
8
8
151. −2
2
11
=
3
5
5
13
=−
15
6
2
5
=
3
6
1
2
166. d +
174. x −
147.
−5
7
5
=− +a
6
6
2
3
1
10
=b+
3
3
143. h + 1 =
− 34
19
20
185.
1
4
7
1
= −2
10
10
− 11
4
135.
2
3
−3
7
5
131. x = + −
6
2
139. y −
162. n −
1
5
=c−
21
7
177. y +
158. −
−2
− 38
1
8
11
3
=
2
2
154. u +
− 32
1
3
=
5
4
1 2
127. k = −2 −
3 3
1
4
10
5
=y+
3
3
150. z − 3 = −
4
7
1
3
=c−2
8
8
165. x −
146. −
−2
4
10
=−
3
3
1
3
=x+
2
8
142. w + 3 = 2
3
7
3
1
=p+2
4
4
157. z −
138.
7
10
− 13
149. −1 + m = −
5
3
+ −
2
2
126. p =
7
9
134. − + p =
4
4
2
2
=z+4
3
11
123. −3 − −
=y
3
3
3
130. − − −
=d
4
8
5
4
3
2
+a=
10
5
145. x +
−2
5
6
6
− 34
5
4
7
5
=−
188. t − −
6
6
−2
1
11
192. p − −
=
2
14
2
7
7
5
=
196. u + −
20
8
39
40
− 12
193.
3
3
=k+ −
14
4
27
28
194.
2
1
+a
=− −
9
6
1
18
7
2
195. m − −
=
10
15
− 17
30
ALG catalog ver. 2.6 – page 122 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
GA
197. g + 5 = 3.75
198. 2.6 + w = 7
−1.25
201. −6 = y − 12.3
6.3
202. −0.99 + c = −1
205. m + 3.7 = 5.2
1.5
206. 0.5 = r − 10.5
210. v + 4.6 = 1.44
209. −0.61 = 0.05 + x
199. 9.2 = b + 12
4.4
16.4
204. −7.7 + a = −9
−1.3
11
207. t + 0.08 = 0.42
0.34
208. 0.16 = k − 2.04
2.2
−3.16
211. −18.4 + y = −22.3
212. d + 7.28 = 0.28
−7
−3.9
213. −5.8 = 14.6 + w
214. k − 0.027 = −0.003
−20.4
215. −3.05 = f + 1.25
−4.3
218. 18.03 + y = 16.5
0.64
221. n − 0.45 = −0.19
0.26
222. −3.06 = −5.21 + h
219. m + 0.001 = 0.0009
23.4
0.01
226. −1 = p + (−0.86)
−0.14
229. −16.05 = y + (−8.75)
−7.3
233. t + (−0.027) = 0.008
0.035
237. −28.23 = −(−30.6) + r
230. x − (−3.04) = 4.47
1.43
234. −10.15 = w − (−12.55)
−22.7
238. 0.0078 = n + (−0.003)
0.0108
220. 10.4 + h = 6.07
−4.33
−0.0001
223. k − 11.1 = 10.07
21.17
224. −0.104 + d = −0.99
−0.886
2.15
−58.83
216. n − 50.3 = −26.9
0.024
−1.53
225. f − (−1.99) = 2
0.06
203. z − 30.4 = −14
−0.01
−0.66
217. 2.3 = 1.66 + p
200. 4 = 3.94 + x
−2.8
227. −(−4.25) + z = 3
−1.25
231. 0.032 = g + (−0.068)
0.1
235. r + (−1.772) = −0.44
1.332
239. a + (−15.86) = 15.24
31.1
ALG catalog ver. 2.6 – page 123 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
228. w + (−8.13) = −5
3.13
232. −8.09 = −(−2.91) + u
−11
236. c − (−20.01) = 24.91
4.9
240. −3.88 = f − (−6.02)
−9.9
GB
Topic:
Equations involving multiplication and division.
Directions:
15—Solve.
45
5
1.
x=
5.
(−6)(−17) = a
9.
5
u = 18 −
2
9
−45
13. −6 ÷ 24 = h
17. −k = 30
21. 39 = 3d
102
− 14
−30
13
25. −14m = 28
−2
29. −8f = −184
33. −12 = 12y
23
15
41. −14y = −126
45. 448 = −32c
49. 18n = 0
9
−14
53. −5 = 30x
69.
x
−7
14. p =
−15
−35
−38
22
22. 2k = −22
−11
26. −45 = −15y
30. 210 = −6x
1
x=0
14
−35
1
−8
46. −25h = −600
10y
= −15
−3
9
2
−168
4
11. k = − (−60)
5
48
10
13
4.5
24
0
2
3
16
4.
b = −42 ÷ (−7)
8.
(−12)(7) = w
12. 32 ·
9
=n
8
16. z =
27
−18
20. −a = −(−7)
24. −5x = 75
−7
−29
32. 7t = 224
35. −19x = 19
−1
36. −15 = −15b
43. 26u = 182
25
51. −23a = 0
32
40. −9y = 117
1
−13
44. −135 = 15f
7
47. −735 = 21k
−7
−15
31. −145 = 5p
39. −200 = −8h
−35
0
55. −70b = −7
1
10
−9
48. 18p = 324
18
52. 0 = −12w
0
56. −24j = 32
− 43
− 14
59. 24 = −54y
− 49
60. 12 = 96p
62. 24 = −108n
− 29
63. 85k = −50
− 10
17
64. −105x = −30
66
68.
−k
= −3
8
78.
r
−11
z
= 12
5
1
8
24
67. −6 =
x
−10
−100
71.
1
k
13
−65
1
75. −25 = − n
4
100
76.
79.
15
w = −30
8
−16
2
80. − u = 22
3
83.
3z
= 24
−8
9
a = 27
2
6
−6w
11
3n
7
6
90. − x = 8
5
−110
t
=7
7
87. −
0
− 20
3
9
z=0
20
91. 10 =
ALG catalog ver. 2.6 – page 124 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
72. −9 =
49
4n
7
−64
0
35
2
84.
92.
−n
9
81
120
−33
−10x
= −90
3
d
8
2
7
60
1
c=8
15
88. 0 =
−84
− 32
28. −70 = 10a
5
6
36
58. 56m = −14
86. 0 =
0
y = 8(−21)
27. 9b = 45
3
42. 176 = −22j
82. 60 =
49
7.
−3
23. −64 = −4m
−16
74. −5 =
12y
= 84
7
−36
=m
12
19. −4.5 = −r
38. −144 = 9u
70. 10 =
30
3.
15. 40 ÷ 52 = x
3
7
18. −22 = −x
144
−54
−4
80
2
10. − · 57 = r
3
66.
3
77. −18 = − p
5
89.
c = 16(5)
−35
a
= 12
12
85. −
9
5
23
12
1
73. − d = 6
9
81.
6.
54. 10 = 15u
− 16
57. −25m = −45
65. 5 =
2 ÷ (−8) = y
50. 0 = −(−9y)
0
61. 48r = 92
2.
34. 27f = 27
−1
37. 7w = 105
16—Solve and check.
0
9
p = −6
2
− 43
27
GB
93.
3
3
k=−
2
2
−1
94.
4
4
=
t
11
11
97.
10
5n
=
3
7
14
3
98.
8y
−4
=
−9
15
3
10
49
4
102.
3
−5k
=
5
3
9
− 25
−24
106.
10
2
m=−
33
15
110.
1
2
z=3
9
3
2
7
101. − c = −
7
2
105. −
r
8
=
15
5
109. −2
1
1
= a
6
2
− 13
3
−0.03
114. −3.5 = −10d
117. −16 = 0.1z
−160
118. 0.04f = 1
125. −0.054 = −2.7y
129.
1
d = 0.7
6
133. −0.02 =
137. −3 = −
141. −
4.2
x
15
r
2.4
x
= 1.1
0.4
0.02
− 11
25
0.35
126. 0.004x = 0.4
100
1
130. − p = 0.06
5
−0.3
−0.3
134. −0.19 = −
7.2
138.
−0.44
142. 0.08 =
n
10
x
= −5
1.6
y
0.05
1
− 21
1.9
0.004
100.
3
3
x=
10
5
−1
2
100
49
104.
4y
9
=−
9
4
107.
−5x
5
=
12
−2
6
108.
7
21
p
=
16
8
3
2
= t
5
3
115. 2y = 1.4
135. −
143.
0.7
116. −2.04 = 6m
1
f
2
y
= 2.5
20
17
2
−0.34
120. −4.5x = −27
−10
0.05
−6
124. 0.01t = 0.8
−50
0.42
−0.073
6
80
128. 0.006 = −0.03k
132. −0.33 =
0.01
a
0.042
c
= −0.73
0.1
ALG catalog ver. 2.6 – page 125 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1
6
1
1
112. − k = −2
4
8
127. 0.25w = −1.5
131. 0.005 =
− 81
16
12
5
123. −0.11 = −2.2n
139. −10 = −
−8
7a
7
=
4
4
10
7
=
w
7
10
119. 12 = −1.2r
−0.04
96. −
1
103.
111. 1
25
122. −0.02 = 0.5a
18
−2r
−2
=
5
5
7
1
99. − c =
2
6
33
113. −8x = 0.24
121. 0.35w = 6.3
95.
1
136.
z
= 0.3
12
140.
n
=6
2.5
144. −
1
c
3
−0.2
−0.99
3.6
15
e
= −0.003
0.3
0.0009
GC
Topic:
Equations of the form: ax + b = c.
Directions:
15—Solve.
1.
−a = 24 − 17
16—Solve and check.
2.
−7
−21 + 13 = −m
3.
8
−p = −14 − 16
4.
30
−8 − (−20) = −h
−12
5.
−y + 14 = 9
9.
−6 = −20 − k
13. 5k − 35 = 0
6.
5
4 − x = 17
7.
−13
10. −r − (−8) = −12
−14
14. 0 = −9m − 18
7
6
18. 8 + 6x = 50
21. −2x + 51 = −3
27
22. −25 = 4w − (−7)
29. −3a − (−12) = 19
33. −8z − 43 = −11
− 73
−4
37. −39 = −18w − 21
41. −8c − 56 = 0
1
45. −14t + 7 = 49
49. 28 = −14u + 16
53. −95 − 6s = 25
− 67
−20
57. 4f − (−56) = 56
0
16
23. 14 = −2p − 4
15 = −w − 6
−21
12. −28 − z = −32
4
16. 7b − (−70) = 0
−10
20. 23 = 9h + 50
−2
−9
0
−3
24. 3y − 5 = 16
7
28. 9 = 9 − 11k
0
30. 31 = −2s + 24
− 72
31. −10a − 2 = 4
− 35
32. −6w − 4 = 11
34. 9x − 61 = −16
5
35. −26 = 6t − 2
−4
36. −47 = −3x − 8
38. 15 = −8d − (−7)
−1
39. −24h − 9 = 15
7
4
47. 21 − (−15f ) = 96
7
50. −7 − (−12p) = 38
15
4
− 52
40. 28 − 3c = 25
−1
43. 0 = 24x − 42
4
46. 37 − 11k = −40
−3
−13
27. 15c − (−30) = 30
0
42. 64 − 16t = 0
−7
−8
8.
2
19. 8y − (−18) = 2
7
26. 28 + 6q = 28
0
11. −p − 26 = −13
15. −3n + 48 = 0
−2
17. 23 = 3y − (−5)
25. 17 = 17 − 32p
20
10 = −a − (−12)
5
13
1
44. −45 − 27n = 0
− 53
48. −18 = 60 − 13a
6
− 11
5
51. 19 = 10y − 16
7
2
52. −15x − 11 = 22
3
56. 10t − (−95) = 25
54. −9a − 47 = 52
−11
55. 19 = 64 − 15x
58. 9h − 25 = −25
0
59. −43 − (−22m) = −43
60. 39 = 13p + 39
−7
0
0
61. −8 = −4r − 13
62. 15x − (−33) = 27
− 54
65. −128 + 5y = −43
17
66. −25 = −4r − 97
− 35
−18
63. −17 + 3b = −9
64. 46 − 9s = 16
8
3
67. −17 − (−5k) = −37
10
3
68. −7r − 21 = −63
6
−4
69. 12x − (−22) = 10
−1
70. −28 + 17w = −11
1
71. 61 = 5y + 56
72. −41 − (−7m) = −48
1
−1
73. −14d + 15 = 21
77. 83 − 2x = 179
− 37
−48
74. 20z − (−9) = 25
78. −5a − 68 = 92
75. −30x − 28 = 20
4
5
−32
76. 28h − 15 = 15
− 85
79. 133 = −7a − (−21)
15
14
80. 156 = 8u − 132
36
−16
81. −4z − (−122) = 62
82. 5y + 147 = 37
83. −24s + 39 = −105
−22
6
84. 13a − (−76) = −119
−15
15
85. −111 = 9a − 12
−11
86. −13d − 18 = −70
4
87. −44 = 26d − 200
6
88. −35y − 134 = −29
−3
89. −20y − 3 = −7
1
5
90. −8 = −21b + 10
6
7
91. 5 − 12m = −3
2
3
92. −44k − 2 = −6
1
11
Fractions
93. 0 =
r
+8
2
1
97. 3 = 4 − w
5
−16
5
94. −
98.
3n
−6=0
5
k
+ 6 = −3
8
−10
−72
95. 0 = 9 −
1
k
10
90
96.
2
p − 10 = 0
7
a
7
−7
100.
1
d + 20 = 16
10
99. 15 = 14 −
ALG catalog ver. 2.6 – page 126 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
35
−40
GC
101.
2y
+ 5 = −9
5
8
102. − c − 2 = 6
3
−35
4h
= 16
9
12z
109. 15 = 7 +
7
105. 16 +
3
106. − t − 8 = −8
4
6y
110.
− 6 = 3 152
5
0
14
3
103. 20 =
−3
4n
+8
7
21
1
107. 5 = − y + 5 0
5
8
111. −33 = − h − 35
11
0
3
104. 15 − x = 12
2
2
108. −10 = −10 +
7c
10
112.
14a
− 17 = 18
9
0
45
2
− 11
4
113. 0 = 10k −
1
5
117. −7x + 3 = −
121. 9c +
125.
15
2
1
5
=
4
8
1
24
1
− 72
129. 0 =
145.
3
2
9
11
− 15y =
12
8
2m 2
−
9
3
2 1
133. − r = 2
5 5
x 7
137. − = 3
3
6
7a 4
1
141.
+ =
6
3
3
9
= 0 − 38
4
9
118. 2z + 5 =
− 85
5
11
1
122.
= − + 7d
9
3
5
7 13
126. 6t − =
6
15 60
2
9
−12
25
2
− 67
146. −
−3
3
5k
1 7
=−
−
2
7
4 4
c
5
7 1
153.
+ =
12 6
8 2
7
2
1
157. − y − =
−1
6
3
2
3
1
1
=− z−
5
8
10
150.
3
14
3
− 10
5
72
132. 0 = −
131.
147.
4
5
1
5
=−
y+
3
12
4
4x 11
5
154. −
−
=
7
14
6
r
3
1
=
1
158. +
5 10
2
149. −
4
+ 2m = 2 57
7
1
5
1
120. = 10w +
− 15
6
6
7
17
124. −
= − + 18x
10
5
1
3
1
− 40
128. −2x + =
4
10
116.
− 29
5
d
−
= 0 25
2 10
1
2
135. y − 1 =
10
6
3
x 23
139. 3 = − +
− 14
5
4
10
6
3
1
143. p +
=−
− 23
5
10
10
8
4
r + = 0 −6
15
5
x 3
134. 4 = − −
−22
4
2
1
15
= 2 54
138.
s+
10
8
2
7 2
5
142. − = p −
9
2
9 9
130.
3
1
5
1
q+ =
2
3
6
1
3
8
2
119. 4h − = −
7
7
14
7
123. 7k +
=
5
10
5
5
127. = 14c −
9
12
115. 3 = −15w −
114. 6u +
1
50
− 25
− 17
6
4
4k
8
+
=−
5
15
3
1
1
q−
15
3
3
20
−5
n 5
+ = 3 14
8
4
1
7
140. −4 − y =
− 23
2
2
4
7
c
11 8
144. −
= −
12
8 12 3
136.
148. −
−1
7
9 4
1
151. − a + = −
4
2
10 5
5
10
5
5
155. = − y −
− 12
6
3
9
c
1
2
159.
+ =
7
14 6
3
5r
7
1
+ =
8
8
4
1
9d
3
3
=−
−
4
10
5
1
1
1
156. w +
=
6
18
4
3
7
3
160. = − t −
2
10
5
152.
2
− 15
7
6
−3
Decimals
161. −1.5d − 13.5 = 0
−9
165. −2h − 0.3 = 1.4
−0.85
169. 0.2 = 5 + 12w
162. 0 = 4.8 + 0.3x
166. 2 = 5y + 1.7
163. 5m − 0.4 = 0
−16
167. 2.8 − 10x = 3.8
0.06
170. −3a − 3.2 = −0.5
−0.4
178. 11.4 + 8t = 7.4
175. 1.2 = −5y − 3
0.09
−0.5
179. 2s + 3.9 = 0.3
180. −1.3 = −10m − 4.7
−1.8
−0.34
182. −0.76 = 4n − 0.4
−2.7
183. −14q − 1.3 = −4.1
−0.09
185. 4 = 1.2 − 0.2a
189. 0.4y + 3.8 = 3.8
1.3
176. −0.9 + 9r = −0.09
−0.84
−0.07
181. 3x + 8 = −0.1
2.25
172. −6y + 10.4 = 2.6
−0.04
177. −5u − 0.4 = −0.05
1.2
−0.02
174. −11c − 0.42 = 0.02
1.8
168. 4k − 6.1 = 2.9
−0.1
171. 7d − 0.16 = −0.04
−0.9
173. 3p − 6 = −0.6
164. 0 = −9z + 10.8
0.08
186. −1.5w + 5 = 0.5
−14
187. 0.08z + 0.8 = 1
3
190. −9r + 0.02 = 0.02
0
184. 0.6 − 6a = −3
0.6
0.2
0
2.5
191. −14.5 − 1.2p = −14.5
188. 0.24c + 3 = 0.6
−10
192. −1.9 = 3x − 1.9
0
0
193. −1.6s − 0.4 = −10
6
194. −3.4 − 0.5k = −0.8
195. −0.79 = 3.6r − 0.07
−5.2
197. 2.7 = −4.8 + 0.01b
198. 2.7h + 1.2 = −1.5
196. 0.02t − 5 = −3.3
85
200. −2.2d − 3 = 2.5
−2.5
−0.2
−1
199. −0.8k + 0.3 = −0.5
750
ALG catalog ver. 2.6 – page 127 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1
GD
Topic:
Mixed practice and review (first-degree equations).
Directions:
15—Solve.
16—Solve and check.
1.
y + 23 = −9
−32
2.
22 = x − (−17)
5.
18 = m − 38
56
6.
r − 16 = −25
9.
a − (−14) = 28
−9
14
10. y + 35 = 70
13. −15 = −x − 32
−17
14. 21 − (−p) = 20
2
3
17. k − (− ) = −
5
5
−1
21. c +
1
5
=
6
3
3
2
7
1
25. p − 2 = −6
8
4
−3 38
29. 13.6 = d − (−18.2)
5
7.
k + (−39) = 41
−13
1
2
= −5
3
3
7
1
22. x − −
=
10
2
80
−7
8.
−44 = w − 55
11
20.
1
2
=
5
15
− 13
24. h +
− 15
23. −z −
22
16. −33 = −c − (−7)
32
−4
19. −3
3
11
27. − = w −
4
12
30. y + 2.45 = −5.88
a − (−12) = 5
1
3
=r+
4
4
8
− 59
4.
12. z + 18 = 40
9
15. 8 − y = −24
−1
18. −w + 2
−4.6
8 = c + 21
11. 17 = m + 8
35
5
5
26. k + =
6
18
3.
7
1
−v =
6
6
1
28. t + −3
2
31. k + (−7.7) = −3.7
4
1
7
2
=−
3
12
1
6
40
− 54
=1
3
10
32. −1.4 = x − 0.8
4 45
−0.6
−8.33
33. −z − 0.13 = −0.84
34. 8.2 = −a − (−2.2)
−6
35. 3.59 = 2.09 − u
−1.5
36. 10.1 = 16.8 − m
6.7
0.71
37. −56 = 7r
38. −12p = −108
−8
41. −18x = −45
45. −
49.
h
= 30
6
9
n = 36
2
53. −
42. 9n = −33
5
2
46. 8 =
−180
8
20
8u
=
21
3
5
− 14
1
d
4
39. 45 = −15x
9
47.
32
50. −
5z
= 100
4
54. −
10
10
w=−
11
3
55.
11
3
−13
58. 0.07 = 1.4f
61. 36 − 4a = 0
9
62. −27 + 5m = −27
1
70. −24 = 24 − 4a
73. −19 = 9 − 7a
4
74. −3d − (−42) = 12
77. 9z − 3 = −30
−3
78. −11 − 7x = 3
81. 14 + 10k = 9
− 12
89. −12.3 = 3y + 20.7
3
9
y=
8
2
98
12
−2.1
−36
75. 3r + 37 = −11
91. 3.3 = 7.9 + 2z
56.
15
5
=− a
4
4
− 13
60. −3.2r = −9.6
68. 25 = 4f + 17
76. −7 = 18 − 5c
−16
6
20
3
87. −9b − (−35) = 38
9
4
−50
3
−2.3
− 13
0
2
72. 4s − (−42) = 6
4
83. 8 = 3m − 12
3
7
3x
= −15
10
64. 12 = 3k − (−12)
−5
79. −13 = 7r + (−55)
−2
52.
−9
5
80. −2p − 11 = −35
12
84. 22 = 5x − (−16)
6
5
88. 4m − 21 = −27
− 32
92. −6t − 0.8 = 5.2
−1
0.07
93. 1.2t − 0.06 = −0.9
94. −2.5 = 0.1s + 3.8
−0.7
97. 8 −
10
90. 5n − 0.16 = −0.51
−11
64
71. 5 = 29 − 6y
12
82. 10 − 63r = −17
86. 24 = 16c − 12
1
48. − y = −8
8
67. 2w + 19 = −53
7
−8
17
3
−45
63. 0 = 25 + 5b
0
69. 5x + 56 = 16
85. −6x + 13 = −21
− 67
59. −0.042 = 0.02x
0.05
66. 6x − 14 = 28
44. 36 = −42p
4
51. −56 = − h
7
−80
57. −0.6p = 7.8
65. 31 = 4y − (−27)
a
= −15
3
17
4
13
43. 78w = 24
− 11
3
40. 4y = 68
−3
−63
95. 0.03x − 5.7 = −2.1
96. 0.03 = 3.5c − 1.02
120
y
= 17
2
−18
98.
1
x − 12 = −3
5
45
1
99. 35 = 32 − a
3
ALG catalog ver. 2.6 – page 128 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−9
100.
k
− 11 = −20
6
−54
0.3
GD
101.
2
k − 42 = −22
3
105. 4r + 4 =
2
3
102. 17 = 29 +
30
− 56
109.
4
16
= −2m +
5
5
113.
3
11
a−
= −4
8
2
117.
7
11
− 5s =
4
2
− 34
121.
1
2
3
− y=
5 10
5
2
3
6
5
5
− 5w
2
110. 4t −
45
17
=−
7
7
122.
125. (−4)(−7c) = 140
5
129. 12(4x) = −45
− 15
16
133. 48 − 4(6a) = 0
2
1
4
=
6
3
1
4
3
104. −36 = − p − 30
5
52
108. 4d −
3
8
2
7
− 10h =
− 16
3
3
x
17
115. − −
=8
6
3
112. −
111.
1
− 23
3
3x 9
=
−
2
4
2
5c
− 46 = 84
2
107. 1 = 2a +
1
2
7
5c
=4+
3
2
118. −6a −
103.
−16
106. 0 =
114.
4
3m
4
1
4
8
119.
5
7
= 10y +
6
2
123.
n 9 7
− −
2
8 8
−1
130. 42 = 16(−3u)
− 78
131. −5(6y) = −50
5
3
−1
9
9
= 3r +
2
2
−3
9
3
= − 12n
2
8
1
2
13
32
5
5
5
a+ =−
3
6
6
−1
128. 8(5y) = −120
−3
124.
1
2
127. 42 = 3(−14r)
134. 41 = 14 + 9(−3p)
120. −
4
− 15
2
1
32
7
11
116. − y − 1 = −
2
4
14
126. −9(2k) = −36
1
=0
8
10
132. 33 = (−6)(−12k)
135. 7 = (−7)(−5b) + 112
11
24
136. 2(7w) − 22 = 48
5
−3
137. −14 = −(31 + p)
141. 68 = −4(m + 9)
−17
−26
145. 8(z − 3) = −25
− 18
149. −12(1 − 2y) = 108
5
153. 24 = −4(3k − 11) 53
2
1
3
157. 6 x = −
− 16
9
4
1
161. −27 = 6 d +
−5
2
3
165. 5 3r −
= 0 14
4
138. −(28 − n) = −9
19
139. 45 = −(w + 45)
−90
140. −(x − 16) = 16
142. 0 = −7(14 − w)
14
143. 3(a − 8) = −24
0
144. −52 = −2(19 + c)
146. 21 = −6(5 − h)
17
2
147. −9(d − 6) = 33
7
3
148. −45 = 10(n + 3)
− 15
2
152. 3(6p − 13) = −3
2
156. 0 = −21(5s − 4)
4
5
150. 9(7h + 8) = 72
151. −24 = 4(5t + 19)
0
154. 10(3r + 8) = 45 − 76
2
2
1
158. −12 z =
− 20
3
5
3
162. 8 = 14 h −
1
7
1
166. −20 5 − p = −15
4
155. 2(9a + 7) = −13
−5
− 32
5
1
=
(−20y) 38
8
12
2
163. −10
− f = 86
5
2
167. 48 = 18 w − 1
3
159. −
169. 2 t +
173.
5
8
=7
1
4
9
(10 + 6y) = 42
2
7
3
3
= − (−5k) 25
5
10
5
164. 8 x −
= −34 −3
4
8
= 66
168. −15 3y +
5
160.
9
23
4
−2
17
0
7
12
4
3
3
170. 8 m −
− 19
2
174. −18 = − (15n − 45)
5
=
3
4
171. 3
175.
1
−b
6
=3
1
2
3
(30 + 4x) = 18
4
−1
172.
1
3
= 10 k +
2
4
− 32
176.
3
(2p − 64) = −24
8
180.
−r − 8
= 14
2
7
− 10
0
6
177. −5 =
7−x
4
−27
181.
15m + 3
= −6
2
185.
−5(2 − x)
= 90
3
189.
3
3(2k + 3)
=
5
5
−1
56
−1
178.
a+3
=4
7
182.
7x + 28
=0
12
186. −20 =
190.
179. 0 =
25
10(1 − p)
4
=−
3
3
−11
183.
3u − 20
= 11
2
88
187.
−2(p + 12)
=4
9
7
5
191. −
−4
−(r − 8)
4
n + 11
3
−30
9
−3(x + 10)
=
10
10
−7
ALG catalog ver. 2.6 – page 129 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
184. −1 =
14
4a + 5
7
−36
−3
188.
3(c − 19)
=0
5
192.
−5(3y − 1)
11
=
7
7
19
− 25
GE
Topic:
Combining terms on one side of an equation.
Directions:
15—Solve.
1.
2y + 7y = 36
16—Solve and check.
2.
4
−85 = 11c − (−6c)
3.
42 = m + 6m
7.
5p − (−13p) = −28
6
4.
3a − (−8a) = −99
8.
14 = 7s + 15s
−9
−5
5.
3a + 12a = −21
6.
− 75
50 = 16w + 14w
5
3
7
11
− 14
9
9.
24 = 8k + (−5k)
10. 19h − 12h = −42
8
−6
13. 4y − 9y = −95
19
14. 78 = 7a − 10a
17. 31t − 27t = −6
− 32
18. −33 = 5k + (−20k)
11. −32 = 15w − 7w
15. 6d − 14d = 56
−26
12. 14y − 5y = 9
−4
16. −64 = x + (−9x)
−7
19. 10 = 25y − 17y
1
20. 13h − 19h = 8
5
4
8
− 43
11
5
21. 18 = −24d − (−6d)
22. −13m + 7m = −72
−1
25. 6p + 18p = 0
23. −11k + 16k = 85
17
24. −99 = −8c + 17c
27. 0 = 12r + (−16r)
0
28. 0 = 24m − 9m
0
31. −4a − 3a = −84
12
32. −2p − 7p = 54
−6
−11
12
26. −7u + 7u = 0
0
Anyrealorirrationalnumber.
29. −36 = −5h − 4h
30. 11 = −10b − b
4
−1
33. −z + z = 21
Ø
34. 12y − 12y = −5
2
37. n − n = 12
5
20
1
38. x + x = 16
3
9
41. 2x − x = 15
2
−6
42. −
3
6
a + a = 18
7
7
14
46. 4 =
45.
4
39. −22 = s + s
7
12
11
q + 2q = −7
5
4
8
y− y
9
9
35. −9n − (−9n) = 18
Ø
35
7
43. 5b − b = 39
4
47.
9
Ø
36. 4t + (−4t) = −2
3
40. z − z = −9
4
−14
Ø
−36
8
44. −24 = − w + 3w
3
12
3
9
d + d = −48
11
11
48.
4
2
x− x=6
5
5
52.
2
4
1
d− d=
5
5
10
72
−15
−44
49. −
53.
5
2
1
= d− d
6
3
3
1
3
y + y = −20
2
4
3
3
6
k+ k=−
7
7
14
5
2
50.
−16
54. 1 =
5
57. 3p − p − 14 = 0
4
8
58.
2
3
r− r
3
5
− 16
15
2
k + 3 − 2k = 11
3
−6
51.
4
4
8
= y+ y
3
9
9
55.
3
1
x + x = 26
4
3
1
56. −15 =
24
1
59. −13 = 8 + a + a
2
5
1
a− a
6
3
1
4
−30
9
60. c − c − 3 = −8
4
4
64. 3.6y + 2.9y = 13
2
−14
61. 5.2z − 8.6z = −34
65. −9.3 = p + 2.1p
10
−3
69. 8u + 6 − 3u = 41
62. −18 = 1.3x + 1.7x
66. 3.5s − 2s = 10.5
−6
67. r − 0.2r = −6.4
7
70. −2n + 10 − 5n = 17
7
63. 7 = 2.2c − 0.8c
5
−8
71. 4s − 8 − 15s = −30
2
68. −5.2 = 3w − 3.4w
13
72. 7n − 49 + 3n = −9
4
76. 0 = −7x + 12 − 7x
6
7
−1
73. 11y + 14 − 7y = 0
− 72
74. 0 = −8z − (−15z) − 35
−9
5
77. 32 = 2a − (−14) + 7a
78. 3x + 9x + 47 = 11
−3
−6 17
85. 14 = 12r − 4 + (−12r)
Ø
79. −80 = 8b − 15 + 5b
−5
2
81. 4 − 6x + (−8x) = 90
75. 9u − 27 + (−12u) = 0
82. 38r − 14 − 17r = 16
10
7
86. −22n − 11 + 22n = 33
Ø
83. 28 = 34 + 7y − (−y)
− 34
87. −9t − 30 − (−9t) = 0
Ø
ALG catalog ver. 2.6 – page 130 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
80. −38 + 9t − (−14t) = 54
4
84. 46 = −4n − 9 + 19n
11
3
88. 15y − 8 − 15y = 17
Ø
GE
89. 2k − 23 − 11k = −17
90. 4h + (−10h) + 21 = −15 91. −55 = 21x − 37 − 3x
− 23
93. 7x − (−9x) − 15x = 8
94. 75 = −5a + 2a − 22a
97. 0 = −8z + 27z − 19z
101. 37 − (w + 8) = 32
−2
99. 15 − (−7r) − 7r = 15
IR
100. 8 = −12k − 8 + 12k
IR
IR
102. 0 = −(p − 22) − 9
−3
96. −3q − 25q + 5q = 46
−5
98. 45z − 9z + (−36z) = 0
IR
− 15
95. 8p + 7p − (−3p) = −90
−3
8
92. −1 = 64m + (−14m) + 9
−1
6
13
103. −(31 − k) + 24 = −5
104. 12 − (x + 2) = 25
−15
2
105. 5 = −(11 − 4s) − 14
106. 36 − (5a + 19) = −3
109. −3u − (5u + 7) = −2
3
116. −19 − 6(w − 8) = 35
−1
118. 21m + 6(m − 2) = 12
−8
119. −11(4 − n) − n = −60
8
9
121. 6a + 2(4a + 1) = −5
−6
126. −8 = 12h − 4(2 + 3h)
IR
3
(3k + 2) − 2k
4
14
138.
7
132. 2 =
1
5
134. 5 =
1
2
Ø
1
131. 4x + (12x + 2) = 3
2
5
4
1
2
(4y − 3) − = −1
3
3
128. −8p − 8(3 − p) = −15
Ø
2
130. − (6z − 15) − 4z = 0
3
−3
− 56
127. −2(9d + 4) + 18d = 10
IR
3
129. 1 − (5w − 10) = 16
5
135.
11
3
1 2
+ (5 − 3a) = −3
5 5
3
136. −6x + (x + 8) = 0
2
13
3
3
(p + 3) − 4 = 2
4
5
4
139. −6 = q − (3 − q)
5
3
(8p − 20) − 5
4
8
3
−2
2
(2a − 1) − 3a = 11
3
140.
−7
141. −2p + (−7p) − (−5) = 59
143. 14 = 24 − (−3s) + (−9s)
145. 19a − 7 − 17a + 23 = −14
−6
5
3
−15
147. −29 = −36 + 14n + 7 + 21n
149. 17 − c − 5c + 22 = 20
0
142. 6 = −11 − (−2k) + (−12k)
− 17
10
144. −8p − (−27) + (−15p) = 27
0
146. 22 − 25y + 9 − (−13y) = 43
−1
148. 16d − (−4) − 11d + 10 = 49
7
150. −14 + (−t) + 44 − 7t = −28
− 19
6
151. −2m − 6 + 19 + (−4m) + 9 = 0
11
3
153. 58 = x + 7 + (−15x) + 9 − 17x
− 42
31
155. 4y − 14 + 18y − 5 − 10y = 29
5
124. 13 = 4(−7r − 3) − 2r
123. 55 + 5(3x + 7) = 0
−2
125. 3(x − 8) − 3x = −24
120. 48 + 9(y − 5) = 48
− 85
122. −3(5 − 2t) + 28 = 1
− 12
1
(2x − 5) + 1 = 4
3
115. 22 = 5b + 7(b − 2)
3
2
8
117. 57 = 22 − 7(k + 3)
137.
112. −18 = 8p − (33 − 2p)
0
114. −3(z + 5) + 29 = −10
−6
−7
111. −(5c − 30) − 2c = 30
− 34
113. 4(y − 9) − 10y = 0
108. −(8d − 7) − 63 = 0
− 94
110. 6r − (−2r − 27) = 21
−58
133.
107. −8 − (12y + 2) = 17
4
15
2
4
152. −38 = −9r + 18 + r − 26
1
4
15
4
154. 5 − 8x − 49 + 14x + 26 = 0
3
156. 8 + 12w − 17w + w + (−29) = 19
157. −12 − 9r + 53 + 17r − 6 = 41
3
4
158. −4w + 17 − 3w − 3 + 9w = −5
159. 5 + 6k − 9k − 40 − 11k = −33
− 17
160. 13m + 16 − 4m − 29 + 7 = 0
ALG catalog ver. 2.6 – page 131 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
2
3
−10
− 19
2
GE
161. 10y − (−18) + 3y − 7 − 13y = 0
163. 2z + 14z + 14 + (−16z) + 6 = 20
165. 44 = −(11r − 20) + 2r + 36
162. 17 − 2p + p − 20 − (−p) = −5
Ø
164. −14 − 9x − (−11x) + 6 − 2x = −8
IR
4
3
166. 8x − (26 + 6x) + 60 = 21
− 13
2
167. 21 + 9t − (5t − 7) = −8
−9
168. 37 − (4d − 6) + 9d = 108
13
169. 5 + 3(4 − 3d) + 6d = −7
8
170. 32 − 5c − 2(c − 1) = −50
12
171. 6(2p + 3) − 4p + 37 = 95
5
172. 22 = −2k + 11 + 4(6 + k)
173. (6n − 9) − (2n − 19) = 0
− 52
174. −18 = −(7 − 2a) − (a + 12)
175. (8 − 3z) − (12 + 3z) = 10
177. 3(4t − 6) − 10(t + 1) = 4
Ø
− 13
2
1
176. −(14s − 8) + (9s + 4) = −13
− 73
5
178. −4(7y − 12) + 6(2 + 3y) = 12
16
24
5
179. 0 = −2(m + 10) − 7(4 − 2m)
4
180. 3(7z − 1) − 5(3 + z) = −14
181. 30 = −(18k − 1) − 9(4 − 2k)
Ø
182. 2(3 − 10x) + 4(5x + 2) = −6
Ø
184. −3(6r + 5) + 9(2r + 3) = 12
IR
183. 8(a + 3) − 4(2a + 1) = 20
185.
IR
2
1
(21 + 3a) + 6( + a) = −15
3
2
1
1
187. − (12h − 4) − 6( − h) = −1
4
3
189.
1
3
(n + 6) − (n − 3) = 0
4
2
1
3
188. 7 = 16( w − 2) − (8w + 4)
4
4
0
−24
5
5
191. 3( − 2k) + (3k − 2) = 8
6
2
1
4
2
1
186. 10( x − 2) − (30 − 5x) = −11
2
5
−4
7
193. 7(2 − x) − 3(x + 1) + 3 [9 − (4 − 2x)] = 18
190.
4
1
(c − 2) − 2(c + ) = −6
5
5
192.
1
1
(r + 1) + (2r − 5) = −1
3
6
3
4
10
3
− 34
194. 4(a + 6) + 2(5 − a) − 2 [3 − (a − 2)] = 4
2
195. −5(2c + 3) − 8(1 − 3c) + 4 [5 − (1 + c)] = −15
IR
− 45
−5
196. −2(1 − 3y) − 3 [(2 + y) − 5] + 3(y − 4) = 13
Quadratic terms
197. (x2 − 3x + 10) − (x2 − 9x + 6) = −4
199. (5c2 + 4c − 5) − (5c2 + 11c + 8) = 8
− 43
−3
198. (2y 2 + y + 7) − (2y 2 + 4y − 5) = 12
200. (3p2 − 6p + 2) + (−3p2 + 8p − 2) = −26
201. −4a(3a − 3) − 6a(1 − 2a) = 36
6
202. −2(n2 + 3n − 2) − n(2n − 7) = −5
203. y(5 − 3y) + 3(y 2 − 9y + 2) = 2
2
11
204. 6u(u + 4) − 2u(3u − 8) = 10
205. n(4n − 5) + (1 − 2n)(1 + 2n) = −9
207. −p(p − 3) + (p + 6)2 = 42
2
209. (y + 3)2 − (y + 5)(y − 6) = −10
206. (c + 4)(c − 6) − c(c + 1) = 15
208. (4 − r)(4 + r) + (4 + r)2 = 0
2
5
210. (x − 4)2 − (x + 4)2 = 24
−7
211. (k + 8)(k + 1) − (k + 2)(k − 2) = 3
−1
0
1
4
−13
−4
− 32
212. (a + 7)(a − 1) − (a − 2)(a + 5) = 15
ALG catalog ver. 2.6 – page 132 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−9
4
−13
3
GF
Topic:
Variables on both sides of an equation.
Directions:
15—Solve.
16—Solve and check.
1.
4y − 10 = 3y
10
2.
16 + 2x = 10x
5.
5 − 3w = 2w
1
6.
5n + 4 = 7n
9.
4p + 2 = 11p
2
7
10. 8 − 5a = −2a
13. 5x − 3 = −x
1
2
14. 2 − 4c = 2c
17. 7r − 5 = 2r + 5
8
3
1
3
6
22. 2k + 14 = 6k − 22
7
25. 7m + 15 = 3m + 3
29. 3t + 5 = 3t + 11
2
18. 3m − 2 = 2m + 4
2
21. 4a + 2 = 6a − 12
2
−3
26. 4y + 3 = 2y − 7
9
−5
30. 4 − 6h = −6h + 5
Ø
33. 9r + 4 = 7r + 4
0
34. 3w − 5 = 8w − 5
37. 3k − 1 = 3k − 1
IR
38. 4m + 2 = 2 + 4m
Ø
0
IR
3.
4a = 3 + 7a
−1
4.
2c = 4c − 6
7.
6d = 4d − 8
−4
8.
12 − 4y = −2y
6
11. 7g = 1 − 6g
1
13
12. 9k − 12 = −5k
6
7
15. 3 + 5y = 9y
3
4
16. 7n = 1 − 4n
19. 8w − 15 = 4w − 3
3
3
1
11
20. 9t − 5 = 6t + 10
5
23. 5x + 2 = 7x − 2
2
24. 4p + 7 = 5p + 4
3
27. 9c + 7 = 4c − 3
−2
28. 5x − 6 = 8x + 6
−4
31. 7 + 2y = 2y − 3
Ø
32. 5 + 11a = 11a − 6
35. 6 − 2g = 3g + 6
0
36. 8x + 2 = 2 − 4x
0
40. 9 + 6a = 6a + 9
IR
44. 7c + 4 = 10c − 5
3
− 19
2
39. 11 − 3y = −3y + 11
Ø
IR
41. 4x − 1 = 6x + 2
45. 3a − 7 = a + 8
− 32
15
2
49. 3z + 15 = 6z − 13
28
3
53. 7w − 6 = 6w − 7
−1
57. 4f + 5 − 2f = 3f
5
42. 5p + 2 = 3p − 5
− 72
43. 9n − 5 = 3n + 11
46. 4z + 3 = 7z − 2
5
3
47. u − 7 = 8u + 3
− 10
7
48. 5d + 8 = 3d − 11
50. 8g − 7 = 3g + 7
14
5
51. 6r + 4 = r − 8
− 12
5
52. 8s + 9 = 5s − 4
− 13
3
54. 8y − 5 = 5 − 8y
5
8
55. 3a − 5 = 4a − 3
56. 9k + 4 = 4k − 9
−135
58. 5p − 6 + 2p = 4 + 3p
5
2
61. 6x − 2 = 5x − 7 − 3x
62. 4p + 5 = 3p − 11 − 8p
− 54
− 16
9
8
3
−2
59. 3c + 5 − 7c = 2c − 11
60. 8y − 9 + 2y = 5y − 3
8
3
6
5
63. 3t + 8 = 5t + 3 − 9t
64. 7k − 6 = k − 5 + 4k
− 57
1
2
65. 6u + 7 − 3u = 8 + 5u − 11
5
66. 4x + 3 − 2x = 15 − 5x + 9
67. 8a − 11 + a = 3 + 4a − 19
−1
68. −2w + 13 + 5w = 8 − 4w − 9
69. 5p + 2 − 3p = 8 + 4p − 6
70. 6c − 11 + c = 5 − 2c − 16
0
71. 2x + 5 + 3x = 14 − 2x − 2
73. 8d + 5 − 3d = 2 + 5d + 3
72. 7 − 3t + 4 = 5t − 1 + 4t
1
3
0
1
74. 2g − 6 + 5g = 3g + 2 − 8 + 4g
IR
−2
IR
75. 3 − 2r + 8 = 6r + 11 − 8r
IR
76. 12 + a − 5 + 2a = 4 − a + 3 + 4a
77. 9y − 1 − 7y = 7 − 6y − 15
− 78
78. 8t − 5 + 2t = 5 + 5t − 12
79. 4 − 2p + 25 = 7p − 8 − 5p
37
4
80. 6 + 4k − 10 = 5k − 7 − 8k
81. 5w + 2 − 8w = 5 − 3w + 1
Ø
82. 4 + 6c + 13 = 7c − 5 − c
83. 9y − 5 + y = 8 + 10y − 3
Ø
85. 9 + 6a − 14 = 8 − a − 6 + 4a
87. 13 + 6y − 8 = −3y + 6 − 2y
7
3
1
11
− 25
− 37
Ø
84. 9 + 3x + 12 = 8x − 9 − 5x
Ø
86. 7 − 3s − 9 = 2s + 11 + 5s
− 13
10
88. 20 − 4u + 3 = 5u − 8 + 2u
31
11
ALG catalog ver. 2.6 – page 133 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
IR
GF
89. 4(y + 8) = 7(y + 2)
6
90. 5(n − 1) = 2(n + 8)
91. 2(4 − c) = 3(2 − c)
−2
92. 5(3k − 6) = 6(2k − 3)
93. 4(2a − 2) = −2(1 − 5a)
95. −3(8 + 2y) = 2(y − 8)
7
4
94. 2(3u + 10) = 5(4 − 2u)
−3
0
96. −5(p + 3) = 4(2p − 7)
−1
1
97. 6(4 − 3j) = −2(3j − 5)
7
6
98. −8(4 − 5x) = −2(6x + 8)
99. 7(3k − 8) = −4(6k + 3)
44
45
100. −5(6 − r) = 3(9r − 2)
4
13
− 12
11
101. 6(4 − 3s) = −2(5 + 9s)
Ø
102. 3(8r − 5) = −4(7 − 6r)
Ø
103. 5(4 + 6h) = 3(10h − 7)
Ø
104. −2(5 − 8d) = 4(7 + 4d)
Ø
105. 3(4y + 12) = 6(2y + 6)
IR
106. −5(2m − 8) = 10(4 − m)
IR
108. 6(5y − 1) = −2(3 − 15y)
IR
107. 8(5 − 3r) = −4(6r − 10)
IR
109. 3(g − 5) + 8g = 18 − (3 − 10g)
111. 5 + 2(2n − 13) = 8n + 5(6 − n)
115. 3(x + 6) + 2x − 6 = 8x − 2(2x − 4)
117. 2(3x − 5) + 4 = 15 − (9x − 4)
−3
114. 4(s + 2) − 6s + 15 = 10 − (s + 8)
2
−4
13
6
123. 5d − 2(3 − d) = 4(2d − 3) − d
Ø
1
2
120. 3(2k − 4) + 3 = −5k − 2(k + 4)
122. 3(x + 2) − x = 2(x + 4) + 11
Ø
21
116. 3w + 6(w + 1) − 10 = 11w − 3(4 − 2w)
118. 7p − 2(3 − 4p) = 12p − (p + 4)
5
3
119. −2(c − 4) + c = 5c − 3(6 − 2c)
3
112. 6(z − 2) + 2z = 7z + 3(z − 2)
51
113. 18 − (2a − 5) − 2(a + 2) = 3a + 5
121. 1 − 3(t + 2) = 2(5 − 2t) + t
110. 7k − (4 − 2k) = 3(k + 5) − 1
30
1
13
Ø
124. 3(8p − 5) + 3 = 22p + 2(p − 6)
IR
125. 14 − 2(1 − 5r) = 4(3 + r) + 6r
IR
126. 5(2a − 3) + 2 = 4a + 2(3a − 6) − 1
127. 2 − 7(3t + 2) = −5(4t + 2) − 2
0
128. 3 + 4(5 − 7x) = 5(2x + 5) − 2
129.
1
3
(6 − 15w) = (40w − 8)
3
8
1
4
3
3
131. − (5 − 10k) = (24k + 4)
5
4
− 12
5
2
133. − (30 + 18k) = (45 − 15k)
6
3
135.
2
1
(3m + 3) = (4m + 28)
3
4
137.
3
3
(16y − 32) = − (56 − 28y)
4
7
139.
3
2
(4p − 6) = (27p − 45)
2
9
−11
5
Ø
IR
0
130.
2
1
(6x + 9) = (10x − 2)
3
2
132.
1
7
(3 + 9r) = (28 − 12r)
3
4
134.
3
1
(12z − 42) = − (4 − 16z)
6
4
136.
5
1
(3c − 3) = (20c + 5)
3
5
138.
4
2
(28n − 35) = (24n − 30)
7
3
IR
140.
3
1
(25x + 35) = (30 + 45x)
5
3
Ø
ALG catalog ver. 2.6 – page 134 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
7
2
− 25
6
IR
1
GF
5
3
141. 10 − h = h − 6
8
8
5
1
142. 1 + m = m − 7
4
4
16
−8
143.
6
1
c−2=− c+5
7
7
4
1
144. − w + 2 = 14 − w
3
3
7
−12
145.
1
3
y = 19 − y
10
5
2
1
149. 4 − m = m
3
2
1
1
146. 24 + n = n
6
3
38
24
7
153.
1
7
11
u+
=− u
4
4
16
157.
1
1
1
p+5= p− p
8
3
12
−4
150.
3
3
s+9= s
5
4
154.
4
7
1
a− = a
12
3
3
158.
2
3
1
m−3= m+ m
3
8
4
40
161.
144
60
96
7
3
148. 24 − s = s
6
2
151.
3
1
y =1+ y
10
3
−30
152.
2
2
r−4= r
3
7
60
11
156.
1 7
4
− w=
w
3 5
15
1
1
1
h − h = 13 + h
5
6
4
160.
1
3
1
x−6=
x+ x
2
10
8
159.
−60
7(15 − 3c)
= −5c + 6
8
162.
5
5
1
x − 12 = x
8
2
3
5
1
155. − z =
z−
8
12
2
16
3
72
2(5p − 1)
= 3p + 1
3
147.
163. 6a + 3 =
4(5x + 6)
= 2x + 5
3
9
− 14
169. 12c + 2.6 = 6.8c
167. 9r + 2 =
2
− 17
5(4a + 1)
3
164. 2x + 17 =
2
173. 0.07y + 2.3 = 0.3y
−104
177. 0.7d − 512 = 0.5d + 288
181. 5.02h + 4.3 = −6.08 + 5.02h
191. 0.4(6a + 1) = 3.5(a + 2)
175. 2.5x = 0.05x − 2.94
−17
4.5
0
20
Ø
193. 0.75(1.2w − 1.6) = 0.8(w + 2)
195. 1.5(c − 0.02) = 0.6(2.0c − 0.03)
500
188. 0.15x − 0.4 − 0.05x = 0.05 − 0.4x
0.9
192. 2.1(3 − p) = 0.7(7p − 6)
−6
1.5
−0.3
196. 0.25(0.08 − 0.4y) = 0.2(0.3y + 0.1)
0.04
197. 2(2.6p − 3.4) − 5.2p = 7.1 − (p + 12.3)
2
194. 0.2(0.1 − 5x) = −0.8(x − 0.1)
28
IR
186. 2.4 + 0.03z = 0.04z + 7.4 − 0.02z
190. 0.05(r − 18) = 0.2(2r − 1)
0.3
1.6
199. 0.02(0.5c + 0.3) = 0.03c − 0.07(0.3 − 0.1c)
1
0
198. 14.3(2d − 1) − 28.6d = 11.6 − (d + 14.2)
11.7
200. −0.4(0.5k − 0.4) = 1.3 − 0.6(0.3 − 0.2k)
−3
ALG catalog ver. 2.6 – page 135 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−2
176. 0.09m = 0.72 − 0.07m
184. 4.15 − 0.9m = −0.9m + 4.15
1.5
2(4y + 7)
5
172. 0.8 − 5u = −5.4u
6
182. 4.8 − 0.65m = 7.3 − 0.65m
IR
189. 0.4(4k − 0.9) = 0.1(k + 0.9)
171. 20.4 + 3.6p = 7p
180. 0.55y − 4.5 = 0.05y + 5.5
Ø
185. 0.3n + 0.4 = 0.6n + 0.7 − 0.5n
3(1 − 5x)
7
26
23
178. 0.06x + 11.2 = 11.2 − 0.04x
−6000
187. 3.2r + 10.1 − 2.1r = 5 + 0.8r
168. 8 − 3y =
−1.2
4000
179. 0.05r + 260 = 0.03r + 140
183. 0.05y + 3.1 = 3.1 + 0.05y
2.5
174. 12.48 + 0.2a = 0.08a
10
7(r − 5)
3
− 41
20
170. −3.3x = 3.9x − 18
−0.5
1
5
−4
6(3 − 2m)
=m+4
5
166.
21
2
80
−3
165.
9
GG
Topic:
Advanced first-degree equations.
Directions:
15—Solve.
16—Solve and check.
1.
1
1
− (4w + 6) = (9 − 3w)
2
3
−6
2.
3
3
(8x + 10) − (10x − 5) = 0
2
5
3.
2
2
(12p − 9) = (7p + 14)
3
7
5
3
4.
4
3
(30 − 15m) = (6m + 28)
5
2
5.
3
1
(8z − 12) = (36z + 12)
4
6
6.
5
1
(2c + 18) = (25c + 30)
2
5
7.
1
4
(12d + 36) − (96 + 32d) = 0
3
2
8.
1
1
(49 − 14x) = − (12x − 42)
7
6
9.
3
2
(5y − 4) − (2y + 8) = 0
3
5
10.
4
4
(3a + 2) = (5a − 4)
5
7
11.
7
3
(2r − 17) = (3r − 11)
3
4
12.
6
3
(3w + 14) = (5w + 16)
7
4
14.
1
8
(6u + 7) − 8u = (4 − u) + 11
3
2
− 23
16.
3
3
(2a + 3) + (8a − 2) − 9a = 0
4
5
7
18
Ø
IR
7
2
13
1
2
13. 8 + (3h − 1) = −2 + (20 − 3h)
5
2
1
4
15. 7 − (5y − 3) = 10 − (3 − 4y)
2
5
4
27
5
− 11
−3
6
7
Ø
IR
17
2
0
17. 7 + 3 [4(2x − 3) − 8x] = 3x − 8
−7
18. 11 − 2y = 2 [3y − 8(y − 2)] − 13y
19. 5k − 2 [3k + 2(k − 4)] = 4k − 5
7
3
20. r + 4 = 4 [2(8r − 7) + 5r ] − 3r
21. 3(n + 2) − 1 = 5 − 2 [6n + 2(n − 5)]
23. 2 − 5(a + 3) = 3a + 4 [5 − 2(3a + 11)]
1
3
4
22. 5x + 3 [4(2x − 5) − (−7x)] = 6 − 2(3x + 1)
20
19
8
7
24. 7 − 5 [3(2c − 5) − (−4c)] = 2 − 3(5c − 6) + 2
− 55
16
25. 7 [4w − 2(3w + 8)] − w = −4(3w − 2) + 6
−42
26. 2b − 3(4b − 6) = 10 − 3 [4(2b − 5) + 4b]
27. 5 [3(4y − 1) + (−2)] + 2y = 7 − 2(3y − 1)
1
2
28. 5x − (x + 4) = 5 − 2 [3x − 2(x + 8)]
29. 3(2a − 1) + 4 = 3a − 7 [2(3a − 1) − (−6)]
− 35
30. 9n + 4 [3n − 4(2n − 3)] = 2 − 3(2n + 4)
31. 3(m + 2) + 5m = −3 [(4m + 3)3 + 5] + 2m
32. 5k + 2 [−4(k + 3) − k ] = 4 − (2k + 3)
− 13
14
33. 6 [2(−5y − 3) + 1] − 15y = 33 + 3 [7 − 2(8y + 5)]
35. 3 [4(2r − 7) − r ] + 3r = 2 + 2 [8 + 3(2r − 1)]
−2
8
10
7
2
41
6
58
5
− 25
3
34. 4 − 8 [2(x − 3) − (−2x)] = 5 [9 − 2(3x + 1)] + 7x
17
9
36. 5 + 2 [3(2w − 5) − (−4w)] = −3 [4 − 5(w − 1)] + 2w
− 25
37. 8 + 2 [3b − 2(b − 8b + 5)] = 3(7b + 3) − 3b
39. 3 [2 − 3(2n + 4 − 3n)] + 5n = 4n − 2(n − 5)
38. 5(4c − 3) + 11 = 3 [2 − 3(2c + 8c − 1) + 5c]
3
10
3
41. 2x + 7(3x + 1) = 4 − 3 [2x + (−5) + 3(2x − 5x + 3)]
40. 7m + 2(3m − 5) = 6 [5 + 2(2m − 5 + 2m)]
1
5
4
7
42. 2 − [3k + 2(20k − 4 + 24k + 1)] = 6(2k − 3) − 3k
− 15
2
43. 3r − 4(r + 4) = 5 − 2 [3r − 2(4r − 2r + 3) − (−5)]
44. 7 + 3 [2a − 4(8 + 2a) − 3 + a] = 7a − 3(a + 1)
− 23
3
ALG catalog ver. 2.6 – page 136 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−5
13
50
GG
Quadratic terms
45. r2 + 4(r2 + 1) + 7r = 2r2 + 3(r2 + 6)
47. 12 − 4(3 − 2x2 ) + 3x = 6(x2 + 4) + 2x2
49. p(p + 4) = p2 + 6p − 8
48. 3(y 2 + 3y) − y 2 = 3y + 2(1 + y 2 ) + 16
8
50. y 2 − 5y − 12 = y(y − 3)
4
51. (w + 1)(w − 4) = w2 + 9w + 17
53. 3r2 + r − 17 = 3r(r − 11)
55. 5h2 + 3h = h(5h − 4)
46. 2(a2 − 1) + 3(a2 + 1) + 6a = 5(a2 + 2)
2
54. 2m(4m − 7) = 8m2 − 10
1
2
56. 2p(3p − 1) = 6p2 + 5p − 4
0
57. (3x + 5)(2x − 1) = x(6x + 1) − 16
59. (7a − 1)(a + 5) = 7a(a + 5) − 2
61. (k − 5)(3k + 4) = 3k 2 + 4k − 10
2
65. 2m(6m − 1) = (3m − 2)(4m + 3)
67. 4c(4c + 9) = (8c − 11)(2c + 1)
2
− 11
50
69. (9p − 4)(p + 2) = (3p + 1)(3p − 5)
71. (14b − 9)(2b + 3) = (7b − 5)(4b + 5)
5
3
62. 5w2 + 10w − 7 = (5w − 7)(w + 2)
− 23
63. (4x − 3)(x + 3) = 4x2 + 7x − 5
4
7
60. (2 − n)(4 − n) − n2 = 5(1 − n)
−3
20
13
5
7
58. (1 − y)2 + (y − 1)2 = 2y 2 − 18
− 11
6
64. 2y 2 − 9y − 4 = (y − 6)(2y + 5)
13
66. (6x + 1)(4x − 5) = 3x(8x − 7)
−1
68. (4b − 2)(5b + 4) = 10b(2b − 3)
2
9
−1
70. (5h + 4)(6h + 1) = (3h + 1)(10h + 3)
3
26
72. (2x + 1)(9x − 4) = (3x + 2)(6x − 1)
2
9
3
−6
52. n2 + 6n − 10 = (n − 5)(n − 2)
− 74
3
2
1
− 10
− 14
73. (3x + 5)(2x − 7) + 2x2 = (4x + 1)(2x − 5)
30
7
74. (5y − 3)(2y + 1) = y 2 + (3y + 5)(3y + 2)
− 13
22
75. (4u + 3)(3u + 4) = (2u − 1)(7u − 5) − 2u2
− 16
76. 2c2 + (6c − 5)(3c + 1) = (4c + 3)(5c − 2)
1
16
77. 4u − 7 − (u − 1)(1 + u) + (u − 2)(u − 3) = −3
79. (k + 1)(2k − 1) − (k − 1)(2k + 1) = −18
−9
3
78. 8 + 3(4n − 1) + n2 + (2 − n)(n + 1) = 6n + 42
80. 2(5a − 3a2 ) + (3a − 1)(2a + 3) = 8a + 12
ALG catalog ver. 2.6 – page 137 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
5
3
5
GH
Topic:
Solving for other variables (first-degree equations). See also categories JE (factorable equations) and LM
(rational equations).
Directions:
19—Solve for the indicated variable.
1.
x − a = b; for x
a+b
2.
A = i + p; for p
A−i
3.
C = K + 273; for K
4.
C − 273
5.
P = a + b − c; for c
a+b−P
9.
6.
e = v + f − 2; for f
2a − z = a; for z
a
7.
180 − X
K = x − (y − 1); for y
8.
x+1−K
e−v+2
10. 3b + y = b; for y
−2b
12.
−c
13. ax = b; for x
b
a
F
m
17. F = ma; for a
21. I =
E
; for R
R
14. y = kx; for x
y
k
A
; for W
W
A
L
23. R =
V
`h
27. C = 2πr; for r
I
Pt
26. V = `wh; for w
25. I = Prt; for r
V
h
15. V = Bh; for B
W
F
22. L =
19. V = IR; for I
W
I2
V
; for V
I
16. d = rt; for t
20. A = LW ; for W
IR
24. d =
C
; for C
π
28. A =
1
bh; for h
2
C
2π
1 2
r w; for w
2
V
πr2
30. W = I 2 R; for R
33. S =
πdN
; for d
12
12S
πN
34. s =
gt2
; for g
2
2s
t2
35. V =
b2 h
; for h
3
37. C =
mv 2
; for r
r
mv 2
C
38. t =
pD
; for s
2s
pD
2t
39. F =
gm1 m2
; for m1
d2
2i
r2
3V
b2
2c − 11
3
45. 2ax + 1 = ax + 5; for x
4
a
5
a
2
46. 3by − 2 = 2by + 1; for y
n
+ b = c; for n
a
ac − ab
57. v = V + gt; for t
v−V
g
r
= 2s; for r
3
3p − 6s
58. p = 2w + 2`; for w
p − 2`
2
a+b+c
; for c
4
4A − a − b
43. 3w =
1
z − 5; for z
2
6w + 10
47. 3rw + 1 = rw − 7; for w
62. Z =
X −Y +1
; for Y
W
1 + X − ZW
55. 3x + a = b; for x
b−a
3
59. A = P + Prt; for t
A−P
Pr
v2 − v1
; for v1
t
v2 − at
ALG catalog ver. 2.6 – page 138 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
e
10t2
36. R =
k`
; for `
d2
40. F =
kE1 E2
; for E1
d2
Rd2
k
44.
1
x + 3y = −6; for y
3
1
− x−2
9
48. 6bct − 1 = 4bct + 1; for t
52. 2nk + 3m = nk + 1; for k
1 − 3m
n
1+t
9p
63. a =
2A
b
1
bc
50. 5ax − 2b = ax − c; for x 51. 1 − 3pr = 6pr + t; for r
54. p −
πd
F d2
kE2
4
−
r
2b − c
4a
b+d
2c
61. A =
3−
3
b
49. 3cy − b = cy + d; for y
53.
42. 5a = −2b + 6; for b
A
L
32. e = 10kt2 ; for k
F d2
gm2
41. 2c − 3d = 11; for d
d
r
V
R
29. V = πr2 h; for h
31. i =
2
8
d = d − w; for w
3
3
2d
18. W = Fd; for d
E
I
T = 1 − (m + n); for n
1−m−T
c
3c
= − ; for x
2
2
11. x −
X = 180 − Y ; for Y
56. a − 2y = 5b; for y
a − 5b
2
60. E = Ir + IR; for r
E − IR
I
64. v =
s2 − s1
; for s2
t
vt + s1
GH
65.
a
+ 1 = 0; for a
t
−t
66. 0 =
a
− 1; for t
t
a
67.
ax
= c + d; for x
b
68.
bx
− a = d; for x
c
bc + bd
a
69. F =
9
C + 32; for C
5
70. S = P + prt; for r
S−P
pt
5
(F − 32)
9
73. I = p(1 + rt); for t
9
C + 32
5
81. s = vt + 16t2 ; for v
82. s =
s − 16t2
t
2
; for w
−w
m + 2n
= t; for n
rx
90. S =
v(P − r)
; for P
f −r
98. V =
1
(b1 + b2 )h; for b1
2
91. s =
95. r =
h(B + b)
; for B
H
99. A =
111. 5e(e + f ) = e(e − 3f ); for f
−
e
2
2V − IR
; for R
2I
h(b1 + b2 )
; for b1
2
103. x = 2 + y(w − 3z); for z
88. a = 2c(s1 − s2 ); for s2
2cs1 − a
2c
92. I =
2b
a
96. K =
100. d =
a(2t − r)
; for t
2
2d + ar
2a
104. R = r(r − 2s) + k; for s
r2 + k − R
2r
108. I =
1
c(c + m) = c(c − m); for m
2
2V
; for r
R + 2r
2V − IR
2I
110. (2x + b)(2x − b) = 4x(x − 1); for x
ALG catalog ver. 2.6 – page 139 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
kb + B
; for b
3
3K − B
k
2V − 2Ir
I
112.
nE
; for R
R + nr
nE − Inr
I
2 − x + wy
3y
1 2
2V
πh (3r − h); for r 107. I =
; for R
3
R + 2r
sm
H
A − 2πr2
2πr
2A − hb2
h
3V − πh3
3πh2
109. (a + b)(y − 2) = (y + 2)(b − a); for y
H
; for t1
m(t1 − t2 )
H
(t1 − t2 ); for t2
m
84. A = 2πr2 + 2πrh; for h
2V − 2Ir
I
b + c − 4d
4
2A − b2 h
h
t1 −
H + smt2
sm
102. c = 4(a + d) − b; for a
106. V =
80. s =
S − πrs
πs
HV − hb
h
S−a+d
d
105. A =
87. S = π(r + R)s; for R
cd + bn
a
G(f − r)
+r
v
101. S = a + (n − 1)d; for n
a
; for r
1−r
am − bn
= d; for m
c
94.
3c − d
3c
A − πr2
πr
S−a
S
trx − m
2
97. G =
83. A = πr2 + πr`; for `
A − 2πr2
2πr
T n2 − 2
T
93.
1 2
gt + vo t; for vo
2
86. A = 2πr(r + h); for h
L − Lo · L
Lo · t
n
(a + 50); for a
2
IR
2
76. 3c(1 − y) = d; for y
2t
− 50
n
2s − gt2
2t
85. L = Lo (L + ct); for c
n2
79. t =
V
R
− ; for V
I
2
Ir +
S − 2an
2a
5
(F − 32); for F
9
78. C =
2A
−b
h
89. T =
75. S = 2a(n + `); for `
c − ab
a
1
h(b + c); for c
2
72. r =
v 2 − u2
2s
74. a(x + b) = c; for x
I −p
pr
77. A =
71. v 2 = u2 + 2as; for a
ac + cd
b
c
3
b2
4
HA
Topic:
Number problems (simple).
Directions:
0—(No explicit directions.) 16—Solve and check.
20—Solve using one variable.
39—Translate and solve. 40—Write an equation and solve.
2.
−18 is the sum of p and −65. Solve for p.
4.
The sum of x and −20 is 13. Find x.
The sum of a number and 85 is 305. What is the
number? 220
6.
The sum of a number and 148 is 110. What is the
number? −38
7.
152 equals a certain number plus 75. Find the
number. 77
8.
−3 equals a certain number plus 28. Find the
number. −31
9.
242 is equal to a number increased by 117. Find the
number. 125
10. 37 is equal to a number increased by 65. Find the
number. −28
1.
29 is the sum of r and 54. Solve for r.
3.
The sum of a and 12 is −7. Find a.
5.
−25
−19
47
33
11. When a number is increased by 29, the result is 76.
What is the number? 47
12. When a number is increased by 96, the result is 42.
What is the number? −54
13. 24 more than a number is 80. Find the number.
14. 81 more than a number is 100. Find the number.
56
19
15. 16 is five more than n. Find n.
16. 42 is eight more than y. Find y.
11
17. 36 more than a number is 25. What is the number?
−11
34
18. 19 more than a number is 6. What is the number?
−13
19. Twelve more than a number is −12. Find the
number. −24
20. Five more than a number is −32. Find the number.
21. When 24 is subtracted from b, the result is −80.
Find b. −56
22. When 56 is subtracted from c, the result is 18.
Find c. 74
23. A number decreased by 110 is 54. Find the number.
24. A number decreased by 25 is −78. Find the number.
−37
−53
164
25. A number reduced by 70 is −35. What is the
number? 35
26. A number reduced by 41 is 119. What is the
number? 160
27. When x is decreased by 28, the result is −5. Find x.
28. When r is decreased by 95, the result is −40. Find r.
23
55
29. When y is subtracted from 300, the result is 144.
Solve for y. 156
30. When a is subtracted from 106, the result is −200.
Solve for a. 306
31. 75 minus some number is −113. What is the
number? 188
32. 50 minus some number is −121. What is the number
33. 21 less than a certain number is 79. What is the
number? 100
34. 34 less than a certain number is 46. What is the
number? 80
35. −20 is 75 less than some number. Find the number.
36. −12 is 110 less than some number. Find the number.
55
98
37. 105 less than b is equal to 85. Solve for b.
39. 12 less than k is −8. Find k.
171
190
40. 300 less than a is −190. Find a.
4
41. The difference of 35 and p is 15. Find p.
38. 14 less than f is equal to 57. Solve for f .
20
110
42. The difference of 100 and c is 66. Find c.
ALG catalog ver. 2.6 – page 140 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
71
34
HA
43. −5 is the difference between 21 and some number.
What is the number? 26
44. −29 is the difference between 11 and some number.
What is the number? 40
45. −200 is the difference between a number and 500.
What is the number? 300
46. 35 is the difference between a number and 125.
What is the number? 160
47. The difference between n and −9 is 25. Find n.
48. The difference between c and 17 is −3. Find c.
16
49. 4 times a number is 128. What is the number?
51. −4 times a number is 100. Find the number.
32
−25
14
50. 12 times a number is 156. What is the number?
52. −6 times a number is 72. Find the number.
−12
53. Some number multiplied by negative five is 145.
Find the number. −29
54. Some number multiplied by negative three is 81.
Find the number. −27
55. When a number is multiplied by −6, the result
is −90. What is the number? 15
56. When a number is multiplied by −11, the result
is −154. What is the number? 14
57. Twice a number is 448. Find the number.
58. Three times a number is −246. Find the number.
224
13
−82
59. When a number is tripled, the result is −51. What
is the number? −17
60. When a number is doubled, the result is 46. What is
the number? 23
61. Five times a number is 65. What is the number?
62. Seven times a number is −105. What is the number?
−15
13
63. 1000 equals a number times 160. Find the number.
25
4
65. −21 times r is −35. Find r.
− 15
2
66. −20 times k is 65. Find k.
5
3
67. Half of a number is −13. What is the number
69. −30 equals one-third of n. Solve for n.
−26
− 13
4
68. Half of a number is 45. What is the number?
70. 21 equals one-third of u. Solve for u.
−90
71. One-fourth of a number is 16. What is the number?
90
63
72. One-tenth of a number is −50. What is the number?
−500
64
73. Two-thirds of x is equal to 72. Solve for x.
75. −21 is three-fourths of m. Find m.
74. Three-fifths of y is equal to 60. Solve for y.
108
76. −80 is one-half of k. Find k.
−28
77. The product of −26 and c is 156. Solve for c.
79. 120 equals the product of z and 45. Find z.
81. The quotient of x and −6 is 15. Find x
83. The quotient of p and 2 is 21. Find p.
6
8
3
−90
42
85. A number divided by 4 is −13. What is the number?
−52
100
−160
78. The product of −14 and d is −266. Solve for d.
80. −90 equals the product of h and 18. Find h.
82. The quotient of y and 10 is −3. Find y.
84. The quotient of w and −5 is −8. Find w.
19
−5
−30
40
86. A number divided by −12 is 3. What is the number?
−36
87. −6 equals a number divided by −9. Find the
number. 54
89. A number divided by −22 is
64. −750 equals a number times 100. Find the number.
1
2.
88. 14 equals a number divided by 8. Find the number.
112
What is the number? 90. A number divided by 10 is − 15 . What is the number?
−11
91. When a number is divided by 7, the result is −7.
What is the number? −49
−2
92. When a number is divided by −25, the result is −1.
What is the number? 25
ALG catalog ver. 2.6 – page 141 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HA
93.
2
3
is equal to a divided by 50. Solve for a.
94. − 52 is equal to r divided by 45. Solve for r.
100
3
95. The quotient c divided by −18 equals 1. Find c.
−18
− 225
2
96. The quotient w divided by 5 equals 36. Find w.
180
97. 24 minus the product of x and −8 is 96. Solve for x.
9
98. −10 minus the product of 12 and a is −154. Solve
for a. 12
99. −13 plus the product of y and 10 is −53. Solve
for y. −4
100. 100 plus the product of p and −2 is 50. Solve for p.
101. Seventeen plus half a number is 12. What is the
number? −10
102. Fourteen minus half a number is −31. What is the
number? 90
103. 48 is equal to eight times the quantity r minus 9.
Find r. 15
104. −115 is equal to five times the quantity c plus 3.
Find c. −26
105. The sum of 27 and five times a number is 38. Find
the number. 11
106. The sum of −18 and three times a number is 10.
Find the number. 28
25
5
3
107. 80 more than twice a number is −10. What is the
number? −45
108. 23 more than twice a number is 15. What is the
number? −4
109. 111 is 75 more than six times k. Find k.
110. 200 is 90 more than five times d. Find d.
6
22
111. −16 is 21 more than half of a number. Find the
number. −74
112. 5 is 25 more than one-third of a number. Find the
number. −60
113. Eleven less than four times a number is forty-nine.
What is the number? 15
114. Twelve less than five times a number is thirty-three.
What is the number? 9
115. Four less than one-fifth of a number is −16. What is
the number? −60
116. Ten less than two-thirds of a number is 20. What is
the number? 45
117. 19 less than three-fourths of n is 32. Find n.
118. 33 less than half of a is −5. Find a.
68
56
119. If 35 is added to three times a number, the result
is 101. Find the number. 22
120. If 32 is added to four times a number, the result
is 100. Find the number. 17
121. 46 is the difference of 18 and 7 times a number.
What is the number? −4
122. 106 is the difference of 50 and 8 times a number.
What is the number? −7
123. When a number is multiplied by −5 and then added
to 16, the result is −24. Find the number. 8
124. When a number is multiplied by −8 and then added
to −6, the result is 82. Find the number. −11
125. When 86 is subtracted from ten times a certain
number, the result is 34. What is the number? 12
126. When 70 is subtracted from four times a certain
number, the result is −10. What is the number?
127. When 22 is subtracted from three times a number,
the result is −1. Find the number. 7
128. When 45 is subtracted from nine times a number,
the result is 0. Find the number. 5
ALG catalog ver. 2.6 – page 142 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
15
HB
Topic:
Number problems (advanced).
Directions:
0—(No explicit directions.) 16—Solve and check.
20—Solve using one variable.
21—Solve using two variables. 22—Solve using a system of equations.
39—Translate and solve. 40—Write an equation and solve.
1.
Four times a certain number is the same as the
number increased by 78. Find the number. 26
2.
Five times a certain number is the same as the
number decreased by 52. Find the number. −13
3.
35 less than half of a number is the same as three
times the number. What is that number? −14
4.
27 more than half of a number is the same as twice
the number. What is that number? 18
5.
If you add three-fourths of a number to the number
itself, you get 49. What is the number? 28
6.
If you add three-fifths of a number to the number
itself, you get −32. What is the number? −20
7.
A number is decreased by 21. The result is
multiplied by 2. The result is 8 more than the
original number. Find the number. 50
8.
A number is increased by 45. The result is multiplied
by 2. The result is 18 less than the original number.
Find the number. −108
9.
When p is decreased by 36, the result is 8 less than
one-third of p. Find p. 42
10. When m is increased by 6, the result is 3 less than
one-fourth of m. Find m. −12
11. Five times the sum of a number and 12 is 32 less
than four times the number. What is that number?
−92
12. Six times the sum of a number and 15 is 99 more
than three times the number. What is that number?
3
13. If you triple a number and then add 10, you get
one-half of the original number. What is the
number? −4
14. If you double a number and then subtract 75, you
get one-third of the original number. What is the
number? 45
15. Four times a number, increased by 36, is the same as
32 decreased by twice the number. Find the number.
16. Three times a number, decreased by 14, is the same
as 22 decreased by five times the number. Find the
number. 9
− 23
2
17. If 3 more than twice a number is decreased by 8 less
than 4 times the number, the result is the same as
the orginal number decreased by 16. What is the
number? 9
18. If 10 less than twice a number is decreased by
5 less than 3 times the number, the result is same
as the original number increased by 7. What is the
number? −6
19. Eight times the sum of k and 5 is one more than the
product of k and 5. Find k. −13
20. Five times the sum of a and 3 is one more than the
product of a and 3. Find a. −7
Pairs of numbers
21. One number is 14 less than another. Their sum
is 300. Find each number. 157, 143
22. One number is 21 more than another. Their sum
is −423. Find each number. −222, −201
23. One number is six times another number. Their sum
is −147. Find the numbers. −21, −126
24. One number is ten times another number. Their sum
is 253. Find the numbers. 23, 230
25. One number is five times another number. Their
sum is 3. What are the numbers? 1 , 5
26. One number is three times another number. Their
sum is 3. What are the numbers? 3 , 9
27. One number is 12 less than three times another
number. Their sum is 188. What are the numbers?
28. One number is 15 less than half of another number.
Their sum is 6. What are the numbers? 14, −8
2
2
4
4
50, 138
29. One number is 14 more than another. When half of
the smaller number is added to the larger number,
the sum is 59. What are the numbers? 30, 44
30. One number is 8 more than another. When twice
the larger number is added to the smaller number,
the sum is 55. What are the numbers? 13, 21
ALG catalog ver. 2.6 – page 143 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HB
31. One number is 9 less than another. When 4 times
the larger is subtracted from 12 times the smaller,
the difference is 36. Find each number. 0, −9
32. One number is 17 less than another. When 3 times
the smaller is subtracted from 4 times the larger, the
difference is 21. Find each number. −30, −47
33. One number is twice another number. When the
larger is diminished by 10, the result is 5 greater
than the smaller. Find the numbers. 15, 30
34. One number is half of another. When the larger
is diminished by 20, the result is 4 less than the
smaller. Find the numbers. 16, 32
35. The sum of two numbers is −117. One number is
11 less than the other. Find the numbers. −53, −64
36. The sum of two numbers is 556. One number is
124 more than the other. Find the numbers. 216, 340
37. The sum of two numbers is −103. One number is
37 larger than the other. What are the numbers?
38. The sum of two numbers is 106. One number is
44 smaller than the other. What are the numbers?
−33, −70
39. The sum of two numbers is −18. One number
exceeds the other by 38. What are the numbers?
31, 75
40. The sum of two numbers is 43. One number exceeds
the other by 75. What are the numbers? −16, 59
−28, 10
41. The sum of two numbers is 284. One number is
three times the other. What are the numbers?
71, 213
42. The sum of two numbers is −216. One number is
five times the other. What are the numbers?
−36, −180
43. The sum of two numbers is −49. Twice the first
number is equal to five times the second number.
Find the two numbers. −35, −14
44. The sum of two numbers is 72. One-third of the first
number is equal to two-thirds of the second number.
Find the two numbers. 48, 24
45. The sum of two numbers is 95. The larger number
increased by 21 equals the smaller number increased
by 32. Find the numbers. 42, 53
46. The sum of two numbers is −10. The larger number
decreased by 18 equals the smaller number increased
by 18. Find the numbers. −23, 13
47. The sum of two numbers is 68. Six times the smaller
number is 8 less than half the larger number. What
are the numbers? 4, 64
48. The sum of two numbers is 14. Half the larger
number is 7 more than twice the smaller number.
What are the numbers? 0, 14
49. The sum of two numbers is 1000. The first number
is 350 less than two-thirds of the other number.
What are they? 190, 810
50. The sum of two numbers is 450. Twice the first
number is 65 more than half the other number.
What are they? 116, 334
51. The sum of two numbers is 45. If 4 times the smaller
is increased by 3 times the larger, the result is 150.
Find the numbers. 15, 30
52. The sum of two numbers is 99. If 10 times the
smaller is decreased by 2 times the larger, the result
is 30. Find the numbers. 19, 80
53. The difference of two numbers is 12. When twice the
larger is subtracted from 5 times the smaller, the
difference is −9. Find each number. 5, 17
54. The difference of two numbers is 5. When twice the
larger is subtracted from 6 times the smaller, the
difference is 26. Find each number. 14, 9
55. The difference of two numbers is 13. Four times the
larger number is 68 more than twice the smaller.
What are the two numbers? 8, 21
56. The difference of two numbers is 18. Twice the
smaller number is 56 less than three times the larger.
What are the numbers? 2, 20
57. The second of two numbers is 8 more than twice
the first number. Their sum is 29. Find the two
numbers. 7, 22
58. The second of two numbers is 4 more than 7 times
the first number. Their sum is 76. Find the two
numbers. 9, 67
59. The larger of two numbers is 24 more than half of
the smaller. The sum of the two numbers is −3.
Find the two numbers. −18, 15
60. The smaller of two numbers is 1 more than
three-fifths of the larger. The sum of the numbers
is 17. Find the two numbers. 7, 10
ALG catalog ver. 2.6 – page 144 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HB
61. The larger of two numbers is 10 less than twice the
smaller. If half the larger is increased by 4 times the
smaller, the result is 50. What are the numbers?
11, 12
63. The first of two numbers is 11 more than the other.
If the first is increased by three times the other
number, the result is 75. Find the two numbers.
62. The smaller of two numbers is 2 less than half the
larger. If twice the larger is decreased by 5 times the
smaller, the result is zero. What are the numbers?
8, 20
64. The first of two numbers is 9 less than the other. If
the first is decreased by four times the other number,
the result is −18. Find the two numbers. −6, 3
27, 16
65. Three numbers have the sum 207. The second
number is 9 more than the first, and the third is
3 less than the second number. Find the three
numbers. 64, 73, 70
66. Three numbers have the sum 107. The second
number is 12 less than the first, and the third is
5 more than the second number. Find the three
numbers. 42, 30, 35
67. Three numbers have the sum 81. The second is twice
the first, and the third number is 6 more than the
second. Find the three numbers. 15, 30, 36
68. Three numbers have the sum 129. The second is
three times the first, and the third number is 11 less
than the second. Find the three numbers. 20, 60, 49
69. The second of three numbers is 6 times the first. The
third is 6 less than the first. The sum of the second
and third numbers is 98. Find all three numbers.
70. The second of three numbers is 4 times the first.
The third is 21 more than the first. The sum of the
second and third numbers is 156. Find all three
numbers. 27, 108, 48
14, 84, 8
71. The second of three numbers is 3 times the first.
The third is 5 more than the second. If the third is
decreased by twice the second, the result is −1. Find
all three numbers. 2, 6, 11
72. The second of three numbers is 4 times the first.
The third is 13 less than the second. If twice the
first is decreased by the third, the result is −21.
Find all three numbers. 17, 68, 55
ALG catalog ver. 2.6 – page 145 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HC
Topic:
Consecutive integers.
Directions:
0—(No explicit directions.) 16—Solve and check.
20—Solve using one variable.
21—Solve using two variables. 22—Solve using a system of equations.
39—Translate and solve. 40—Write an equation and solve.
1.
Find two consecutive integers whose sum is 139.
2.
Find two consecutive integers whose sum is −91.
−46, −45
69, 70
3.
The sum of two consecutive integers is 211. Find the
integers. 105, 106
4.
The sum of two consecutive integers is −49. Find
the integers. −25, −24
5.
The sum of three consecutive integers is −342. What
are the integers? −115, −114, −113
6.
The sum of three consecutive integers is 108. What
are the integers? 35, 36, 37
7.
Find three consecutive integers whose sum is −99.
8.
Find three consecutive integers whose sum is 135.
−34, −33, −32
9.
The sum of two consecutive integers is −75. What is
the smaller integer? −38
44, 45, 46
10. The sum of two consecutive integers is −59. What is
the smaller integer? −30
11. The sum of three consecutive integers is −171. Find
the largest of the three integers. −56
12. The sum of three consecutive integers is 216. Find
the largest of the three integers. 73
13. The sum of three consecutive integers is 282. What
is the middle integer? 94
14. The sum of three consecutive integers is −501. What
is the middle integer? −167
15. The of sum of four consecutive integers is −130.
What is the smallest integer? −34
16. The sum of four consecutive integers is 182. What is
the smallest integer? 44
17. Find four consecutive integers such that the sum of
the second and fourth is l32. 64, 65, 66, 67
18. Find four consecutive integers such that the sum of
the first and fourth is −35. −19, −18, −17, −16
19. Find three consecutive integers such that the sum of
second and third is −17. −10, −9, −8
20. Find three consecutive integers such that the sum of
the first and third is 40. 19, 20, 21
21. Find two consecutive integers such that the larger
minus twice the smaller is −13. 14, 15
22. Find two consecutive integers such that the smaller
plus twice the larger is −61. −21, −20
23. Find two consecutive integers such that half the
smaller plus three times the larger is −32. −10, −9
24. Find two consecutive integers such that four times
the larger minus half the smaller is 53. 14, 15
25. Find three consecutive integers such that three times
the first, added to the third, is 102. 25, 26, 27
26. Find three consecutive integers such that four times
the second, added to the third, is 66. 12, 13, 14
27. Find three consecutive integers such that the sum of
the first and second is 9 more than half of the third.
28. Find three consecutive integers such that the sum of
the second and third is 33 more than half of the first.
6, 7, 8
20, 21, 22
29. Find two consecutive integers such that the difference
of their squares is 201. 100, 101
30. Find two consecutive integers such that the difference
of their squares is 151. 75, 76
31. Find three consecutive integers such that the product
of the first and second is 10 less than the square of
the third. 2, 3, 4
32. Find three consecutive integers such that the square
of the first is 14 less than the product of the second
and third. 4, 5, 6
ALG catalog ver. 2.6 – page 146 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HC
Odd & Even integers
33. The sum of two consecutive even integers is 114.
What are the integers? 56, 58
34. The sum of two consecutive even integers is −86.
What are the integers? −44, −42
35. Find two consecutive odd integers whose sum is −36.
36. Find two consecutive odd integers whose sum is 72.
−19, −17
35, 37
37. The sum of three consecutive even integers is 174.
Find the integers. 56, 58, 60
38. The sum of three consecutive even integers is −126.
Find the integers. −44, −42, −40
39. The sum of three consecutive odd numbers is −219.
What are they? −75, −73, −71
40. The sum of three consecutive odd integers is 189.
What are they? 61, 63, 65
41. The sum of four consecutive even integers is −420.
Find the smallest integer. −108
42. The sum of five consecutive even integers is 520.
Find the smallest integer. 100
43. The sum of four consecutive odd integers is 336.
Find the largest integer. 87
44. The sum of five consecutive odd integers is −305.
Find the largest integer. −57
45. Find two consecutive odd integers such that 4 times
the first integer is 29 less than 7 times the second.
46. Find two consecutive even integers such that twice
the larger is 14 less than 5 times the smaller. 6, 8
5, 7
47. An odd integer is added to 3 times the next
consecutive odd integer. The sum is 66. Find the
two integers. 15, 17
48. An even integer is doubled. The result is 16 more
than the next consecutive even integer. Find the two
integers. 18, 20
49. Find three consecutive odd integers such that twice
the sum of the smaller two integers is 25 more than
3 times the largest. 33, 35, 37
50. Find three consecutive even integers such that twice
the sum of the larger two is 4 less than 5 times the
smallest. 16, 18, 20
51. If the sum of three consecutive odd integers is
decreased by 25, the result is twice the middle
integer. Find the three odd integers. 23, 25, 27
52. If the sum of three consecutive even integers is
decreased by 80, the result is equal to half the
middle integer. Find the three even integers.
30, 32, 34
53. Find four consecutive even integers such that twice
the sum of the first and fourth is 124. 28, 30, 32, 34
54. Find four consecutive odd integers such that twice
the second, added to the last, is 61. 17, 19, 21, 23
55. Find four consecutive even integers such that the
largest is 2 more than half the sum of the first three
integers. 2, 4, 6, 8
56. Find four consecutive odd integers such that the
smallest is 25 less than twice the sum of the largest
two integers. 5, 7, 9, 11
Special
57. Are there four consecutive integers whose sum is −6?
If so, what are they? −3, −2, −1, 0
58. Are there five consecutive integers whose sum is 10?
If so, what are they? 0, 1, 2, 3, 4
59. Are there four consecutive even integers whose sum
is 32? If so, what are they? no
60. Are there two consecutive odd integers whose sum
is 58? If so, what are they? no
61. Are there four consecutive even integers whose sum
is zero? If so, what are they? no
62. Are there five consecutive odd integers whose sum is
zero? If so, what are they? no
63. Are there four consecutive odd integers whose sum is
zero? If so, what are they? −3, −1, 1, 3
64. Are there five consecutive even integers whose sum is
zero? If so, what are they? −4, −2, 0, 2, 4
ALG catalog ver. 2.6 – page 147 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HC
65. Find three consecutive integers such that the sum of
the first and third, decreased by twice the second,
is 0. any three con. integers
66. Find four consecutive integers such that the sum of
the second and third equals the sum of the first and
fourth. any four con. integers
67. Find three consecutive integers such that second is
2 more than half the sum of the first and third.
68. Find three consecutive integers such that the sum of
the first and third is 6 less than twice the second.
not possible
not possible
69. Julio said, “I am thinking of two consecutive even
integers. If I take the first one, multiply it by 4 and
subtract 22, I get the next even integer.” What
integers was Julio thinking about? 8, 10
70. Rosanna said, “I am thinking of two consecutive
integers. If I take first one, divide it by 2 and
add 18, I get the three times the next integer.”
What integers was Rosanna thinking about? 6, 7
71. Henry made up a problem about three consecutive
odd integers. He said, “The first decreased by the
third equals the second decreased by 9.” What
integers did he have in mind? 3, 5, 7
72. Theresa made up a problem about three consecutive
integers. She said, “The first decreased by half of
the third equals the second decreased by 15.” What
integers did she have in mind? 26, 27, 28
73. Explain why two consecutive integers always have an
odd sum. n + (n + 1) = 2n + 1
74. Explain why four consecutive integers always have an
even sum. n + (n + 1) + (n + 2) + (n + 3) = 4n + 6 = 2(n + 3)
75. Explain why three odd integers can never have an
even sum. answers will vary
76. Explain why the sum of five odd integers can never
equal zero. answers will vary
77. Jessica thought of a relationship between two
consecutive odd integers. She said, “If you subtract
the second from the first and divide by 2, you get the
number 1.” What integers did Jessica have in mind?
78. Marcus thought of a relationship between three
consecutive even integers. He said, “The second even
integer is half the sum of the first and third.” What
integers was Marcus thinking about?
any two cons. odd integers
79. Show that the following statement is true.
“Given any two consecutive integers, the square of
the smaller subtracted from the square of the larger
is one more than twice the smaller.”
(n + 1)2 − n2 = 2n + 1
any two cons. even integers
80. Show that the following statement is true.
“Given any two consecutive even integers, the square
of the smaller subtracted from the square of the
larger is four times the smaller, plus 4.”
(n + 2)2 − n2 = 4n + 4
ALG catalog ver. 2.6 – page 148 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HD
Topic:
Coins, stamps, etc.
Directions:
0—(No explicit directions.) 16—Solve and check.
20—Solve using one variable.
21—Solve using two variables. 22—Solve using a system of equations.
39—Translate and solve. 40—Write an equation and solve.
1.
A boy has 7 more dimes than quarters. The total
value of the coins is $4.90. Find the number of dimes
and quarters. 12 q, 19 d
2.
Patty has a total of $1.60 in pennies and nickels.
There are 10 more pennies than nickels. Find the
number of each. 25 n, 35 p
3.
Denyse spent $10.60 on stamps. She bought 15 more
10/
c stamps than 25/c stamps. How many 10/
c stamps
did she buy? 41
4.
There are 3 more dimes than nickels in a sack of
coins. The value of the coins is $8.85. How many
nickels are there? 57
5.
There are 81 coins in a parking meter. There are
15 fewer quarters than dimes in a parking meter.
The value of the coins is $13.05. How many dimes
are there? 48
6.
Some loose stamps are worth $2.10. There are
24 fewer 5/
c stamps than 1/
c stamps. How many 5/
c
stamps are there? 31
7.
In his coin box, Brian has 12 fewer nickels than
dimes. The value of his nickels and dimes is $2.40.
How many each type does he have? 20 d, 8 n
8.
Thien has quarters and nickels in his pocket. He has
ten fewer nickels than quarters, and their total value
is $6.10. How many of each does he have? 22 q, 12 n
9.
In a spy movie, agent 707 sits at the casino table
with a pile of chips worth $30,000. There is an equal
number of $100 and $50 chips. Figure out the total
number of chips. 400
10. In a western film, the desparado sits at a poker table
with a stack of coins worth $50. There are as many
silver dollars as there are quarters. Figure out the
total number of coins. 80
11. In Robin’s stamp collection, there is an equal number
of 45/
c and 22/c stamps. Their total face value is
$10.72. How many of each are there? 16
12. Frank collects baseball cards. He has the same
number of $5 cards as $2 cards, and their total value
is $252. How many of each does he have? 36
13. A parking meter contains 4 times as many nickels as
quarters. The meter contains $4.05 total. How many
coins of each type are there? 9 q, 36 n
14. A piggy bank contains 10 times as many pennies as
nickels. The total value of the coins is $1.35. How
many coins of each type are there? 9 n, 90 p
15. Sally has $1.50 worth of change in her pocket—all
nickels and dimes. She has three times as many
nickels as dimes. Find the number of each. 6 d, 18 n
16. Marty, the hot dog vendor, has $11.20 in change—all
quarters and dimes. He has six times as many
quarters as dimes. Find the number of each. 42 q, 7 d
17. Jaime has one-third as many nickels as quarters.
Their value is $2.40. What is the total number of
coins? 12
18. Louise has one-fourth as many 8/
c stamps as 14/
c
stamps. Their value is $4.48. What is the total
number of stamps? 35
19. At the end of the day, a postal worker has 1 21 times
as many 25/
c stamps as 45/c stamps. Their combined
value is $19.80. How many 45/c stamps does she
have? 24
20. After closing down his lemonade stand, David finds
that he has 3 12 times as many dimes as quarters.
Their combined value is $15.60. How many dimes
does he have? 91
21. A change purse contains $6.55 in dimes and quarters.
If the number of dimes is 5 more than 3 times the
number of quarters, how many of each coin are
there? 11 q, 38 d
22. A coin-operated washing machine contains $19.95 in
nickels and quarters. If there is 1 less nickel than
3 times the number of quarters, how many of each
coin are there? 50 q, 149 n
23. Bernice has $2.25 in dimes and nickels. She has
4 fewer nickels than 5 times her number of dimes.
Find out how many dimes she has. 7
24. Tony has $1.50 in pennies and nickels. He has
10 more pennies than twice his number of nickels.
Find out how many nickels he has. 20
25. A change purse contains 120 coins worth $10. They
are all nickels and dimes. How many of each kind
are there? 40 n, 80 d
26. A cash register contains 53 coins worth $4.40. They
are all nickels and dimes. How many of each kind
are there? 35 d, 18 n
ALG catalog ver. 2.6 – page 149 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HD
27. Earl has 28 coins, all dimes and quarters. How many
of each does he have, if their total value is $5.20?
16 q, 12 d
28. Mr. Loren has 36 coins, all 50/
c pieces and quarters.
How many of each does he have, if their total value
is $15.50? 26 h, 10 q
29. Mrs. Nunez has nickels and pennies in her purse.
She has a total of 40 coins, and they are worth
$2.40. How many of each does she have? 25 n, 15 p
30. Manny has dimes and pennies in his pocket. He has
a total of 38 coins, and they are worth $1.91. How
many of each does he have? 17 d, 21 p
31. A stack of 130 half-dollars and quarters is worth $40.
Find the number of each type of coin. 30 h, 100 q
32. A stack of 100 half-dollars and quarters is worth $45.
Find the number of each type of coin. 80 h, 20 q
33. A newspaper girl collects $13.45 in dimes and
quarters. If there are 70 coins in all, how many
quarters does she have? 43
34. An ice cream vendor collects $11.85 in dimes and
quarters. If there are 60 coins in all, how many
dimes does he have? 21
35. Ms. Swenson puts quarters and nickels aside for
paying tolls. She has a total of 18 coins, and they
are worth $3.45. How many quarters does she have?
36. Bob saves quarters and dimes for the laundromat.
He has a total of 32 coins, and they are worth $7.10.
How many dimes does he have? 6
11
37. The value of 92 stamps is $14. There are all 10/
c and
25/
c stamps. Find the number of 10/c stamps. 60
38. The value of 21 stamps is $1.65. There are all 5/
c
and 10/
c stamps. Find the number of 5/
c stamps. 9
39. Paul has 64 coins in his piggy bank. They are all
nickels and dimes, and their total value is $3.65.
How many nickels does he have? 55
40. Sheila has 26 coins in her toy bank. They are all
nickels and dimes, and their total value is $2.45.
How many dimes does she have? 23
41. Martina has twice as many dimes as nickels, and
5 fewer quarters than nickels. The value of all her
coins is $2.25. Find how many nickels she has. 7
42. Celeste has 4 times as many nickels as dimes, and
6 more quarters than dimes. The value of all her
coins is $4.25. Find how many nickels she has. 20
43. A coin-sorting machine contains nickels, dimes and
quarters worth $5.50. There are 3 times as many
nickels as dimes, and 2 more quarters than dimes.
How many of each type of coin are there?
44. A parking meter contains pennies, nickels and dimes
worth $2.50. There 2 fewer pennies than nickels, and
3 times as many dimes as nickels. How many of each
type of coin are there? 5 p, 7 n, 21 d
30 n, 10 d, 12 q
45. Toby has 3 more nickels than dimes and 8 fewer
nickels than pennies. If the value of his coins is
$2.66, how many dimes does he have? 15
46. Mr. Dubois has 6 more dimes in his pocket than
nickels, and twice as many pennies as dimes. If the
value of his coins is $1.40, how many pennies does
he have? 20
47. A vending machine is filled with $12 worth of
change. There are twice as many nickels as dimes,
and 6 more dimes than quarters. Find the number
of each type of coin. 24 q, 30 d, 60 n
48. One night a waiter received $11.30 in tips—all coins.
He had 3 times as many dimes as nickels, and
10 more quarters than dimes. Find the number of
each type of coin. 34 q, 24 d, 8 n
ALG catalog ver. 2.6 – page 150 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HE
Topic:
Age problems.
Directions:
0—(No explicit directions.) 16—Solve and check.
20—Solve using one variable.
21—Solve using two variables. 22—Solve using a system of equations.
39—Translate and solve. 40—Write an equation and solve.
1.
Bob is six years older than his sister, and the sum of
their ages is 32. How old is Bob? 19
2.
Marie is 13 years older than Joanna. The sum of
their ages is 55. How old is Joanna? 21
3.
Romulo is twice as old as his nephew. The sum of
their ages is 108. What are their ages? R 72, n 36
4.
Mrs. Mitchell is 3 times older than her daughter.
The sum of their ages is 52. How old is each person?
M 39, d 13
5.
Frank is 12 years old, and his mother is 39. In how
many years will Frank be half as old as his mother?
6.
Bridgette is 10 years old, and her father is 42. In
how many years will she be one-third his age? 6
Emil, who just celebrated his 17th birthday, has a
50 year old aunt. In how many years will Emil half
as old as his aunt? 16
15
7.
Mr. Valmonte, who is 30 years old, has a one year
old baby girl. In how many years will the girl be a
quarter of her father’s age? 10
8.
9.
Serina is nine years old. Her mother is 39 years old.
In how many years will her mother be three times as
old as Serina? 6
10. Imagine that you are 15 years old and your father
is 45. In how many years will your father be twice
as old as you? 15
11. A woman is now 26 years old and her baby boy is
2 years old. In how many years will the mother be
five times as old as her son? 4
12. A man is 61 years old and his grandson is 7. In how
many years will the man be four times as old as his
grandson? 11
13. Gabriella is 29 years old. Her mother is 50. How
many years ago was the mother twice as old as
Gabriella? 8
14. Sean is 12 years old. His father is 40. How many
years ago was the father five times as old as Sean?
15. Mike is 18 years old and his grandmother is 66 years
old. How many years ago was the grandmother
9 times as old as Mike? 12
16. Marilyn just turned 30 years old. Her uncle is
64 years old. How many years ago was the uncle
three times as old as Marilyn? 13
17. Barbara is 5 years older than Roxanne. In 6 years,
the sum of their ages will be 35 years. How old is
each person now? R 9, B 14
18. Mr. Rodriguez is 25 years older than his son. In
10 years, the sum of their ages will be 59. How old
is each now? R 32, s 7
19. Anna is twice as old as Bettye. Seven years ago, the
sum of their ages was 13. How old are they now?
20. Duane is twice as old as Erik. Five years ago, the
sum of their ages was 26. How old is each person
now? D 24, E 12
A 18, B 9
5
21. Kelly’s age plus her father’s age is 32 years. In
12 years, Kelly will be one-third as old as her father.
Find her age now. 2
22. The sum of the ages of a father and son is 46 years.
In 2 years, the son will be one-fourth as old as the
father. What is the father’s age now? 38
23. In 4 years, Kathy will be 3 times as old as Andy.
The sum of their ages is now 56. How old is each
person? K 44, A 12
24. In 10 years, Mr. Yuen will be twice as old as his
daughter. Their ages now add up to 61. How old is
each person? Y 44, d 17
25. Lindsay is now 3 times as old as Greg. In five years,
she will be twice as old as Greg. Find their ages
now. L 15, G 5
26. Sheila is 5 times as old as her nephew. In four years,
she will be 3 times as old as her nephew. Find their
ages now. S 20, n 4
27. Lee is 4 times older than Nick. In 10 years, he will
be twice as old as Nick. What is each person’s age
now? L 20, N 5
28. Mrs. Salazar is 6 times older than her grandson.
In eight years, she will be 4 times as old as her
grandson. What is each person’s age now? H 72, g 12
ALG catalog ver. 2.6 – page 151 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HE
29. Susan is twice as old as her brother. Five years ago
Susan was 3 times as old as her brother. Find her
age now. 20
30. George is 5 times as old as his son. Two years ago
George was 7 times as old as son. Find his son’s age
now. 6
31. Cassie is 9 years older than her brother. Three years
ago, Cassie was 4 times as old as her brother. How
old is each person now? C 15, b 6
32. Ms. Reid is eighteen years older than her son. One
year ago, she was three times older than he was.
How old is each person now? R 28, s 10
33. Luke is 2 years younger than his friend. Five years
ago, Luke was three-fifths of his friend’s age. How
old is Luke now? 8
34. Nina is 5 years younger than Fred. In 6 years, Nina
will be three-fourths as old as Fred. How old is Fred
now? 14
35. Dierdre’s age is one-fourth of her aunt’s age. In
12 years, Dierdre will be half as old as her aunt.
Find their ages now. D 6, a 24
36. Sherri is half as old as her mother. Twelve years ago,
she was 31 of her mother’s age. Find their ages now.
37. Ben is 8 years older than Frank. In 3 years, Ben will
be 1 21 times older than Frank. Find Ben’s present
age. 21
38. Martha is 7 years younger than George. In 7 years,
George will be 1 13 times older than Martha. Find
Martha’s present age. 14
39. Katy is 5 years younger than Margaret. In 4 years,
four times Margaret’s age will equal six times Katy’s
age. How old is Katy now? 6
40. Joe is 15 years younger than his brother. Five years
from now, three times his own age will equal twice
his brother’s age. How old is Joe now? 25
41. Mr. Jeffries is twice as old as his daughter. In six
years Mr. Jeffries’ age will be three times what his
daughter’s age was six years ago. How old is each
person now? J 48, d 24
42. Ms. Nakao is three times older than her cousin. In
two years, Ms. Nakao will be four times as old as her
cousin was two years ago. How old is each person
now? N 30, c 10
43. The sum of Stuart’s and Tracy’s ages is 27. Stuart’s
age five years from now will be twice what Tracy’s
age was two years ago. How old is each person now?
44. The sum of Martin and Shalene’s ages is 20.
Shalene’s age one year from now will be nine times
Martin’s age one year ago. How old is each person
now? S 17, M 3
S 15, T 12
45. The ages of a mother and daughter add up to
56 years. Four years ago, the mother was 2 times
older than the daughter. How old is daughter now?
S 24, m 48
46. The ages of a father and son add up to 59 years.
13 years ago, the father was 10 times older than the
son. How old is the father now? 43
20
47. A woman was 30 years old when her daughter was
born. The mother’s present age is 6 years more than
3 times the daughter’s age. How old is the mother?
48. A man was 25 years old when his son was born. The
father’s present age is 3 years less than 5 times the
son’s age. How old is the son? 7
42
ALG catalog ver. 2.6 – page 152 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HF
Topic:
Time, distance, rate.
Directions:
0—(No explicit directions.) 16—Solve and check.
20—Solve using one variable.
21—Solve using two variables. 22—Solve using a system of equations.
39—Translate and solve. 40—Write an equation and solve.
1.
Two trains left the same station at the same time.
Train A traveled north at 28 mph and Train B
traveled south at 52 mph. How much time passed
before they were 320 miles apart? 4 hrs
2.
Two boys on bicycles start from the same place at
the same time. One rides at 15 km/hr, and the other
at 9 km/hr. They travel in opposite directions. How
much time will pass before they are 48 km apart?
2 hrs
3.
Two cars pass each other on the highway, going in
opposite directions. If their rates are 95 and 85 kph,
how long before they are 240 km apart? 1 1 hrs
4.
3
Two airplanes take off at approximately the same
time, but fly in opposite directions. One goes west at
450 mph, the other goes east at 500 mph. How long
after they take off will they be 2280 miles apart?
2 25 hrs
5.
Two fishing boats pass each other going in opposite
directions. Their rates are 20 km/hr and 16 km/hr.
After how many minutes will they be 21 km apart?
6.
A car and a motorcycle pass each other going in
opposite directions. Their rates are 48 and 42 mph.
After how many minutes will they be 75 miles
apart? 50
Two trains pass each other on parallel tracks. One
train is going west at 90 kph, the other is going east
at 42 kph. After how many minutes will they be
121 km apart? 55
35
7.
A car and a truck leave a highway rest area at the
same time. The car travels south at 55 mph and
the truck travels north at 40 mph. After how many
minutes will they be 38 miles apart? 24
8.
9.
Two ships pass each other at 2:00 in the afternoon.
One ship is going north at 12 km/hr and the other is
going south at 9 km/hr. At what time will they be
105 km apart? 7:00 pm
10. Two trains, going in opposite directions, pass each
other at 10:30 in the morning. Their rates are 65
and 85 mph. At what time will they be 450 miles
apart? 1:30 pm
11. At 1:30 in the afternoon, two cars pass each other on
the interstate. One car is heading west at 60 mph.
The other is heading east, pulling a trailer at 48 mph.
At what time will they be 252 miles apart? 3:50 pm
12. Two Girl Scout troups leave the same campground
at 8:30 in the morning. One troup hikes north at a
rate of 2 21 mph, the other hikes south at 3 12 mph. At
what time will they be 15 miles apart, if they keep
walking? 11:00 am
13. Two boats, originally 35 km apart, sail toward each
other. The first goes at a speed of 8.5 kph. The
second goes at a speed of 5.5 kph. How long until
they meet? 2 1 hrs
14. An Air Force jet and a refueling plane, which are
initially 210 km apart, fly toward each other. If their
rates are are 800 and 250 km/hr, how long will it
take for them to meet? 1 hr
15. At 11:30 am, two cars start traveling toward each
other from cities that are 170 km apart. One car’s
rate is 45 km/hr, and the other car’s rate is 40 km/hr.
At what time of day will they meet? 1:30 pm
16. At 9:00 in the morning, two fishing boats begin
heading toward each other on the open sea. Their
rates are 8 km/hr and 14 km/hr, and they start out
121 km apart. At what time of day will they meet?
2
5
2:30 pm
17. How many minutes will it take an airplane, flying
at 370 mph, to intercept another airplane that is
95 miles away, if the second airplane is heading
toward the first at 200 mph? 10
18. How many minutes will it take a helicopter, flying
at 120 km/hr, to reach a naval vessel that is 45 km
away, if the vessel is heading toward the helicopter
at 30 km/hr? 18
19. Two people, who are 315 meters apart, start walking
toward each other at the same time. If they walk at
rates of 1.5 and 2 m/sec, how much time will pass
before they meet? 90 sec
20. At the same time and place, two people start running
around a 440 yard track. They go in opposite
directions, and their speeds are 15.5 and 17.5 ft/sec.
How long before they pass each other? 40 sec
ALG catalog ver. 2.6 – page 153 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HF
“Round trips”
21. A fisherman takes his boat out to sea and then back
to the harbor. The entire trip lasts 8 hours. If he
travels out at 18 mph and follows the same route
back at 14 mph, how many miles does he travel
altogether? 126 mi
22. Mrs. Pomroy takes her Sunday drive into the country
at an average rate of 60 km/hr. She returns over the
same road at an average rate of 48 km/hr. If the
round-trip takes 3 hours, how many kilometers does
she travel altogether? 160 km
23. A round trip in a helicopter lasts 4 12 hours. The
average rate going is 160 km/hr, and the average rate
returning is 80 km/hr. Find the total distance flown
by the helicopter. 480 km
24. A barge travels up the Mississippi River at 12 mph
and then back to its original port at a rate of
15 mph. The round trip takes 7 21 hours, What is the
total distance traveled by the barge? 100 mi
25. Susan started her bicycle ride at 1:30 in the
afternoon. After riding a certain distance, she got a
flat tire and had to walk home. She arrived home at
4:30. If Susan averaged 21 km/hr on her bicycle and
6 km/hr walking, how far did she ride before getting
the flat tire? 14 km
26. At 12:30, Martin left his house to go on a bike ride.
After riding a certain distance, he got a flat tire
and had to walk home. He arrived home at 3:00
the same afternoon. If Martin rode at 12 mph and
walked at 3 mph, how far did he ride before getting
the flat tire? 6 mi
27. Jason forgot that his bicycle was locked up at school.
So he ran to the school to get the bicycle, and then
rode it back home. His running speed was 6 mph
and his riding speed was 18 mph. Find the distance
between his home and school, if the round trip took
20 minutes. 1.5 mi
28. A girl runs to the park to get her bicycle. Then she
rides the bicycle back home. The total time for the
round-trip is 45 minutes. Her running speed is 8 kph
and her riding speed is 24 kph. How far is the park
from her home? 4.5 km
29. At 10:30 in the morning, Laura’s mom drove her to a
friend’s house, where Laura stayed for 2 hours. Then
she walked back home, and arrived there at 2:00 in
the afternoon. If her mom averaged 48 km/hr in the
car, and she walked at 6 km/hr, find the distance
between Laura’s house and her friend’s. 8 km
30. A gasoline truck leaves the refinery at 11:45 am.
After making a round trip to deliver the gasoline, the
truck returns at 2:15 in the afternoon. The truck’s
average speed going is 30 mph, and returning is
45 mph. If it takes one hour to unload the gasoline,
find out how far it is from the refinery to the delivery
point. 27 mi
31. As part of Jane’s workout, she runs to the store, buys
a snack, and walks home. She spends 10 minutes in
the store, and the whole trip takes one hour. Her
running speed is 6 mph, and her walking speed is
3 mph. How far is the store from her home? 1 2 mi
32. Dewayne ran from his house to the gym, exercised
for 45 minutes, and ran home. His speed running to
the gym was 12 km/hr, and returning was 13 km/hr.
If the entire workout, including the running, lasted
2 hours, figure out the distance between Dewayne’s
house and the gym. 7.8 km
33. On a trip to the mountains, Ms. Jacobs drove her
car at an average rate of 50 mph. Due to bad road
conditions, the return trip took 3 hours longer, and
she averaged only 30 mph. Find out how much time
she spent driving home. 7.5 hrs
34. John drove from his college to his parents’ house at
an average rate of 55 kph. On the return trip there
was not as much traffic, so it took an hour less and
and John’s average rate was 80 kph. How much time
did he spend driving to his parent’s house? 3.2 hrs
35. Freddie walks to the top of a hill, then rides his
skateboard back down. He walks at 4 mph and
skateboards at 10 mph. If it takes 21 hour less time
to go down the hill, find out how much time Freddie
spends walking. 5 hr
36. Freddie gets a car ride to the top of a hill, then rides
his skateboard back down. The downhill ride takes
15 minutes longer than the uphill. If the car averages
32 mph, and Freddie rides his skateboard at 8 mph,
find out how much time he spends skateboarding.
3
6
1
3
hr
“Catching up”
37. A police car receives a radio call to catch a vehicle
that is speeding down the highway at 70 mph. The
police car, which is 14 miles behind the vehicle,
drives after it at 98 mph. How long will it take for
the police car to overtake the vehicle? 1 hr
2
38. Two trains head in the same direction on parallel
tracks, but Train E is 30 km in front of Train F. If
Train E averages 42 km/hr and Train F averages
60 km/hr, how long will it take for the faster train to
catch up with the slower? 1 2 hrs
ALG catalog ver. 2.6 – page 154 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
3
HF
39. Steve and Helen ride their mountain bikes on a dirt
road when Steve gets a flat tire. He stops to fix it.
By the time he starts riding again, Helen is one mile
ahead. If Helen’s average speed is 8 mph and Steve’s
is 10 mph, find the distance from where he got the
flat tire to where he eventually catches up. 5 mi
40. Mrs. Quintaro started driving to an out-of-town
conference. When she was 5 km away from her office,
she called her assistant and asked him to meet
her there. Mrs. Quintaro and her assitant drove
along the same route and averaged 40 and 48 kph,
respectively. If they arrived at the same time, find
out how far the conference was from their office.
30 km
41. A car leaves a highway rest area 1 21 hours after a
truck. Both are traveling in the same direction. In
how many hours will the car pass the truck, if their
respective rates are 90 and 60 kph? 3
42. One hour after Pete left on a canoe trip down the
river, his friend Sam started after him in kayak. Pete
traveled at 16 kph. Sam, who left from the same
place, traveled at 22 kph. How long did it take Sam
to catch up with Pete? 2 2 hr
3
43. Thirty minutes after Eileen leaves on a bicycle trip,
her brother gets in a car and drives after her. Eileen
rides at average rate of 16 kph, and her brother
drives the car at 48 kph. How long will it take for
him to catch up? 1 hr
44. Martin left the harbor in a sailboat and headed
north at 24 km/hr. Twenty minutes later, Leslie left
the harbor in a motorboat and headed in the same
direction at 30 km/hr. How long before she met up
with Martin? 1 1 hrs
45. Marshall rides his bike to school, and forgets his
lunch. Ten minutes after he leaves the house, his
mother starts to drive after him in a car. The speed
of the car is 22 mph. Marshall rides his bike at
12 mph. How many minutes will it take his mother
to catch up? 12
46. A jogger, starting at a certain place, runs at a
constant speed of 10 kph. Ten minutes later, someone
else starts jogging from the same place. If the second
jogger runs at a rate of 14 kph and follows the same
course, how many minutes will it take to catch up
with the first jogger? 25
47. Freddie and I have a race. He rides his skateboard
and I ride my bicycle. I give him a lead time of
15 minutes. His speed is 6 mph and mine is 21 mph.
How far will I have to ride before I catch up? 2.1 mi
48. A truck crosses the county line at 60 mph. Five
minutes later, a sheriff crosses the border in hot
pursuit, driving at 90 mph. How far from the border
will the sheriff overtake the trucker? 15 mi
49. At 9:30 am, Josh and Marlee leave on a bicycle trip.
Their average speed is 15 mph. At 11:30 am, Richard
leaves from the same place and starts driving after
them in his car. His average speed is 35 mph. At
what time of day will Richard meet up with the
bicyclists? 1:00 pm
50. Brian leaves Eagle Nest Campground at 8:30 am,
and hikes along a trail at 4 kph. One hour later,
Rob leaves the same campground, and hikes along
the same trail at 6 kph. At what time of day will he
catch up with Brian? 11:30 am
51. At 12:30 in the afternoon, Francis left his house to
go on a bike ride. One hour later, his sister set out
on the same route on her bicycle. If Francis’ speed
was 14 mph and his sister’s was 18 mph, find the
time when she caught up with him. 4:00 pm
52. A small plane left the airport at 11:15 am, and flew
west at 280 km/hr. at 1:15 pm, a jet left the airport
and flew in the same direction at 760 km/hr. At
what time did the jet overtake the small plane?
4
3
2:25 pm
Special
53. A cyclist had been traveling 24 km/hr for 8 hours
when he was overtaken by a motorist who left the
same starting point 5 hours after the cyclist. Find
the motorist’s speed. 64 kph
54. Mr. Applebaum had been hiking for 5 hours when
he was passed by by Ms. Brooks, who left the same
starting point 21 hour later. If Mr. Applebaum hiked
along at 3 mph, find the rate of Ms. Brooks. 3 1 mph
3
55. A motorcycle leaves a highway rest area 2 hours after 56. A passenger train leaves a station 2 12 hours after a
a car, and heads in the opposite direction. After
freight train, and heads in the opposite direction.
traveling for 1 12 hours at 64 kph, the motorcyle is
After traveling for one hour at 54 mph, the passenger
376 km away from the car. Find the speed of the car.
train is 173 miles away from the freight train. Find
the speed of the freight train. 34 mph
80 kph
ALG catalog ver. 2.6 – page 155 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HF
57. At 11:00 in the morning, Fred and Stuart leave town
in their cars and travel in opposite directions. At
5:00 in the afternoon, they are 378 mi apart. Find
Stuart’s average rate, if it is twice as fast as Fred’s.
42 mph
59. Two cars travel in opposite directions from the same
starting point. The rate of one car is 10 km/hr faster
than the rate of the other car. After 4 hours, the
cars are 472 km apart. Find each car’s rate.
54 and 64 kph
58. Two jets leave Vancouver at the same time, one
flying east and one flying west. The rate of the
second jet is 100 km/hr faster than the first. In
3 hours, they are 4200 km apart. Find the rate at
which the faster jet is traveling. 750 kph
60. At 1:30 in the afternoon, a commuter train and an
express train pull out of the station and travel in
opposite directions. The express train goes three
times faster than the other. at 2:45 in the afternoon,
they are 150 miles apart. Find the speed of each
train. 30 and 90 mph
61. Mr. Brunnelle went on a 114 km bike ride over a
mountain pass. He averaged 24 km/hr going up the
mountain and 36 km/hr going down the other side.
If the time spent going down was one hour less, find
how many kilometers were traveled in the uphill part
of the ride. 60
62. Vannessa sailed her boat a total of 50 miles in a race
from the harbor to the lighthouse and back again.
She averaged 9 mph to the lighthouse, and 12 mph
on the return trip, which took a half-hour less time.
How long was the route to the lighthouse (it was not
necessarily a straight line). 24
63. A car traveled from one city to another at an average
speed of 90 km/hr. The car returned via the scenic
route, which was 20 km longer and took an extra
half-hour. If the rate returning was 80 km/hr, find
the total distance traveled by the car. 380 km
64. On a trip to the mountains, Lori drove at a rate of
55 mph. On the trip back home, she decided to try a
route that was 10 miles shorter. As it turned out,
she took an extra hour to get home and averaged
only 40 mph. Find the total distance she traveled.
356 23 mi
ALG catalog ver. 2.6 – page 156 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HG
Topic:
Mixture problems.
Directions:
0—(No explicit directions.) 16—Solve and check.
20—Solve using one variable.
21—Solve using two variables. 22—Solve using a system of equations.
39—Translate and solve. 40—Write an equation and solve.
1.
How much water should be added to 2.2 liters of
pure acid in order to obtain a solution that is 55%
acid? 1.8 L
2.
How much water should be added to 75 mL of pure
lemon juice in order to get a mixture that is 20%
lemon juice? 300 mL
3.
Most paints have to be thinned before they can
be applied with a roller. How many ounces of
turpentine should be added to 1 gallon (128 ounces)
of paint, so that the final mixture is 60% turpentine?
4.
Most indoor plant foods have to be diluted before
they can be used. How many ounces of water should
be added to 2.5 ounces of plant food, so that the
final mixture is 95% water? 47.5 oz
192 oz
5.
How much water must be added to 42 liters of a
liquid fertilizer that is 0.8% nitrogen to get a mixture
that is 0.3% nitrogen? 70 L
6.
How much water must be added to one liter of
a window cleaner that is 45% ammonia to get a
solution that is 20% ammonia? 1.25 L
7.
How much water must be added to 48 mL of a metal
cleaner that is 63% acid to get a solution that is 48%
acid? 15 mL
8.
How much water must be added to 6 quarts of
radiator fluid that is 4% glycol to get a mixture that
is 2.5% glycol? 3.6 q
9.
A restaurant has some pancake batter that is too
thin. It weighs 12 kilograms and contains 85% flour.
How many kilograms of flour should be added so
that the final mixture is 90% flour? 6 kg
10. A bakery has some cookie dough that is not sweet
enough. It weighs 10 kilograms and contains 30%
sugar. How many kilograms of sugar should be
added so that the final mixture is 37.5% sugar?
1.2 kg
11. Imagine you have 6 gallons of fruit punch. It is 25%
soda and 75% juice. If you wanted it to contain only
60% juice, how much soda would you have to add?
1.5 gal
12. Imagine you have 15 quarts of fuel for your moped.
It is 85% gas and 15% oil. If you wanted the fuel to
contain only 75% gas, how much oil should you add?
2q
13. A material scientist has 35 grams of an alloy that is
76% copper and 24% zinc. How much pure copper
should be added, so that the content is raised to
79%? 5 g
14. An aircraft parts company has 130 kg of an alloy
that is 80% aluminum and 20% magnesium. How
much pure magnesium should be added, so that the
content is raised to 35%? 30 kg
15. A 12-liter antifreeze solution is made up of water and
alcohol. How much pure alcohol must be added to
raise the concentration from 20% to 50%? 4.8 L
16. A 6-liter cleaning solution is made of up water and
detergent. How much pure detergent must be added
to raise the concentration from 5% to 10%? 0.2 L
17. A concentrated drink is 22% fruit juice. If the proper
amount is 10% fruit juice, how much water must be
added to 15 quarts of the concentrated drink? 18 q
18. A concentrated cleaner is 25% alcohol. If the proper
amount is 7.5% alchohol, how much water must be
added to 12 ounces of the concentrated cleaner?
28 oz
19. A wall covering compound is made up of water and
plaster-of-paris. How many kilograms of water must
be added to 7 kilograms of the compound, in order
to reduce the concentration of plaster from 45% to
35%? 2 kg
20. Some bricks are joined together with a mixture of
water and clay. How many kilograms of water must
be added to 14 kilograms of the mixture, in order to
reduce the concentration of clay from 62% to 56%?
1.5 kg
Three different percentages
21. Chemical A is 60% acid and chemical B is 36% acid.
How much of each should be mixed together to get
120 oz of a chemical that is 52% acid? 80 oz–A, 40 oz–B
22. Two metal alloys, containing 6% zinc and 16% zinc
are to be mixed together. How much of each is
needed to get 150 kg of an 14% zinc alloy?
30 kg–6%, 120 kg–16%
23. Two kinds of milk, which have butterfat contents of
1% and 3.5%, are to be mixed together. How many
liters of each kind are needed to produce 10 liters of
low-fat milk (2% butterfat)? 6 L–1%, 4 L–3.5%
24. A jeweler works with two kinds of metals, the first
contains 72% silver and second contains 87% silver.
How many grams of each should be melted together
to get 50 grams that is 84% silver? 10 g–72%, 40 g–87%
ALG catalog ver. 2.6 – page 157 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HG
25. A painter has two types of solvent, which contain 4%
and 10% methanol. How much of each should be
mixed together to obtain 6 liters of a solvent that is
7.5% methanol? 2.5 L–4%, 3.5 L–10%
26. A pharmacist has two anti-bacterial solutions, which
contain 14% and 24% iodine. How much of each
should be mixed to obtain 36 mL of a solution that
is 16% iodine? 28.8 mL–14%, 7.2 mL–24%
27. Reformate is used to boost the octane rating of
gasoline. Assume there are two kinds of gasoline,
which contain 20% and 12% reformate. How much
of each should be mixed together to produce 3000
gallons of gasoline that is 18% reformate?
28. Pine-oil is often used in household cleaning products.
Assume there are two types of detergent, which
contain 3% and 8% pine-oil. How much of each
should be mixed together to produce 240 ounces of
detergent that is 6% pine-oil? 144 oz–8%, 96 oz–3%
2250 gal–20%, 750 gal–12%
29. A beaker contains 20 mL of a solution that is 12%
cetyl alcohol (or CEA), a moisturizer often found in
shampoo. How much of a solution that is 45% CEA
should be added, so that the final mixture is 30%
CEA? 24 mL
30. A beaker contains 6 mL of a chemical that is 35%
glycine, an important amino acid. How much of
a another chemical that is 50% glycine should be
added, so that the resultant chemical is 46% glycine?
31. A road repair company has two kinds of patching
material, which contain 30% and 60% asphalt. How
much of 30% asphalt should be added to 230 kg of
the 60% asphalt, so that the mixture contains 42%
asphalt? 345 kg
32. A jewelry designer works with two types of precious
metal, which are 34% and 62% gold. How much of
the 62% gold should be added to 7.5 grams of the
34% gold, so that the mixture contains 50% gold?
16.5 mL
10 g
Special
33. A 21 gram mixture of water and potassium is 0.8%
potassium. How much water must be evaporated so
that the mixture is 1.4% potassium? 9 g
34. A 45 ounce mixture of water and calcium is 4%
calcium. How much water must be evaporated so
that the mixture is 18% calcium? 35 oz
35. How much water must be evaporated from 630 grams
of a 25% salt solution to get a 45% salt solution?
36. How much water must be evaporated from 120
gallons of a 2.5% sulfate solution to get a 20% sulfate
solution? 105 gal
280 g
37. A truck radiator is filled with 18 gallons of an
antifreeze solution. The concentration of glycol in
the antifreeze is 15%. How much of the solution
should be drained and replaced with pure water so
that the new solution is 10% glycol? 6 gal
38. An automobile radiator has a volume of 16 liters.
It is filled to capacity with 80% water and 20%
antifreeze. How much of the mixture must drained
off and replaced with pure antifreeze to get a mixture
that is 30% antifreeze? 2 L
39. A 75 mL jar is filled with an water-alcohol solution
that is 30% alcohol. How much of the solution
should be poured out and replaced with pure water
so that the jar contains 25% alcohol? 12.5 mL
40. A 228-liter drum is filled with a solution that is 24%
acid. How much of the solution should be poured
out and replaced with pure acid so that the drum
contains 34% acid? 30 L
41. Frank makes a cleaning solution out of 1.5 grams of
bicarbonate and 21 grams of water. If the mixture
is supposed to be 75% water, how much more
bicarbonate is needed? 5.5 g
42. 66 kilograms of cement is mixed with 22.5 kilograms
of water. However, the mixture is supposed to
be 40% water in order to dry properly. Find the
amount of water that should be added. 17.5 kg
43. 10 liters of water is mixed with 400 mL of lawn
fertilizer. If a 2% concentration of fertilizer is
required, how much more water must be added?
44. Nurse Levin added 50 mL of dextrose to 1.8 liters
of intravenous fluid. But the physician said a 10%
concentration of dextrose is needed. Find the amount
of dextrose that the nurse should add. 150 mL
9.6 L
45. James has a 10 liters of fuel for his moped, which
contains gas and oil in a 3 : 2 ratio. How much gas
should he add so that the ratio is 5 : 1, gas to oil?
14 L
47. Mrs. Chang has just mixed 3 gallons of semi-gloss
enamel. The mixture contains paint and thinnner in
a 1 : 1 ratio. How much paint should she add, if she
wants a 4 : 3 ratio of paint to thinner? 0.5 gal
46. Sarah has made 72 ounces of fruit drink, which
contains apple and pear juice in a 5 : 1 ratio. How
much pear juice should she add so that the ratio is
5 : 3, apple to pear? 24 oz
48. A shoe repair shop has 120 mL of leather cleaner,
which contains soap and conditioner in a 2 : 1 ratio.
How much conditioner must be added to reverse the
ratio (1 : 2 soap to conditioner)? 120 mL
ALG catalog ver. 2.6 – page 158 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HG
Unit pricing
49. Assume you have 25 pounds of cookies that cost
96/
c/lb. How many pounds of cookies costing 72/
c/lb
should you add to them, in order to get a mixture
that costs 80/c/lb? 50 lb
50. Assume you have 15 kilograms of coffee that costs
$4.75/kg. How many kilograms of coffee costing
$4.00/kg should you add to it, in order to get a
mixture that costs $4.45/kg? 10 kg
51. How much fresh orange juice at $3.00 per liter must
52. How many kilograms of cashew nuts worth $6.60/kg
be combined with 6 liters of frozen orange juice at
should be combined with 10 kilograms of pecans
$1.20 per liter to get a mixture worth $1.80 per liter?
worth $8.50/kg to get a mixture worth $7.10/kg?
3L
28 kg
53. Premium grade apples cost $2.25/lb and fancy apples
cost $1.45/lb. How many pounds of premium apples
should be mixed with 11 pounds of fancy apples to
get a mixture that costs $1.70/lb? 5 lb
54. A golf course supervisor wants to mix two types
of grass seed. How much regular seed at $2.65/kg
should he mix with 42 kg of bluegrass seed at
$3.50/kg so that the overall price is $3.00/kg? 60 lb
55. Chemical A, which costs $5.30/mL, is to be combined
with 40 mL of Chemical B, which costs $8.30/mL.
How much of Chemical A should be used, in order
for the overall cost to be $6.90/mL? 35 mL
56. Cheese Doodles, which cost $1.92/lb, are to be
combined with 10 pounds of Corn Crisps, which cost
$1.20/lb. How many pounds of Doodles should be
used, if the mixture is supposed to cost $1.60/lb?
57. Darjeeling tea costs $8.20 per pound. Orange Pekoe
tea costs $6.10 per pound. How much of each kind
must be used to make a 42-pound mixture that costs
$7.60/lb? 30 lb–D, 12 lb–O
58. Clover honey, priced at $4.75/kg, is blended with
Orange Blossom honey, at $6.25/kg. The resulting
mixture sells for $5.05/kg. How much of each kind
of honey is needed for 24 kilograms of the mixture?
12.5 lb
19.2 kg–C, 4.8 kg–O
59. French roast coffee costs $5.20/lb and Colombian
coffee costs $4.00/lb. How much of each kind should
be used for a 100 kilogram mixture that can be sold
for $4.72 a kilogram? 60 kg–F, 40 kg–C
60. Ground beef
pork costing
sausage that
each type of
61. A cheese platter weighs 3.5 kilograms and sells for
$22.75. It is made up of Havarti cheese costing
$6.90/kg and Munster cheese costing $5.90/kg. How
many kilograms of each type of cheese are used for
the platter? 2.1–H, 1.4–M
62. A deli tray that consists of ham and turkey weighs
10.5 pounds and costs $46.15. If the ham sells for
$3.90/lb and the turkey for $4.70/lb, how many
pounds of each meat are used? 6.5–T, 4–H
63. A shopowner plans to combine $3.00-per-pound nuts
and $5.00-per-pound nuts to produce 10 pounds of a
mixture that sells for $36.00. How many pounds of
each are required? 7–$3, 3–$5
64. A chemical engineer plans to mix together two
gasolines that cost 59/
c per gallon and 51/
c per gallon.
His goal is to produce 6000 gallons at a total cost of
$3180. How many gallons of each kind are required?
costing $2.10/lb is mixed with ground
$1.60/lb to produce 60 pounds of
sells for $1.75/lb. How many pounds of
meat are used? 18 lb–B, 42 lb–P
1500–59/
c, 4500–51/
c
65. A $14.85 box of chocolates contains cream- and
caramel-filled varieties. The creams cost $5.80/kg
and the caramels cost $7.00/kg If there are twice
as many caramels (by weight), find the number of
kilograms of each kind of chocolate. 0.75–Cr, 1.5–Ca
66. Peanuts cost $8.75/kg and cashews cost $13.00/kg.
In a package consisting of both kinds, there are
3 times as many peanuts as cashews (by weight).
If the package costs $15.70, find the number of
kilograms of each kind of nut. 0.4–C, 1.2–P
67. A stationary store plans to sell a package of writing
paper and envelopes for a total cost of $2.65. The
paper costs $3.20 kg and the envelopes cost $4.20 kg.
If each package is supposed to contain twice as much
paper as envelopes (by weight), how many kilograms
of each are required? 0.5–P, 0.25–E
68. A health food store wants to sell a large glass of
apple-strawberry juice. Apple juice costs 8/
c per
ounce, and strawberry juice 23/
c per ounce. If each
serving is supposed to cost $1.89 and contain five
times as much apple as strawberry juice, how many
ounces of each should be used? 15–A, 3–S
ALG catalog ver. 2.6 – page 159 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HH
Topic:
Interest and investment.
Directions:
0—(No explicit directions.) 16—Solve and check.
20—Solve using one variable.
21—Solve using two variables. 22—Solve using a system of equations.
39—Translate and solve. 40—Write an equation and solve.
1.
Josephine invests a total of $35,000 in two different
accounts, which pay 10% and 8% annual interest.
After one year she earned $3100 in interest. How
much did she invest in each account?
2.
20,000–8%, 15,000–10%
3.
Arthur invests his inheritance of $24,000 in two
different accounts, which pay 6% and 5% annual
interest. After one year he received $1340 in interest.
How much did he invest in each account?
14,000–6%, 10,000–5%
Sophia invests her money in a regular savings acount
and a money market account, which pay 6% and
12% annual interest. If she invested a total of
$20,000, and she received $1890 in interest after one
year, how much did she put into each account?
4.
Mr. and Mrs. Berman put their savings into two
accounts, which pay 6% and 9% annual interest. If
they invested a total of $22,000, and they received
$1440 in interest after one year, how much did they
put into each account? 18,000–6%, 4000–9%
11,500–12%, 8500–6%
5.
How should $9000 be divided into two accounts,
paying 5.5% and 8% interest, so that the total
interest after one year is $630? 5400–8%, 3600–5.5%
6.
How should $27,500 be divided into two accounts,
paying 7.5% and 5% interest, so that the total
interest after one year is $1660? 11,400–7.5%, 16,100–5%
7.
A man invests $16,500 in two kinds of treasury
notes, which yield 7 21 % and 6% annually. After one
year, he earns $1221 in interest. How much does he
invest at the 6% rate? 1100
8.
A woman invests $3000 in two savings accounts,
which yield 5 12 % and 8% annually. After one year,
she earns $184 in interest. How much does she invest
at the 8% rate? 760
9.
Emily invested $6000, part at an interest rate of
7.2% and the rest at 9%. A year later, her interest
income was $497.70. How much did she she invest at
each rate? 3650–9%, 2350–7.2%
10. The Van Burens invested $3650, part at an interest
rate of 6.4% and the rest at 5%. After one year,
the interest earned was $218.20. Find the amount
invested at each rate. 2550–6.4%, 1100–5%
11. $10,000 is invested, part at 6 14 % and the remainder
at 7%. The yearly income from both investments is
$649. Find the amount of invested at 6 14 %. 6800
12. $42,000 is invested, part at 9% and the remainder at
7 34 %. The yearly income from both investments is
$3605. How much is invested at 7 43 %? 14,000
13. A person borrows a total of $20,000 from two banks.
Citywide Bank charges 12% annual interest and
Nationwide Bank charges 13.5%. If the total interest
owed after one year is $2625, how much does the
person borrow from each bank? 15,000–N, 5000–C
14. Ms. Simmons borrows a total of $15,000 from two
banks. 1st Federal Bank charges 12.5% annual
interest and Commerce Bank charges 14%. If the
total interest owed after one year is $1920, how much
does Ms. Simmons borrow from each bank?
12,000–F, 3000–C
15. Partners in a business agreed to take out two loans
totaling $65,000. The annual interest rates were 11%
and 13%, and the interest paid during the first year
was $7950. Find the amount of each loan.
40,000–13%, 25,000–11%
16. A software developer took out two business loans
totaling $40,000. The annual interest rates were 12%
and 14%, and the interest paid during the first year
was $4960. Find the amount of each loan.
32,000–12%, 8000–14%
17. Mrs. Laird invests a total of $14,000 in two kinds of
municipal bonds, which yield annual profits of 6 21 %
and 8%. How much does she invest at each rate, if
her profit after one year is $970? 10,000–6 1 %, 4000–8%
18. A woman invests a total of $25,000 in two different
mutual funds, which yield annual profits of 6% and
8 12 %. How much does she invest at each rate, if her
profit after one year is $1920? 16,800–8 1 %, 8200–6%
19. The total value of a collection of jewelry and rare
coins is $26,000. It is estimated that in a year the
rare coins will increase 3% in value, while the jewelry
will increase 6%. Their total value will then be
$27,275. What is the present value of each part of
the collection? 9500–C, 16500–J
20. The total value of a collection of antique furniture
and paintings is $17,500. It is estimated that in a
year the furniture will increase 2% in value, while
the paintings will increase 5%. Their total value will
then be $18,300. What is the present value of each
part of the collection? 2500–P, 15,000–F
2
2
ALG catalog ver. 2.6 – page 160 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HH
21. A college student puts his $10,000 grant in two
different savings accounts, which pay 5% and 7%
annual interest. If the total interest is equivalent to
6.4%, how much is deposited in each account?
7000–7%, 3000–5%
22. Mrs. Billet puts her inheritance of $60,000 in two
different savings accounts, which pay 6% and 9%
annual interest. If the total interest is equivalent to
7.7%, how much is deposited in each account?
34,000–9%, 26,000–6%
23. A redevelopment agency is going to make loans
totaling $1.4 million, part of it at 9% annual interest
and the rest at 12.5%. Find the amount that should
be loaned at each rate, so that the total interest
after one year is equivalent to 10.6%.
640,000–12.5%, 760,000–9%
25. Roger has $5000 more invested at 7 14 % than he has
at 6%. If his total profit at the end of one year is
$659.30, how much is invested at each rate?
24. A venture capital group is going to make loans
totalling $3.2 million, part of it at 8.8% annual
interest and the rest at 12%. Find the amount that
should be loaned at each rate, so that the total
interest after one year is equivalent to 9.4%.
600,000–12%, 2,600,000–8.8%
26. Katy has $800 more invested at 6% than at 8 21 %.
After one year, her total interest income is $168.35.
How much is invested at each rate? 830–8.5%, 1630–6%
2240–6%, 7240–7 14 %
27. Mr. Bowman invested one sum of money at 5 12 % and
another sum at 6%. He invested $400 less at the 6%
rate than the 5 12 % rate, and his total interest after
one year was $114. Find the amount he invested at
each rate. 1200–5 1 %, 800–6%
28. Ms. Richards invested one sum of money at 6% and
another sum at 7 43 %. She invested $1200 less at the
6% rate than the 7 34 % rate, and her total interest
after one year was $285.50. Find the amount she
invested at each rate. 2600–7 3 %, 1400–6%
29. Mrs. Lin invested some money at 5% annual interest
and twice as much at 8 14 % After one year, she makes
a profit of $322.50. Find how much is invested in
each. 1500–5%, 3000–8 1 %
30. Josh invested some money at 6% annual interest and
three times as much at 8%. The total interest after
one year was $660. How much did he invest at each
rate? 2200–6%, 6600–8%
31. Janice borrowed some money from her parents at 6%
interest. She also borrowed four times that amount
from a bank, which charged 11.5% interest. If the
total interest after one year was $390, find how much
she borrowered from each source. 750–6%, 3000–11.5%
32. Miguel borrowed some money from friends at 7.5%
interest. He also borrowed twice that amount from
a bank, which charged 12% interest. If the total
interest after one year was $315, find how much he
borrowered from each source. 1000–7.5%, 2000–12%
33. Two equal loans are made at 11% and 14.4%. After
one year, interest on the 14.4% loan exceeded interest
on the 11% loan by $1700. How much money was
loaned at each rate? 50,000
34. Two equal loans are made at 10.8% and 13%. After
one year, interest on the 13% loan exceeded interest
on the 10.8% loan by $3520. How much money was
loaned at each rate? 160,000
35. Part of $4000 was invested at 7% and the other
part at 9%. The 9% investment yielded $80 more in
profits than the other investment. How much money
was invested at each rate? 1750–7%, 2250–9%
36. Part of $5000 was invested at 5% and the other part
at 6%. The 6% investment yielded $135 more in
profits than the other investment. How much money
was invested at each rate? 1500–5%, 3500–6%
37. Mr. Garrett invested twice as much money at 6% as
he did at 7%. After one year, his earnings at 6%
were $95 more than his earnings at 7%. Find the
amount he invested at each rate. 1900–7%, 3800–6%
38. Claudia invested three times as much money at 8%
as she did at 5%. After one year, the yield from
the 8% investment was $1140 more than the 5%
investment. Find the amount she invested at each
rate. 6000–5%, 18000–8%
39. How should $7250 be invested, part at 8% and the
rest at 6.5%, so that both investments produce equal
income? 4000–6.5%, 3250–8%
40. How should $33,000 be invested, part at 9.5% and
the rest at 7%, so that both investments produce
equal income? 19,000–7%, 14,000–9.5%
41. Joanna Kim invested $20,000 in two stock portfolios.
With one portolio, she made a 12% profit; with the
other, she had an 4% loss. Her net profit for the
year was $1200. How much did she invest in the
profitable stocks? 12,500
42. Mr. Bohannon invested $12,000 in two different
mutual funds. After a year, one fund lost 2.5% of its
value; the other yielded profits of 15%. If he made a
net profit of $1100, how much did he invest in the
losing fund? 4000
2
4
4
ALG catalog ver. 2.6 – page 161 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HH
43. Matt borrowed $8400, part at 8% and the rest at
10%. If each rate of interest had been interchanged,
his total interest for the year would have been $32
less. Find the amount borrowed at 8%. 3400
44. Mrs. Stolz invested $6500, part at 7% and the rest at
5%. If each rate of interest had been interchanged,
her total earnings after one year would have been
$50 more. Find the amount invested at 7%. 2000
ALG catalog ver. 2.6 – page 162 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HI
Topic:
Area, perimeter and volume.
Directions:
0—(No explicit directions.) 16—Solve and check.
20—Solve using one variable.
21—Solve using two variables. 22—Solve using a system of equations.
39—Translate and solve. 40—Write an equation and solve.
1.
The perimeter of a rectangle is 78 cm. The width
is w and the length is w + 7. Find the dimensions of
the rectangle. 16 × 23 cm
2.
The perimeter of a rectangle is 122 yards. The
length is ` and the width is ` − 15. Find the
dimensions of the rectangle. 23 × 38 yd
3.
The dimensions of a rectangle are w and 2w + 1.
Solve for w if the perimeter is 50. 8
4.
The dimensions of a rectangle are ` and 3` − 8. Solve
for ` if the perimeter is 60. 9.5
5.
The side of a square is 2x + 3. If the perimeter is 96,
what is x? 10.5
6.
The side of a square is 5x − 2. If the perimeter
is 112, what is x? 6
7.
The perimeter of a square is 16y and each side is
y + 6. Find y. 2
8.
The perimeter of a square is 12y and each side is
y + 10. Find y. 5
9.
A rectangle is four times as long as it is wide. Its
perimeter is 200 cm. Find the length and width of
the rectangle. 80, 20 cm
10. A rectangle is three times as long as it is wide. Its
perimeter is 168 in. Find the length and width of the
rectangle. 63, 21 in.
11. The perimeter of a rectangular tabletop is 384 cm.
Its width is 3/5 of its length. Find the dimensions of
the tabletop. 72 × 120 cm
12. The perimeter of rectagular window is 15 ft. Its
width is 1/4 of its length. Find the dimensions of
the window. 1.5 × 6 ft
13. The length of a rectangle is 5 in. more than the
width. If the perimeter is 70 in., what is the width?
14. The length of a rectangle is 11 cm more than the
width. If the perimeter is 74 cm, what is the length?
15 in.
24 cm
15. The width of a rectangular swimming pool is 8 feet
less than the length. Find the dimensions of the pool
if the perimeter is 104 feet. 22 × 30 ft
16. The width of a rectangular lawn is 15 meters less
than the length. Find the dimensions of the lawn if
the perimeter is 86 meters. 14 × 29 m
17. The perimeter of a rectangle is 450 cm. The length
is 35 cm greater than the width. Find the length of
the rectangle. 130 cm
18. The perimeter of a rectangle is 140 in. The length is
28 in. greater than the width. Find the width of the
rectangle. 21 in.
19. The length of a rectangle is twice the width. Find
the length and width if the perimeter is 96 feet.
20. The width of a rectangle is half the length. Find the
width and length if the perimeter is 126 m. 21, 42 m
32, 16 ft
21. The length of a rectangle is 3 more than twice the
width. If the perimeter is 42, what is the width? 6
22. The length of a rectangle is 6 less than 3 times the
width. Find the width if the perimeter is 28. 5
23. The width of a rectangle is 1 more than half the
length. If the perimeter is 86, what is the length?
24. The width of a rectangle is 3 less than half the
length. Find the length if the perimeter is 54. 20
28
25. The length of a rectangular playground is 4 meters
less than 3 times the width. The perimeter is
64 meters. What are the dimensions of the
playground? 9 × 23 m
26. The length of a rectangular carpet is 8 feet more
than twice the width. The perimeter is 46 feet.
What are the dimensions of the carpet? 5 × 18 ft
27. A rectangular garden has a perimeter of 38 yards.
The length is 2 yards less than twice the width.
What is the length and width? 12, 7 yd
28. A rectangular parking lot has a perimeter of
310 meters. The length is 5 meters more than
4 times the width. What is the length and width?
125, 30 m
29. The sides of a triangle have lengths y, 2y and 3y − 6.
What is the length of each side, if the perimeter is
60 inches? 11, 22, 27 in.
30. The sides of a triangle have lengths x, x + 3 and 2x.
What is the length of each side, if the perimeter is
25 cm? 5.5, 8.5, 11 cm
31. The sides of a triangle are n + 7, n + 7 and 2n.
What is n, if the perimeter is 82? 17
32. The sides of a triangle are are 5a − 6, 3a and 2a + 9.
What is a, if the perimeter is 47? 4.4
ALG catalog ver. 2.6 – page 163 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HI
33. The base of an isosceles triangle is 8 meters longer
than each of the other sides (they are equal). How
long is the base, if the perimeter is 71 meters? 29 m
34. The base of an isosceles triangle is 4 feet shorter
than each of the other sides (they are equal). How
long is the base, if the perimeter is 80 inches? 24 in.
35. Each of the legs of an isosceles triangle is 4 cm longer
than the base. The perimeter of the triangle is
68 cm. Find the length of the base. 20 cm
36. Each of the legs of an isosceles triangle is 6 ft shorter
than the base. The perimeter of the triangle is 84 ft.
Find the length of the base. 32 ft.
37. One side of a triangle is half as long as each of the
other two sides. If the perimeter is 65, how long is
each side? 13, 26, 26
38. One side of a triangle is two-thirds as long as each
of the other two sides. If the perimeter is 120, how
long is each side? 30, 45, 45
39. The perimeter of a triangular lot is 117 m. The
lengths of two sides are each 4/5 of the length of the
third side. Find the length of each side. 36, 36, 45 m
40. The perimeter of a triangular curtain is 255 in. The
lengths of two sides are each 5/7 of the length of the
third side. Find the length of each side. 75, 75, 105 in.
41. The length of the first side of a triangle is 5 less
than the second side. The length of the third side
is one-half the length of the second side. If the
perimeter is 60, how long are the three sides?
42. The length of the second side of a triangle is 8 more
than the first side. The third side is 2 less than
twice the first side. If the perimeter is 54, what are
the lengths of the sides? 12, 20, 22
13, 26, 21
43. The first side of a triangle is 5 cm shorter than the
second side. The third side is 4 cm longer than the
second side. The perimeter is 35 cm. How long is
each side? 7, 12, 16 cm
44. The first side of a triangle is twice as long as the
second side. The remaining side is 9 m longer than
the second side. The perimeter is 53 m. How long is
each side? 11, 20, 22 m
45. The perimeter of a triangle is 43 in. The second side
is half as long as the first side. The remaining side
is 3 in. longer than the second. Find the length of
each side. 10, 13, 20 in.
46. The perimeter of a triangle is 33 feet. The first side
is 2 feet shorter than the second. The third side is
7 feet longer than the first. Find the length of each
side. 8, 10, 15 ft
47. The first side of a triangle is 1 inch shorter than the
second side. The third side is twice as long as the
first. If the perimeter is 29 inches, how long is each
side? 7, 8, 14 in.
48. The first side of a triangle is 2 m longer than half the
second side. The third side is 2 m longer than the
first. If the perimeter is 38 m, how long is each side?
10, 16, 12 m
Special
49. A square and an equilateral triangle have the same
perimeter. Each side of the square is 12 in. Find the
length of each side of the triangle. 16 in.
50. A square and an equilateral triangle have the same
perimeter. Each side of the square is 27 cm. Find
the length of each side of the triangle. 36 cm
51. A rectangle and an equilateral triangle have the same
perimeter. The length of the rectangle is twice its
width. Each side of the triangle is 30 mm. Find the
length and width of the rectangle. 30, 15 mm
52. A rectangle and an equilateral triangle have the
same perimeter. The length of the rectangle is
six inches longer than its width. Each side of the
triangle is 52 inches. Find the length and width of
the rectangle. 42, 36 in.
53. The length of a rectangle is 4 cm less than twice the
width. If the length is decreased by 1 cm and the
width is increased by 2 cm, the perimeter will be
24 cm. Find the dimensions of the original rectangle.
54. A rectangular garden is 3 times as long as it is wide.
If the length is decreased by 8 ft and the width is
increased by 5 ft, the perimeter will be 66 ft. Find
the dimensions of the original garden. 9 × 27 ft
5 × 6 cm
55. When two sides of a square are made twice as long,
and the other two sides are made half as long,
the new perimeter is 100 in. What is the original
perimeter of the square? 80 in.
56. When two sides of a square are increased by 7 cm,
and the other two sides are decreased by 4 cm,
the new perimeter is 70 cm. What is the original
perimeter of the square? 64 cm
ALG catalog ver. 2.6 – page 164 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
HI
57. Find x if the volume
of the box is 54. 1.5
58. Find x if the surface
area of the box is 120.
(Refer to the previous
figure.) 2
59. Find x if the volume
of the box is 126. 4
3DFIG01.PCX
3DFIG03.PCX
3DFIG01.PCX
61. Find x if the volume
of the figure is 25.
0.5
60. Find x if the surface
area of the box is 180.
(Refer to the previous
figure.) 4.5
3DFIG03.PCX
62. Find x if the surface
area of the figure
is 58. (Refer to the
previous figure.) 1
63. Find x if the volume
of the figure is 185.
4
3DFIG10.PCX
64. Find x if the surface
area of the figure
is 168. (Refer to the
previous figure.) 3
3DFIG09.PCX
3DFIG10.PCX
3DFIG09.PCX
65. Find x if the volume
of the figure is 180.
2
66. Find x if the surface
area of the figure
is 269. (Refer to the
previous figure.) 2.5
67. Find x if the volume
of the figure is 185.
3.5
3DFIG15.PCX
3DFIG16.PCX
3DFIG16.PCX
3DFIG15.PCX
69. Find x if the volume
of the figure is 226.
2.75
68. Find x if the surface
area of the figure
is 286. (Refer to the
previous figure.) 5
70. Find x if the surface
area of the figure
is 282. (Refer to the
previous figure.) 3
71. Find x if the volume
of the figure is 162.
4.5
3DFIG18.PCX
3DFIG20.PCX
3DFIG20.PCX
3DFIG18.PCX
ALG catalog ver. 2.6 – page 165 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
72. Find x if the surface
area of the figure
is 204. (Refer to the
previous figure.) 4
HJ
Topic:
Angles and triangles
Directions:
0—(No explicit directions.) 16—Solve and check.
20—Solve using one variable.
21—Solve using two variables. 22—Solve using a system of equations.
39—Translate and solve. 40—Write an equation and solve.
1.
In a triangle, the second angle is 10 ◦ smaller than
the first. The third angle is 55 ◦ larger than the first.
What is the measure of each angle? 45, 35, 100 ◦
2.
In a triangle, the second angle is 100 ◦ larger than
the first. The third angle is 29 ◦ larger than the first.
What is the measure of each angle? 17, 117, 46 ◦
3.
The second angle of a triangle is 25 ◦ larger than the
first. The third angle is 34 ◦ larger than the second.
What are the angle measures? 32, 57, 91 ◦
4.
The second angle of a triangle is 62 ◦ smaller than
the first. The third angle is 17 ◦ smaller than the
second. What are the angle measures? 107, 45, 28 ◦
5.
One angle of a triangle is twice as large as another.
The third angle is 20 ◦ more than the smallest angle.
Find the measure of the largest angle. 80 ◦
6.
One angle of a triangle is three times as large as
another. The third angle is 30 ◦ more than the
smallest angle. Find the measure of the smallest
angle. 37.5 ◦
7.
One angle of a triangle is four times as large as
another. The third angle is 25 degrees more than
the sum of the other two angles. Find the measure
of the smallest angle. 15.5 ◦
8.
One angle of a triangle is three times as large as
another. The third angle is 44 ◦ less than the sum
of the other two angles. Find the measure of the
largest angle. 84 ◦
9.
One angle of a triangle is 15 ◦ smaller than another.
The measure of the third angle is twice the sum of
the other two. Find the measure of each angle.
10. One angle of a triangle is 66 ◦ larger than another.
The measure of the third angle is half the sum of the
other two. Find the measure of each angle. 27, 93, 60 ◦
22.5, 37.5, 120 ◦
11. In a triangle, the first angle is 18 ◦ less than twice the
second. The third angle is one-third of the sum of
the other two. What is the measure of each angle?
84, 51, 45 ◦
13. In the figure, a = 2x,
b = 2x and c = 3x + 5.
Solve for x and give
the measure of each
angle. 25; 50, 50, 80 ◦
12. In a triangle, the second angle is 45 ◦ more than twice
the first. The third angle is one-third of the sum of
the other two. What is the measure of each angle?
30, 105, 45 ◦
14. If a = 4x + 5, b = 4x
and c = 6x, solve for x
and give the measure
of each angle.
12.5; 55, 50, 75 ◦
15. In the figure,
a = x + 15, b = 3x and
c = 11x. Solve for x
and give the measure
of each angle.
11; 26, 33, 121 ◦
16. If a = x − 3,
b = 2x − 12 and
c = 3x + 15, solve
for x and give the
measure of each angle.
30; 27, 48, 105 ◦
ABC-TRI1.PCX
ABC-TRI2.PCX
ABC-TRI1.PCX
ABC-TRI2.PCX
17. In the figure,
a = 2x + 30, b = 4x
and c = 4x − 10. Solve
for x and give the
measure of each angle.
16; 62, 64, 54 ◦
18. If a = 5x + 23,
b = 7x + 3 and
c = 10x, solve for x
and give the measure
of each angle.
7; 58, 52, 70 ◦
19. In the figure,
a = 4x, b = 70 and
c = 10x + 5. Solve
for x and give the
measure of each angle.
20. If a = x + 10, b = 5x
and c = 80, solve for x
and give the measure
of each angle.
15; 25, 75, 80 ◦
7.5; 30, 70, 80 ◦
ABC-TRI4.PCX
ABC-TRI3.PCX
ABC-TRI3.PCX
ALG catalog ver. 2.6 – page 166 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
ABC-TRI4.PCX
HJ
Supplements and complements
21. Find the measure of an angle which is 12 ◦ less than
its complement. 39 ◦
22. Find the measure of an angle which is 53 ◦ more than
its supplement. 116.5 ◦
23. The supplement of an angle is 8 ◦ less than three
times the measure of the angle. Find the angle
measure. 47 ◦
24. The complement of an angle is 15 ◦ more than twice
the measure of the angle. Find the angle measure.
25. Two angles are complementary. Four times the
measure of the smaller is half the measure of the
larger. Find both angle measures. 10, 80 ◦
26. Two angles are supplementary. Five times the
measure of the smaller is three times the measure of
the larger. Find both angle measures. 67.5, 112.5 ◦.
27. Two angles are supplementary. The measure of one
angle is 5 times the other. What are the angle
measures? 30, 150 ◦
28. Two angles are complementary. The measure of one
angle is one-fourth of the other. What are the angle
measures? 18, 72 ◦
29. When the measures of an angle’s complement and
supplement are added together, the result is 238 ◦.
What is the measure of the angle itself? 16 ◦
30. When the measures of an angle’s complement and
supplement are added together, the result is 100 ◦.
What is the measure of the angle itself? 85 ◦
31. The complement of an angle is 8 ◦ less than one-third
of the angle’s supplement. Find the angle measure.
32. The supplement of an angle is 72 ◦ more than
two-thirds of the angle’s complement. Find the angle
measure. 144 ◦
57 ◦
33. The two angles are
supplementary. If
a = 19x + 3 and
b = 21x + 17, solve
for x and find the
measure of each angle.
4; 79, 101 ◦
34. If a = 8x + 10 and
b = 17x − 5, solve
for x and find the
measure of each angle.
7; 66, 114 ◦
SUPPL01.PCX
25 ◦
35. The two angles
are supplementary.
If c = 16x and
d = 5(x − 6), solve
for x and find the
measure of each angle.
10; 160, 20 ◦
20; 15, 75 ◦
32; 123, 57 ◦
SUPPL02.PCX
SUPPL02.PCX
SUPPL01.PCX
37. The two angles are
complementary. If
c = 2x − 25 and
d = 3(x + 5), solve
for x and find the
measure of each angle.
36. If c = 3(x + 9) and
d = x + 25, solve for x
and find the measure
of each angle.
38. If c = 12 x and
d = 2x + 30, solve
for x and find the
measure of each angle.
24; 12, 78 ◦
39. The two angles are
complementary. If
a = 23 (x − 9) and
b = 2x, solve for x and
find the measure of
each angle. 36; 18, 72 ◦
COMPL01.PCX
ALG catalog ver. 2.6 – page 167 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
8; 33, 57 ◦
COMPL02.PCX
COMPL02.PCX
COMPL01.PCX
40. If a = 4x + 1 and
b = 5x + 17, solve
for x and find the
measure of each angle.
IA
Topic:
Prime factorization.
Directions:
11—Factor. 13—Factor completely.
68—Find the prime factors.
54—Rewrite without any exponents.
1.
14
7·2
2.
15
5·3
3.
33
11 · 3
4.
65
13 · 5
5.
12
3·2·2
6.
18
3·3·2
7.
27
3·3·3
8.
30
5·3·2
9.
20
5·2·2
10. 28
7·2·2
11. 44
11 · 2 · 2
12. 45
5·3·3
13. 42
7·3·2
14. 75
5·5·3
15. 63
7·3·3
16. 99
11 · 3 · 3
17. 50
5·5·2
18. 70
7·5·2
19. 110
21. 24
3·2·2·2
22. 32
2·2·2·2·2
23. 36
3·3·2·2
24. 40
5·2·2·2
25. 56
7·2·2·2
26. 88
11 · 2 · 2 · 2
27. 72
2·2·2·3·3
28. 98
7·7·2
30. 80
5·2·2·2·2
31. 90
5·3·3·2
32. 60
5·3·2·2
29. 100
33. 96
5·5·2·2
3·2·2·2·2·2
11 · 5 · 2
20. 130
13 · 5 · 2
34. 144
3·3·2·2·2·2
35. 108
3·3·3·2·2
36. 120
5·3·2·2·2·2
37. 175
7·5·5
38. 200
5·5·2·2·2
39. 250
5·5·5·2
40. 300
5·5·3·2·2
41. 180
5·3·3·2·2
42. 270
5·3·3·3·2
43. 162
3·3·3·3·2
44. 216
3·3·3·2·2·2
47. 203
29 · 7
48. 455
13 · 7 · 5
45. 85
17 · 5
46. 93
31 · 3
49. 585
13 · 5 · 3 · 3
50. 693
11 · 7 · 3 · 3
51. 495
11 · 5 · 3 · 3
52. 364
13 · 7 · 2 · 2
53. 525
7·5·5·3
54. 672
7·3·2·2·2·2·2
55. 756
7·3·3·3·2·2
56. 792
11 · 3 · 3 · 2 · 2 · 2
57. 450
5·5·3·3·2
58. 630
7·5·3·3·2
59. 441
7·7·3·3
60. 726
11 · 11 · 3 · 2
61. 1980
11 · 5 · 3 · 3 · 2 · 2
62. 1296
3·3·3·3·2·2·2·2
63. 1573
13 · 11 · 11
64. 1690
13 · 13 · 5 · 2
65. 1500
5·5·5·3·2·2
66. 2184
13 · 7 · 3 · 2 · 2 · 2
67. 2376
11 · 3 · 3 · 3 · 2 · 2 · 2
68. 5610
17 · 11 · 5 · 3 · 2
69. u7
u·u·u·u·u·u·u
70. y 5
y·y·y·y·y
71. n9
72. p6
p·p·p·p·p·p
n·n·n·n·n·n·n·n·n
73. a4 b
77. 21k 6
a·a·a·a·b
3·7·k·k·k·k·k·k
81. 36x2 y 3
2·2·3·3·x·x·y·y·y
74. c3 d2
c·c·c·d·d
75. xy 3
78. 10x2
2·5·x·x
79. 39a5
82. 40pr3
2·2·2·5·p·r·r·r
x·y·y·y
3 · 13 · a · a · a · a · a
83. 100n2 m2
2·2·5·5·n·n·m·m
ALG catalog ver. 2.6 – page 168 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
76. u2 w4
80. 15c3
u·u·w·w·w·w
3·5·c·c·c
84. 54a5 b
2·3·3·3·a·a·a·a·a·b
IB
Topic:
Missing factors.
Directions:
69—Find the missing factor.
1.
y 5 = ( ? )(y 3 )
5.
8x3 = ( ? )(4x2 )
9.
20c3 d = ( ? )(5c)
y2
2x
4c2 d
2.
a3 = ( ? )(a2 )
6.
20z 3 = ( ? )(5z)
a
4z 2
10. 28a2 b = ( ? )(14a)
2ab
3.
c7 = ( ? )(c3 )
7.
4y 5 = ( ? )(2y 5 )
4.
x4 = ( ? )(x4 )
2
8.
6d5 = ( ? )(3d2 )
4y 3
12. 18mw3 = ( ? )(9mw2 )
c4
11. 8xy 3 = ( ? )(2x)
1
2d3
2w
13. −6p5 = ( ? )(6p4 )
−p
14. −15k 4 = ( ? )(3k 4 )
−5
15. −4r3 = ( ? )(2r2 )
−2r
16. −10u3 = ( ? )(2u)
−5u2
17. −15q 7 = ( ? )(−5q)
18. −10c6 = ( ? )(−5c3 )
3q 6
19. −20h3 = ( ? )(−4h)
2c3
21. 4y 2 = ( ? )(−4y 2 )
−1
20. −12k 8 = ( ? )(−6k 4 )
5h2
22. 10w3 = ( ? )(−2w3 )
2k 4
23. 7x2 = ( ? )(−7x)
−x
24. a2 b3 = ( ? )(−ab2 )
−ab
−5
25. 50a3 b9 = ( ? )(2ab4 )
26. 24p2 w3 = ( ? )(6pw2 )
27. 40g 4 h2 = ( ? )(2g 2 h2 )
4pw
25a2 b5
29. 32m4 p15 = ( ? )(m3 p9 )
20g 2
30. 6q 11 w7 = ( ? )(q 3 w)
32mp6
42. −8a4 b5 c2 = ( ? )(−8ab4 )
45. 100c3 d3 = ( ? )(−10cd2 )
46. 50a6 b3 = ( ? )(−10a3 b2 )
47. 63r4 u4 = ( ? )(−7r2 u3 )
51. 84g 5 h15 = ( ? )(4g 5 h8 )
22m4 r7
57. −w3 xy 7 z 8 = ( ? )(wy 2 z 8 )
61. (a − b) = ( ? )(b − a)
p5 r
−w3 z 4
62. (3 − c) = ( ? )(c − 3)
3 2
4m p
3m
= ( ? )( 14 mp)
60. w4 x9 y 3 z = ( ? )(−w4 x3 y 2 )
−x5 yz
63. (2d − 1) = ( ? )(1 − 2d)
5
2
6 rw
64. (x2 − y) = ( ? )(y − x2 )
−1
66. (9k 2 p)3 = ( ? )(9k 2 p)2
70.
km3 p
−w2 xz 5
−1
67. (−7x2 )3 = ( ? )(−7x2 )2
9k 2 p
69.
5ay
58. −w5 x2 yz 7 = ( ? )(w3 xyz 2 )
−1
65. (p5 r)2 = ( ? )(p5 r)
4b10 c6
56. −12k 2 m7 p = ( ? )(−12km4 )
−w2 xy 5
68. −(2ab)5 = ( ? )(2ab)4
−2ab
−7x2
= ( ? )( 16 w)
5rw
71.
5 6
6x
ALG catalog ver. 2.6 – page 169 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−2m5 p2
52. 96b13 c9 = ( ? )(24b3 c3 )
54. 50a3 x4 y 2 = ( ? )(10a2 x4 y)
−13cdh
−1
48. 80m7 p3 =
( ? )(−40m2 p)
21h7
15xy 4
59. w8 x5 yz 6 = ( ? )(−w5 x5 yz 2 )
r3 p2 u2
−9r2 u
50. 66m7 r10 = ( ? )(3m3 r3 )
55. −26c5 d4 h3 = ( ? )(2c4 d3 h2 )
44. 9r5 p4 u3 = ( ? )(9r2 p2 u)
g 2 h2 k
−5a3 b
53. 45w9 x2 y 5 = ( ? )(3w9 xy)
9y 3 z
43. 5g 4 h3 k 2 = ( ? )(5g 2 hk)
a3 bc2
4k8
40. −45y 5 z 2 = ( ? )(−5y 2 z)
30cb
41. −16c3 q 2 w3 =
( ? )(4cw2 ) −4c2 q2 w
49. 72k 11 n4 = ( ? )(18k 3 n4 )
−2w4 u2
39. −90c3 b4 = ( ? )(−3c2 b3 )
−4qr2
−10c2 d
36. 10w6 u5 = ( ? )(−5w2 u3 )
−3
38. −40q 2 r3 = ( ? )(10qr)
−3b2
7k
35. 12b2 m = ( ? )(−4b2 m)
−3ab2
37. −60ab5 = ( ? )(20ab3 )
32. 7k 10 m8 = ( ? )(k 9 m8 )
9ab6
34. 18a4 b2 = ( ? )(−6a3 )
−10y
2bc4
31. 9a2 b19 = ( ? )(ab13 )
6q 8 w6
33. 20x3 y 2 = ( ? )(−2x3 y)
28. 8b6 c8 = ( ? )(4b5 c4 )
= ( ? )( 53 x4 )
1 2
2x
72.
7 5
8y
= ( ? )( 72 y 2 )
1 3
4y
IB
73. 5pr3 = ( ? )(30pr2 )
1
6r
74. 8a4 b = ( ? )(24a)
1 3
3a b
75. 6h4 k 3 = ( ? )(8h2 k)
76. 10wz 7 = ( ? )(15wz)
3 2 2
4h k
77. − 89 ab4 = ( ? )( 43 b2 )
78.
− 23 ab2
15 5
14 g h
= ( ? )(− 37 g 2 h)
79.
− 52 g 3
81. 2a9 b3 = ( ? )( 15 a4 b2 )
1 7
2r
= ( ? )(2r2 )
1 5
4r
10w3
82. 5c6 d = ( ? )( 13 c)
15c5 d
86.
2 3
3k
= ( ? )(6k 2 )
1
9k
90. 8d4 = ( ? )(0.08d2 )
83. 3y 3 z 8 = ( ? )( 12 y 2 z 2 )
0.1y 7
94. 0.06x12 = ( ? )(0.3x4 )
x2a
98. x7a = ( ? )(x5a )
1 4
3p
= ( ? )(2p4 )
84. 8uw7 = ( ? )( 14 w4 )
88.
1
6
95. 0.005c6 = ( ? )(0.1c)
102. y 4n+1 = ( ? )(y 3n )
y n+1
99. p3m = ( ? )(p2m )
103. ax+2 = ( ? )(ax+1 )
2 5
5a
= ( ? )(4a)
1 4
10 a
92. 0.6a3 = ( ? )(0.03a3 )
20
96. 0.14k 2 = ( ? )(0.2k)
0.7k
0.05c5
x2a
3 4 4
= ( ? )( 10
r s )
32uw3
2r
0.2x8
a3n
87.
9 11 5
20 r s
3 7
2r s
91. 0.01r4 = ( ? )(0.005r3 )
100d2
93. 0.02y 9 = ( ? )(0.2y 2 )
101. x2a+1 = ( ? )(x)
80.
6yz 6
89. 0.1w8 = ( ? )(0.01w5 )
97. a4n = ( ? )(an )
= ( ? )( 45 c3 d9 )
2 3
3d
10a5 b
85.
8 3 12
15 c d
2 6
3z
pm
a
100. y 10x = ( ? )(y 5x )
y 5x
104. w3n+4 = ( ? )(wn+3 )
w2n+1
105. a5x bx+3 = ( ? )(ax bx )
a4x b3
106. ma+5 n3a = ( ? )(m5 na )
107. xy−1 = ( ? )(xy−3 )
ma n2a
ALG catalog ver. 2.6 – page 170 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
x2
108. p2n−3 = ( ? )(pn−5 )
pn+2
IC
Topic:
Factoring monomials.
Directions:
11—Factor.
multiplying.
1.
5a − 35
12—Factor, if possible.
2.
5(a − 7)
4x + 6y
13—Factor completely.
2(2x + 3y)
3.
−6c2 − 3
14—Factor, then check answer by
32km + 24n
4.
−3(2c2 + 1)
8(4km + 3n)
5.
4x2 − 7x3
x2 (4 − 7x)
6.
3d3 + d
9.
22t2 − 11t
11t(2x − 1)
10. −42gh − 30h2
d(3d2 + 1)
14. 12k 3 + 21k 5
2a2 (2a − b)
−rs2 (4r4
u2 w3 (1 + 9w)
21. 27x2 y − 6xy 3
5ac(3b3
25. 108c2 n2 z 5 − 35c4 nz 4
9w2 z 3 (5w4 + 2z)
+ 1)
6m(6m2
+ 5m − 11)
42. 70xy 3 − 20y 4 + 100x2 y 2
10y 2 (7xy
− 13a + 7)
− 2y 2
+ 10x2 )
46. ab2 c + 8a2 bc + 4abc2
kq(4k 2 r2 − krq − 10)
abc(b + 8a + 4c)
49. 22m2 np − 121mn2 + 66mnp2
51. 90h3 j 2 k − 72h2 j + 18hjk 2
31. 21w4 − 14w2 − 35
32. −3p − 9q + 15r
7(3w4 − 2w2 − 5)
+ 4h − 1)
38. 36m3 + 30m2 − 66m
+ 5n4 )
53. 6w2 z 3 − 30w3 z 3 − 9w2 z 2
24. 45w6 z 3 + 18w2 z 4
30k 3 m5 (4m2 + 5)
−r2 (4r4 + r2 − 2)
45. 4k 3 r2 q − k 2 rq 2 − 10kq
23. −14g 3 h3 − 14g 2 h
28. 120k 3 m7 + 150k 3 m5
k(k2 + 9k + 1)
−5a4 (5a2
cde(7c + 4d)
6adf 2 (8a − 9df )
34. −4r6 − r4 + 2r2
41. −25a6 + 65a5 − 35a4
20. 7c2 de + 4cd2 e
27. 48a2 df 2 − 54ad2 f 3
5(x − 2xy + 6y)
37. 16n2 − 8n4 + 20n6
−12p4 (p4 + 2)
−18pr4 s(2p2 r2 + 9s)
8(h2
33. k 3 + 9k 2 + k
16. −12p8 − 24p4
19. 8m2 n2 − 3mnp
−14g 2 h(gh2
30. 8h2 + 32h − 8
29. 5x − 10xy + 30y
15n3 (6n + 5)
26. −36p3 r7 s − 162pr4 s2
c2 nz 4 (108nz − 35c2 )
4n2 (4 − 2n2
− 7c3 )
p2 (5r + 1)
12. 90n4 + 75n3
mn(8mn − 3p)
22. 15ab3 c − 35ac4
3xy(9x − 2y 2 )
11. 12a2 b − 68a2
18w2 (wy − z)
18. u2 w3 + 9u2 w4
+ 5s)
5p2 r + p2
8.
y(3 − 8x)
15. 18w3 y − 18w2 z
3k 3 (4 + 7k2 )
17. −4r5 s2 − 5rs3
3y − 8xy
4a2 (3b − 17)
−6h(7g + 5h)
13. 4a3 − 2a2 b
7.
11mn(2mp − 11n + 6p2 )
18hj(5h2 jk − 4h + k 2 )
3w2 z 2 (2z − 10wz − 3)
55. −42a3 b4 x5 − 56a3 x4 − 168a2 b2 x2
−3(p + 3q − 5r)
35. abc − a2 b + 2ac2
36. 2y 7 + 7y 5 + y 3
a(bc − ab + 2c2 )
y 3 (2y 4 + 7y 2 + 1)
39. −42x2 y 2 − 14x2 y + 21x2 40. 40a2 b + 16ab2 − 32b2
−7x2 (6y 2 + 2y − 3)
8b(5a2 + 2ab − 4b)
43. 63nr2 + 18r2 − 27n2 r
44. 34u9 − 24u7 + 18u5
9r(7nr + 2r
− 3n2 )
2u5 (17u4 − 12u2 + 9)
47. −m2 n3 − 3m2 n2 + m2 n
48. x2 y 5 + 2x4 y 4 − 4xy 5
−m2 n(mn2 + 3n − 1)
xy 4 (xy + 2x3 − 4y)
50. −24r3 x2 − 16r4 x − 64r2 x2
52. 28b2 h2 − 63b3 h2 + 7b2 h
−8r2 x(3rx + 2r2 + 8x)
7b2 h(4h − 9bh + 1)
54. 105k 4 n7 − 42k 3 n5 + 21k 4 n5
21k 3 n5 (5kn2 − 2 + k)
56. 24c2 d2 e2 + 84c2 d2 e − 96cd2 e3
12cd2 e(2ce + 7c − 8e2 )
−14a2 x2 (3ab4 x3 + 4ax2 + 12b2 )
57. 32ab + 16ac − 8bc + 24c
59. 15cd2 − 10d2 − 5c2 + 5cd
61. x8 − 4x6 + x4 − 6x2
8(4ab + 2ac − bc + 3c)
5(3cd2 − 2d2 − c2 + cd)
x2 (x6 − 4x4 + x2 − 6)
63. h2 kn2 − h2 k 2 + h3 n2 − h2 kn
65. 3ax − 6ay − 3a2 − 6aw
h2 (kn2 − k 2 + hn2 − kn)
3a(x − 2y − a − 2w)
58. 4x3 − 2x2 + 14x − 2
2(2x3 − x2 + 7x − 1)
60. −72m3 + 36m2 − 45m − 9
62. −6u4 + 10u3 − 3u2 + 2u
64. 4pr − pr3 + 2p2 r4 − r6
−9(8m3 + 4m2 − 5m + 1)
−u(6u3 − 10u2 + 3u − 2)
r(4p − pr2 + 2p2 r3 − r5 )
66. 4wxy − 8x2 y − 2xy + 10xy 2
2xy(2w − 4x − 1 + 5y)
67. 21a2 c + 3c2 b − 15abc − 6ac2
3c(7a2 + cb − 5ab − 2ac)
68. 8x2 − 16x2 y 2 + 4x2 y − 12x3
4x2 (2 − 4y 2 + y − 3x)
69. −8n7 − 28n6 + 16n5 − 14n4
−2n4 (4n3 + 14n2 − 8n + 7)
70. 36x4 − 15x3 y + 18x2 − 21xy
3x(12x3 − 5x2 y + 6x − 7y)
ALG catalog ver. 2.6 – page 171 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
IC
71. 40t8 + 60t3 − 30t6 + 50t4
10t3 (4t5 + 6 − 3t3 + 5t)
73. 9a3 y − 24ay 2 − 15a2 y + 36ay 3
3ay(3a2 − 8y − 5a + 12y 2 )
75. 15nr2 − 18m2 nr − 12mn2 r + 3nr
3nr(5r
− 6m2
72. 14abc − 35acd + 21abd + 56ace
74. 5pc2 − pc + 15p2 c − 10p2 c2
7a(2bc − 5cd + 3bd + 8ce)
pc(5c − 1 + 15p − 10pc)
76. 4c2 dx − 4cdx2 − 10cd2 x + 3cdx
cdx(4c − 4x − 10d + 3)
− 4mn + 1)
77. 4f 2 gh − 5fg 2 h + 6fgh2 − 8f 2 g 2 h
f gh(4f − 5g + 6h − 8f g)
79. 5km2 n + mn2 − k 2 m2 n + 10mn3
78. 3a2 c2 − ac3 + 3ac2 + a3 c2
ac2 (3a − c − 3 + a2 )
80. x2 y 2 z 2 + x3 y 2 z − xy 2 z + x3 y 3 z 3
xy 2 z(xz + x2 − 1 + x2 yz 2 )
mn(5km + n − k2 m + 10n2 )
81. 51m2 n3 p + 34mp3 − 102m2 np2 + 68mnp
17mp(3mn3
+ 2p2
82. 16x2 y + 50xy 2 − 8x2 y 2 + 8xy
2xy(8x + 25y − 4xy + 4)
− 6mnp + 4n)
83. −18a2 b3 − 6a2 b + 12a3 b3 − 24a2 b2
−6a2 b(3b2 + 1 − 2ab2 + 4b)
84. 32r4 p2 s − 80r3 p3 + 16r4 p − 48r3 p2 s
16r3 p(2rps − 5p2 + r − 3ps)
ALG catalog ver. 2.6 – page 172 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
ID
Topic:
Factoring trinomials of the form: x2 + bx + c.
Directions:
11—Factor. 12—Factor, if possible. 13—Factor completely.
14—Factor, then check answer by multiplying.
1.
w2 + 7w + 6
2.
(w + 6)(w + 1)
5.
c2 + 25c + 24
6.
prime
(b + 15)(b + 5)
37. m2 + 17m + 60
45. u2 + 18u + 80
(u + 10)(u + 8)
49. d2 + 17d + 72
(d + 9)(d + 8)
53. y 2 + 25y + 46
(y + 23)(y + 2)
57. y 2 + 15y + 54
(y + 9)(y + 6)
61. m2 + 36m + 99
(m + 33)(m + 3)
65. t2 + 28t + 96
(t + 24)(t + 4)
69. z 2 + 77z + 150
(z + 75)(z + 2)
73. m2 + 33m + 200
(d + 30)(d + 7)
42. w2 + 28w + 20
46. h2 + 24h + 80
(h + 20)(h + 4)
50. z 2 + 23z + 90
(z + 18)(z + 5)
54. m2 + 20m + 51
(m + 17)(m + 3)
58. x2 + 18x + 56
(x + 14)(x + 4)
62. w2 + 35w + 66
(w + 33)(w + 2)
66. p2 + 22p + 96
(p + 16)(p + 6)
70. w2 + 56w + 108
(w + 54)(w + 2)
74. k 2 + 35k + 250
(k + 25)(k + 10)
78. m2 + 34m + 120
(m + 30)(m + 4)
28. d2 + 12d + 32
(d + 8)(d + 4)
32. a2 + 14a + 40
(a + 10)(a + 4)
36. y 2 + 19y + 48
(m + 24)(m + 2)
(y + 16)(y + 3)
39. x2 + 13x + 42
(d + 15)(d + 4)
prime
(y + 5)(y + 4)
35. m2 + 26m + 48
38. d2 + 19d + 60
(m + 12)(m + 5)
24. y 2 + 9y + 20
(y + 12)(y + 3)
(z + 32)(z + 2)
40. k 2 + 15k + 44
(x + 7)(x + 6)
prime
43. k 2 + 10k + 28
47. y 2 + 22y + 40
(y + 20)(y + 2)
51. r2 + 46r + 88
(r + 44)(r + 2)
55. p2 + 52p + 51
(p + 51)(p + 1)
59. m2 + 19m + 70
(m + 14)(m + 5)
63. b2 + 62b + 120
(b + 60)(b + 2)
67. q 2 + 25q + 144
(q + 16)(q + 9)
71. p2 + 28p + 180
(p + 18)(p + 10)
75. a2 + 35a + 150
(a + 30)(a + 5)
79. y 2 + 46y + 240
(y + 40)(y + 6)
ALG catalog ver. 2.6 – page 173 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
prime
(c + 17)(c + 2)
31. y 2 + 15y + 36
34. z 2 + 34z + 64
16. w2 + 9w + 15
20. c2 + 19c + 34
(y + 10)(y + 3)
(h + 15)(h + 3)
33. b2 + 20b + 75
prime
27. y 2 + 13y + 30
30. h2 + 18h + 45
(m + 6)(m + 3)
(x + 5)(x + 3)
(c + 4)(c + 3)
(w + 15)(w + 2)
29. m2 + 9m + 18
12. x2 + 8x + 15
23. c2 + 7c + 12
26. w2 + 17w + 30
(k + 8)(k + 3)
15. w2 + 7w + 18
x2 + 75x + 74
(x + 74)(x + 1)
(d + 11)(d + 3)
(r + 13)(r + 5)
25. k 2 + 11k + 24
8.
19. d2 + 14d + 33
22. r2 + 18r + 65
(w + 11)(w + 5)
q 2 + 50q + 49
(g + 7)(g + 5)
prime
m2 + 10m + 9
(m + 9)(m + 1)
11. g 2 + 12g + 35
(x + 11)(x + 2)
21. w2 + 16w + 55
41. m2 + 22m + 24
14. c2 + 4c + 10
4.
(q + 49)(q + 1)
18. x2 + 13x + 22
(p + 7)(p + 3)
77. d2 + 37d + 210
7.
(g + 4)(g + 2)
17. p2 + 10p + 21
(m + 25)(m + 8)
d2 + 36d + 35
10. g 2 + 6g + 8
(y + 5)(y + 2)
z 2 + 15z + 14
(z + 14)(z + 1)
(d + 35)(d + 1)
y 2 + 7y + 10
13. m2 + 9m + 12
3.
(a + 2)(a + 1)
(c + 24)(c + 1)
9.
a2 + 3a + 2
(k + 11)(k + 4)
prime
44. p2 + 15p + 30
prime
48. k 2 + 26k + 88
(k + 22)(k + 4)
52. r2 + 40r + 76
(r + 38)(r + 2)
56. t2 + 89t + 88
(t + 88)(t + 1)
60. w2 + 52w + 100
(w + 50)(w + 2)
64. x2 + 26x + 120
(x + 20)(x + 6)
68. x2 + 51x + 144
(x + 48)(x + 3)
72. c2 + 25c + 150
(c + 15)(c + 10)
76. r2 + 37r + 160
(r + 32)(r + 5)
80. u2 + 63u + 180
(u + 60)(u + 3)
ID
81. k 2 + 45k + 200
(k + 40)(k + 5)
85. x2 + 204x + 800
(x + 200)(x + 4)
89. 9 + 10a + a2
(9 + a)(1 + a)
93. 6 + 5a + a2
(3 + a)(2 + a)
97. 20 + 9c + c2
(5 + c)(4 + c)
101. 48 + 14x + x2
(8 + x)(6 + x)
105. 80 + 18p + p2
(10 + p)(8 + p)
109. 57 + 22y + y 2
(19 + y)(3 + y)
113. 100 + 25x + x2
(20 + x)(5 + x)
117. 200 + 30k + k 2
(20 + k)(10 + k)
121. 180 + 27w + w2
(15 + w)(12 + w)
125. k 4 + 9k 2 + 8
(k 2
+ 8)(k 2
+ 1)
129. b8 + 12b4 + 35
(b4
+ 7)(b4
+ 5)
133. 96 + 35m5 + m10
(32 + m5 )(3 + m5 )
137. x2 y 2 + 7xy + 6
82. w2 + 60w + 800
(w + 40)(w + 20)
86. d2 + 105d + 500
(d + 100)(d + 5)
90. 5 + 6y + y 2
(1 + y)(5 + y)
94. 15 + 8c + c2
(5 + c)(3 + c)
98. 12 + 8a + a2
(6 + a)(2 + a)
102. 88 + 26y + y 2
(22 + y)(4 + y)
106. 90 + 21b + b2
(15 + b)(6 + b)
110. 75 + 28h + h2
(25 + h)(3 + h)
114. 60 + 17w + w2
(12 + w)(5 + w)
118. 120 + 22x + x2
(12 + x)(10 + x)
122. 300 + 37x + x2
(25 + x)(12 + x)
126. a6 + 5a3 + 4
(a3
+ 4)(a3
+ 1)
130. a6 + 21a3 + 90
(a3
+ 15)(a3
+ 6)
134. 76 + 23p4 + p8
(19 + p4 )(4 + p4 )
138. 15 + 16ab + a2 b2
83. q 2 + 56q + 300
(q + 50)(q + 6)
87. r2 + 120r + 2000
(r + 100)(r + 20)
91. 24 + 25x + x2
(24 + x)(1 + x)
95. 65 + 18p + p2
(13 + p)(5 + p)
99. 32 + 18b + b2
(16 + b)(2 + b)
103. 54 + 15a + a2
(9 + a)(6 + a)
107. 72 + 22w + w2
(18 + w)(4 + w)
111. 51 + 20c + c2
(17 + c)(3 + c)
115. 96 + 28y + y 2
(24 + y)(4 + y)
119. 140 + 39a + a2
(35 + a)(4 + a)
123. 210 + 37m + m2
(30 + m)(7 + m)
127. 14 + 9r2 + r4
(7 + r2 )(2 + r2 )
131. 55 + 16w5 + w10
(11 + w5 )(5 + w5 )
135. h12 + 22h6 + 105
(h6 + 15)(h6 + 7)
139. c4 d2 + 10c2 d + 24
(xy + 6)(xy + 1)
(15 + ab)(1 + ab)
(c2 d + 6)(c2 d + 4)
141. 36 + 15c3 w + c6 w2
142. x4 y 8 + 20x2 y 4 + 51
143. 12 + 8ac2 z + a2 c4 d2
(12 + c3 w)(3 + c3 w)
(x2 y 4 + 17)(x2 y 4 + 3)
(6 + ac2 d)(2 + ac2 d)
ALG catalog ver. 2.6 – page 174 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
84. m2 + 54m + 200
(m + 50)(m + 4)
88. z 2 + 205z + 1000
(z + 200)(z + 5)
92. 49 + 50y + y 2
(49 + y)(1 + y)
96. 33 + 14w + w2
(11 + w)(3 + w)
100. 45 + 18m + m2
(15 + m)(3 + m)
104. 60 + 19c + c2
(15 + c)(4 + c)
108. 64 + 34k + k 2
(32 + k)(2 + k)
112. 99 + 36p + p2
(33 + p)(3 + p)
116. 144 + 25p + p2
(16 + p)(9 + p)
120. 250 + 55c + c2
(50 + c)(5 + c)
124. 400 + 50y + y 2
(40 + y)(10 + y)
128. 24 + 11c3 + c6
(8 + c3 )(3 + c3 )
132. 80 + 42x2 + x4
(40 + x2 )(2 + x2 )
136. r10 + 30r5 + 144
(r5 + 24)(r5 + 6)
140. 18 + 9mp2 + m2 p4
(6 + mp2 )(3 + mp2 )
144. x4 y 2 z 8 + 12x2 yz 4 + 20
(x2 yz 4 + 10)(x2 yz 4 + 2)
IE
Topic:
Factoring trinomials of the form: x2 − bx + c.
Directions:
11—Factor. 12—Factor, if possible. 13—Factor completely.
14—Factor, then check answer by multiplying.
1.
y 2 − 4y + 3
2.
(y − 3)(y − 1)
5.
y 2 − 28y + 27
6.
21. z 2 − 12z + 20
41. x2 − 13x + 40
(x − 8)(x − 5)
(r − 32)(r − 3)
69. y 2 − 72y + 140
(y − 70)(y − 2)
73. k 2 − 30k + 200
(q − 50)(q − 5)
62. r2 − 28r + 70
78. x2 − 29x + 120
(x − 24)(x − 5)
52. m2 − 20m + 51
(m − 17)(m − 3)
56. d2 − 17d + 66
(d − 11)(d − 6)
60. x2 − 15x + 56
(y − 20)(y − 5)
prime
70. m2 − 30m + 144
(y − 26)(y − 10)
(a − 36)(a − 2)
59. y 2 − 25y + 100
(g − 40)(g − 3)
74. y 2 − 36y + 260
48. a2 − 38a + 72
(c − 11)(c − 9)
66. g 2 − 43g + 120
(m − 24)(m − 6)
(a − 10)(a − 4)
55. c2 − 20c + 99
(h − 35)(h − 2)
prime
44. a2 − 14a + 40
(y − 23)(y − 2)
58. h2 − 37h + 70
(r − 18)(r − 3)
(g − 16)(g − 5)
51. y 2 − 25y + 46
(g − 72)(g − 1)
57. r2 − 21r + 54
40. g 2 − 21g + 80
(c − 45)(c − 2)
54. g 2 − 73g + 72
(c − 54)(c − 1)
(r − 22)(r − 2)
47. c2 − 47c + 90
(x − 19)(x − 4)
53. c2 − 55c + 54
36. r2 − 24r + 44
(p − 11)(p − 8)
50. x2 − 23x + 76
(w − 19)(w − 3)
(t − 16)(t − 4)
43. p2 − 19p + 88
(x − 18)(x − 4)
49. w2 − 22w + 57
32. t2 − 20t + 64
(t − 30)(t − 2)
46. x2 − 22x + 72
(a − 9)(a − 10)
(c − 9)(c − 5)
39. t2 − 32t + 60
(b − 15)(b − 5)
45. a2 − 19a + 90
28. c2 − 14c + 45
(c − 12)(c − 4)
42. b2 − 20b + 75
63. p2 − 12p + 50
67. n2 − 20n + 96
(n − 12)(n − 8)
71. a2 − 68a + 132
(a − 66)(a − 2)
75. h2 − 31h + 150
(h − 25)(h − 6)
79. a2 − 36a + 180
(a − 30)(a − 6)
ALG catalog ver. 2.6 – page 175 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
prime
(a − 6)(a − 5)
35. c2 − 16c + 48
(k − 40)(k − 2)
20. a2 − 12a + 24
24. a2 − 11a + 30
(d − 25)(d − 3)
38. k 2 − 42k + 80
(x − 20)(x − 3)
prime
31. d2 − 28d + 75
(p − 21)(p − 2)
37. x2 − 23x + 60
19. a2 − 10a + 30
(t − 17)(t − 5)
(k − 9)(k − 2)
34. p2 − 23p + 42
(w − 8)(w − 6)
16. t2 − 22t + 85
27. k 2 − 11k + 18
(n − 10)(n − 6)
33. w2 − 14w + 48
(k − 9)(k − 3)
(p − 4)(p − 6)
30. n2 − 16n + 60
(p − 9)(p − 4)
12. k 2 − 12k + 27
23. p2 − 10p + 24
(x − 12)(x − 2)
29. p2 − 13p + 36
65. r2 − 35r + 96
prime
p2 − 46p + 45
(p − 45)(p − 1)
(b − 19)(b − 2)
26. x2 − 14x + 24
(g − 16)(g − 2)
8.
15. b2 − 21b + 38
(x − 6)(x − 2)
25. g 2 − 18g + 32
h2 − 65h + 64
(g − 7)(g − 5)
22. x2 − 8x + 12
(z − 10)(z − 2)
61. r2 − 16r + 40
18. x2 − 8x + 20
x2 − 13x + 12
(x − 12)(x − 1)
11. g 2 − 12g + 35
(y − 11)(y − 7)
prime
4.
(h − 64)(h − 1)
14. y 2 − 18y + 77
(k − 13)(k − 3)
77. q 2 − 55q + 250
7.
(x − 5)(x − 3)
13. k 2 − 16k + 39
(k − 20)(k − 10)
b2 − 21b + 20
10. x2 − 8x + 15
(r − 3)(r − 2)
p2 − 6p + 5
(p − 5)(p − 1)
(b − 20)(b − 1)
r2 − 5r + 6
17. x2 − 5x + 15
3.
(w − 7)(w − 1)
(y − 27)(y − 1)
9.
w2 − 8w + 7
(x − 8)(x − 7)
prime
64. p2 − 20p + 75
68. h2 − 22h + 120
(h − 12)(h − 10)
72. c2 − 40c + 144
(c − 36)(c − 4)
76. g 2 − 37g + 300
(g − 25)(g − 12)
80. k 2 − 38k + 240
(k − 30)(k − 8)
prime
IE
81. y 2 − 46y + 240
(y − 40)(y − 6)
85. h2 − 57h + 350
(h − 50)(h − 7)
89. 14 − 15x + x2
(14 − x)(1 − x)
93. 8 − 6a + a2
(4 − a)(2 − a)
97. 85 − 22x + x2
(17 − x)(5 − x)
101. 24 − 11x + x2
(8 − x)(3 − x)
105. 30 − 17p + p2
(15 − p)(2 − p)
109. 44 − 15a + a2
(11 − a)(4 − a)
113. 90 − 23w + w2
(18 − w)(5 − w)
117. 120 − 26c + c2
(20 − c)(6 − c)
121. 210 − 29m + m2
(15 − m)(14 − m)
125. a4 − 7a2 + 6
(a2
− 6)(a2
− 1)
129. p8 − 18p4 + 45
(p4
− 15)(p4
− 3)
133. 96 − 20a5 + a10
(12 − a5 )(8 − a5 )
137. w2 x2 − 6wx + 5
82. p2 − 44p + 160
(p − 40)(p − 4)
86. k 2 − 53k + 150
(k − 50)(k − 3)
90. 4 − 5y + y 2
(5 − w)(2 − w)
98. 51 − 20y + y 2
(17 − y)(3 − y)
102. 18 − 9r + r2
(6 − r)(3 − r)
(16 − c)(4 − c)
110. 42 − 13x + x2
(7 − x)(6 − x)
(18 − r)(4 − r)
118. 140 − 72y + y 2
(70 − y)(2 − y)
122. 180 − 63p + p2
(60 − p)(3 − p)
126. y 6 − 15y 3 + 14
− 1)
130. m6 − 19m3 + 90
− 9)
134. 56 − 18r4 + r8
(14 − r4 )(4 − r4 )
138. 12 − 7by + b2 y 2
(wx − 5)(wx − 1)
(4 − by)(3 − by)
141. 40 − 13k 3 m + k 6 m2
142. r4 q 8 − 21r2 q 4 + 54
(8 − k 3 m)(5 − k 3 m)
95. 15 − 8c + c2
(5 − c)(3 − c)
99. 91 − 20a + a2
(13 − a)(7 − a)
103. 40 − 14a + a2
107. 48 − 19m + m2
(16 − m)(3 − m)
111. 100 − 29m + m2
(25 − m)(4 − m)
114. 72 − 22r + r2
− 10)(m3
(x − 100)(x − 8)
(10 − a)(4 − a)
106. 64 − 20c + c2
(m3
87. x2 − 108x + 800
(20 + w)(1 + w)
94. 10 − 7w + w2
− 14)(y 3
(x − 60)(x − 5)
91. 20 + 21w + w2
(4 − y)(1 − y)
(y 3
83. x2 − 65x + 300
(r2 q 4 − 18)(r2 q 4 − 3)
115. 96 − 22a + a2
(16 − a)(6 − a)
119. 250 − 55k + k 2
(50 − k)(5 − k)
123. 600 − 50w + w2
(20 − w)(30 − w)
127. 16 − 10c2 + c4
(8 − c2 )(2 − c2 )
131. 75 − 28k 5 + k 10
(25 − k 5 )(3 − k 5 )
135. p12 − 38p6 + 105
(p6 − 35)(p6 − 3)
139. a4 c2 − 14a2 c + 24
(a2 c − 12)(a2 c − 2)
143. 16 − 10wx2 y + w2 x4 y 2
(8 − wx2 y)(2 − wx2 y)
ALG catalog ver. 2.6 – page 176 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
84. p2 − 50p + 600
(p − 30)(p − 20)
88. z 2 − 110z + 1000
(z − 100)(z − 10)
92. 35 − 36r + r2
(35 − r)(1 − r)
96. 20 − 9w + w2
(5 − w)(4 − w)
100. 99 − 36k + k 2
(33 − k)(3 − k)
104. 32 − 12m + m2
(8 − m)(4 − m)
108. 60 − 17y + y 2
(12 − y)(5 − y)
112. 80 − 18x + x2
(10 − x)(8 − x)
116. 144 − 51k + k 2
(48 − k)(3 − k)
120. 160 − 44a + a2
(40 − a)(4 − a)
124. 300 − 37r + r2
(25 − r)(12 − r)
128. 24 − 14w3 + w6
(12 − w3 )(2 − w3 )
132. 80 − 24y 2 + y 4
(20 − y 2 )(4 − y 2 )
136. x10 − 25x5 + 144
(x5 − 9)(x5 − 16)
140. 22 − 13ar2 + a2 r4
(11 − ar2 )(2 − ar2 )
144. b4 d2 h8 − 17b2 dh4 + 30
(b2 dh4 − 15)(b2 dh4 − 2)
IF
Topic:
Factoring trinomials of the form: x2 + bx − c.
Directions:
11—Factor. 12—Factor, if possible. 13—Factor completely.
14—Factor, then check answer by multiplying.
1.
k 2 + 4k − 5
2.
(k + 5)(k − 1)
5.
r2 + 24r − 25
6.
p2 + 17p − 18
(w + 3)(w − 2)
13. t2 + 2t − 15
14. a2 + 6a − 27
(t + 5)(t − 3)
21. r2 + 13r − 30
29. t2 + 4t − 45
(d + 30)(d − 2)
prime
58. m2 + 30m − 60
62. y 2 + 33y − 70
(y + 35)(y − 2)
65. r2 + r − 210
66. u2 + u − 156
(r + 15)(r − 14)
(u + 13)(u − 12)
69. x2 + 70x − 144
(x + 72)(x − 2)
73. p2 + 10p − 200
(p + 20)(p − 10)
77. m2 + 19m − 150
70. a2 + 18a − 144
(a + 24)(a − 6)
74. r2 + 5r − 150
(r + 15)(r − 10)
78. x2 + 13x − 300
(x + 25)(x − 12)
36. t2 + 8t − 65
(t + 13)(t − 5)
40. h2 + 12h − 64
(h + 16)(h − 4)
44. b2 + 15b − 54
(b + 18)(b − 3)
48. y 2 + 27y − 58
(y + 29)(y − 2)
52. q 2 + 14q − 120
(q + 20)(q − 6)
55. k 2 + 18k − 88
56. h2 + 3h − 88
(k + 22)(k − 4)
prime
59. d2 + 34d − 70
63. w2 + 74w − 75
(w + 75)(w − 1)
67. p2 + 15p − 100
(p + 20)(p − 5)
71. d2 + 53d − 110
(d + 55)(d − 2)
75. h2 + 18h − 280
(h + 28)(h − 10)
79. r2 + 45r − 250
(r + 50)(r − 5)
ALG catalog ver. 2.6 – page 177 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
prime
(m + 12)(m − 4)
(p + 40)(p − 3)
(x + 32)(x − 3)
28. c2 + 8c − 30
32. m2 + 8m − 48
51. p2 + 37p − 120
54. x2 + 29x − 96
(r + 12)(r − 8)
prime
(p + 17)(p − 2)
(h + 36)(h − 2)
53. r2 + 4r − 96
27. c2 + 14c − 20
(d + 12)(d − 3)
47. p2 + 15p − 34
50. h2 + 34h − 72
(d + 9)(d − 8)
24. d2 + 9d − 36
(w + 13)(w − 7)
(d + 40)(d − 2)
49. d2 + d − 72
(c + 11)(c − 3)
43. w2 + 6w − 91
46. d2 + 38d − 80
(c + 10)(c − 8)
20. c2 + 8c − 33
(d + 25)(d − 3)
(d + 11)(d − 9)
45. c2 + 2c − 80
(b + 8)(b − 3)
39. d2 + 22d − 75
42. d2 + 2d − 99
(x + 33)(x − 3)
16. b2 + 5b − 24
(r + 30)(r − 3)
(y + 15)(y − 4)
41. x2 + 30x − 99
(y + 5)(y − 4)
35. r2 + 27r − 90
38. y 2 + 11y − 60
(w + 10)(w − 6)
(x + 6)(x − 5)
(b + 16)(b − 3)
(a + 10)(a − 7)
37. w2 + 4w − 60
12. y 2 + y − 20
31. b2 + 13b − 48
34. a2 + 3a − 70
(x + 8)(x − 5)
61. d2 + 28d − 60
prime
(w + 9)(w − 4)
33. x2 + 3x − 40
11. x2 + x − 30
(m + 16)(m − 2)
30. w2 + 5w − 36
(t + 9)(t − 5)
57. m2 + 25m − 40
26. w2 + 8w − 10
b2 + 48b − 49
(b + 49)(b − 1)
23. m2 + 14m − 32
(q + 15)(q − 3)
prime
8.
(g + 35)(g − 1)
(q + 11)(q − 7)
22. q 2 + 12q − 45
(r + 15)(r − 2)
g 2 + 34g − 35
19. q 2 + 4q − 77
(d + 13)(d − 2)
a2 + 7a − 8
(a + 8)(a − 1)
(q + 9)(q − 2)
18. d2 + 11d − 26
(a + 17)(a − 3)
4.
15. q 2 + 7q − 18
(a + 9)(a − 3)
17. a2 + 14a − 51
(m + 25)(m − 6)
7.
10. w2 + w − 6
(p + 4)(p − 3)
c2 + 5c − 6
(c + 6)(c − 1)
(p + 18)(p − 1)
p2 + p − 12
25. w2 + 5w − 12
3.
(w + 3)(w − 1)
(r + 25)(r − 1)
9.
w2 + 2w − 3
(h + 11)(h − 8)
prime
60. d2 + 18d − 45
prime
64. b2 + 44b − 45
(b + 45)(b − 1)
68. h2 + 48h − 100
(h + 50)(h − 2)
72. c2 + 68c − 140
(c + 70)(c − 2)
76. c2 + 25c − 350
(c + 35)(c − 10)
80. w2 + 35w − 200
(w + 40)(w − 5)
IF
81. r2 + 17r − 110
(r + 22)(r − 5)
85. b2 + 43b − 350
(b + 50)(b − 7)
89. p2 + 10p − 600
(p + 30)(p − 20)
82. w2 + 44w − 300
(w + 50)(w − 6)
86. k 2 + 47k − 150
(k + 50)(k − 3)
90. q 2 + 30q − 1000
83. m2 + 34m − 240
(m + 40)(m − 6)
87. c2 + 92c − 800
(c + 100)(c − 8)
91. g 2 + 90g − 1000
(q + 50)(q − 20)
(g + 100)(g − 10)
93. 9 + 8y − y 2
94. 10 + 9r − r2
95. 25 + 24p − p2
(9 − y)(1 + y)
(10 − r)(1 + r)
97. 27 + 6y − y 2
(9 − y)(3 + y)
101. 45 + 12y − y 2
(15 − y)(3 + y)
105. 75 + 10w − w2
(15 − w)(5 + w)
109. 51 + 14y − y 2
(17 − y)(3 + y)
113. 88 + 18w − w2
(22 − w)(4 + w)
117. 130 + 63a − a2
(65 − a)(2 + a)
121. 250 + 45k − k 2
(50 − k)(5 + k)
125. m4 + 6m2 − 7
(m2
+ 7)(m2
− 1)
129. k 8 + 13k 4 − 48
(k 4
+ 16)(k 4
− 3)
133. 44 + 7x5 − x10
(11 − x5 )(4 + x5 )
137. x2 y 2 + 5xy − 6
(xy + 6)(xy − 1)
141. 40 + 6a3 x − a6 x2
(10 − a3 x)(4 + a3 x)
98. 14 + 5r − r2
(7 − r)(2 + r)
102. 36 + 9w − w2
(12 − w)(3 + w)
106. 64 + 12y − y 2
(16 − y)(4 + y)
110. 72 + 34p − p2
(36 − p)(2 + p)
114. 120 + 19a − a2
(24 − a)(5 + a)
118. 180 + 8w − w2
(18 − w)(10 + w)
122. 500 + 40y − y 2
(50 − y)(10 + y)
126. a6 + 11a2 − 12
(a3
+ 12)(a3
− 1)
130. p6 + 27p3 − 90
(p3
+ 30)(p3
− 3)
134. 85 + 12y 4 − y 8
(17 − y 4 )(5 + y 4 )
138. 22 + 9cd − c2 d2
(25 − p)(1 + p)
(g + 60)(g − 4)
88. w2 + 94w − 600
(w + 100)(w − 6)
92. t2 + 80t − 2000
(t + 100)(t − 20)
96. 45 + 44x − x2
(45 − x)(1 + x)
99. 20 + 8a − a2
100. 30 + x − x2
(10 − a)(2 + a)
(6 − x)(5 + x)
103. 70 + 3m − m2
(10 − m)(7 + m)
107. 99 + 2m − m2
(11 − m)(9 + m)
111. 72 + 14a − a2
(18 − a)(4 + a)
115. 156 + w − w2
(13 − w)(12 + w)
119. 150 + 19y − y 2
(25 − y)(6 + y)
123. 200 + 17m − m2
104. 70 + 9a − a2
(14 − a)(5 + a)
108. 56 + w − w2
(8 − w)(7 + w)
112. 96 + 10r − r2
(16 − r)(6 + r)
116. 144 + 7x − x2
(16 − x)(9 + x)
120. 240 + 56p − p2
(60 − p)(4 + p)
124. 600 + 10w − w2
(25 − m)(8 + m)
(30 − w)(20 + w)
127. 10 + 3r2 − r4
128. 24 + 5y 3 − y 6
(5 − r2 )(2 + r2 )
131. 65 + 8w5 − w10
(13 − w5 )(5 + w5 )
135. r12 + 15r6 − 100
(r6 + 20)(r6 − 5)
139. k 4 m2 + 2k 2 m − 35
(11 − cd)(2 + cd)
(k 2 m + 7)(k 2 m − 5)
142. c4 y 8 + 11c2 y 4 − 42
143. 24 + 10nq 2 r − n2 q 4 r2
(c2 y 4 + 14)(c2 y 4 − 3)
84. g 2 + 56g − 240
(12 − nq 2 r)(2 + nq 2 r)
ALG catalog ver. 2.6 – page 178 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(8 − y 3 )(3 + y 3 )
132. 60 + 4a2 − a4
(10 − a2 )(6 + a2 )
136. c10 + 32c5 − 144
(c5 + 36)(c5 − 4)
140. 32 + 14pr2 − p2 r4
(16 − pr2 )(2 + pr2 )
144. x4 y 2 z 8 + 16x2 yz 4 − 36
(x2 yz 4 + 18)(x2 yz 4 − 2)
IG
Topic:
Factoring trinomials of the form: x2 − bx − c.
Directions:
11—Factor. 12—Factor, if possible. 13—Factor completely.
14—Factor, then check answer by multiplying.
1.
x2 − 3x − 4
2.
b2 − b − 2
(b − 2)(b + 1)
3.
(x − 4)(x + 1)
5.
9.
6.
k 2 − 14k − 15
(y − 12)(y + 1)
(k − 15)(k + 1)
p2 − 4p − 21
10. h2 − 2h − 8
(p − 7)(p + 3)
7.
(y − 12)(y + 2)
prime
25. y 2 − 8y − 20
(y − 17)(y + 3)
45. p2 − 7p − 44
(x − 16)(x + 6)
61. q 2 − 38q − 80
(q − 40)(q + 2)
54. y 2 − 15y − 50
58. m2 − 20m − 96
(m − 24)(m + 4)
62. k 2 − 18k − 40
(k − 20)(k + 2)
65. x2 − x − 132
66. r2 − r − 272
(x − 12)(x + 11)
(r − 17)(r + 16)
69. t2 − 7t − 144
(t − 16)(t + 9)
73. w2 − 8w − 180
(w − 18)(w + 10)
77. a2 − 17a − 200
70. x2 − 32x − 144
(x − 36)(x + 4)
74. y 2 − 6y − 160
(y − 16)(y + 10)
78. b2 − 20b − 300
(b − 30)(b + 10)
40. w2 − 30w − 64
(w − 32)(w + 2)
44. g 2 − 21g − 46
(g − 23)(g + 2)
47. b2 − 16b − 80
48. r2 − 11r − 80
(b − 20)(b + 4)
(r − 16)(r + 5)
51. a2 − 58a − 120
(d − 18)(d + 4)
prime
(x − 17)(x + 5)
(x − 19)(x + 2)
50. d2 − 14d − 72
(k − 24)(k + 3)
36. x2 − 12x − 85
43. x2 − 17x − 38
(a − 22)(a + 2)
49. k 2 − 21k − 72
(r − 24)(r + 2)
(d − 15)(d + 5)
46. a2 − 20a − 44
(p − 11)(p + 4)
32. r2 − 22r − 48
39. d2 − 10d − 75
(w − 19)(w + 3)
52. t2 − 19t − 120
(a − 60)(a + 2)
prime
55. p2 − 28p − 80
(t − 24)(t + 5)
prime
59. n2 − 19n − 66
(n − 22)(n + 3)
63. y 2 − 49y − 50
(y − 50)(y + 1)
67. m2 − 21m − 100
(m − 25)(m + 4)
71. n2 − 73n − 150
(n − 75)(n + 2)
75. t2 − 14t − 240
(t − 24)(t + 10)
79. d2 − 25d − 150
(d − 30)(d + 5)
ALG catalog ver. 2.6 – page 179 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
prime
(p − 8)(p + 4)
(w − 18)(w + 5)
42. w2 − 16w − 57
24. a2 − 8a − 24
28. p2 − 4p − 32
35. w2 − 13w − 90
(a − 20)(a + 3)
41. y 2 − 14y − 51
prime
(b − 8)(b + 6)
38. a2 − 17a − 60
(y − 12)(y + 5)
23. x2 − 5x − 10
(k − 21)(k + 3)
31. b2 − 2b − 48
(a − 14)(a + 5)
37. y 2 − 7y − 60
20. k 2 − 18k − 63
(r − 6)(r + 3)
34. a2 − 9a − 70
(d − 10)(d + 4)
(t − 15)(t + 3)
27. r2 − 3r − 18
(c − 12)(c + 3)
33. d2 − 6d − 40
57. x2 − 10x − 96
prime
30. c2 − 9c − 36
(w − 15)(w + 3)
16. t2 − 12t − 45
(n − 18)(n + 3)
(c − 10)(c + 3)
29. w2 − 12w − 45
(t − 5)(t + 2)
19. n2 − 15n − 54
26. c2 − 7c − 30
(y − 10)(y + 2)
53. x2 − 15x − 75
22. w2 − 7w − 12
12. t2 − 3t − 10
(n − 16)(n + 2)
(c − 11)(c + 5)
d2 − 20d − 21
(d − 21)(d + 1)
15. n2 − 14n − 32
18. c2 − 6c − 55
(m − 13)(m + 5)
8.
(a − 7)(a + 2)
(p − 4)(p + 3)
17. m2 − 8m − 65
g 2 − 44g − 45
11. a2 − 5a − 14
14. p2 − p − 12
w2 − 9w − 10
(w − 10)(w + 1)
(g − 45)(g + 1)
(h − 4)(h + 2)
13. y 2 − 10y − 24
(a − 25)(a + 8)
4.
(k − 9)(k + 1)
y 2 − 11y − 12
21. r2 − 9r − 20
k 2 − 8k − 9
56. b2 − 30b − 105
60. b2 − 5b − 66
(b − 11)(b + 6)
64. y 2 − 69y − 70
(y − 70)(y + 1)
68. r2 − 99r − 100
(r − 100)(r + 1)
72. y 2 − 78y − 160
(y − 80)(y + 2)
76. c2 − 15c − 250
(c − 25)(c + 10)
80. k 2 − 55k − 300
(k − 60)(k + 5)
prime
IG
81. b2 − 19b − 120
(b − 24)(b + 5)
85. c2 − 42c − 400
(c − 50)(c + 8)
89. y 2 − 195y − 1000
(y − 200)(y + 5)
82. r2 − 46r − 200
(r − 50)(r + 4)
86. x2 − 7x − 450
(x − 25)(x + 18)
90. y 2 − 10y − 2000
(y − 50)(y + 40)
93. 6 − 5x − x2
94. 4 − 3p − p2
(6 + x)(1 − x)
(4 + p)(1 − p)
97. 15 − 2w − w2
(5 + w)(3 − w)
101. 32 − 14m − m2
(16 + m)(2 − m)
105. 90 − 9k − k 2
(15 + k)(6 − k)
109. 54 − 15a − a2
(18 + a)(3 − a)
113. 120 − 14m − m2
(20 + m)(6 − m)
117. 140 − 68c − c2
(70 + c)(2 − c)
121. 150 − 25p − p2
(30 + p)(5 − p)
125. y 4 − 7y 2 − 8
(y 2
− 8)(y 2
+ 1)
129. p8 − 10p4 − 75
(p4
− 15)(p4
+ 5)
133. 44 − 20y 5 − y 10
(22 + y 5 )(2 − y 5 )
137. a2 b2 − 2ab − 3
(ab − 3)(ab + 1)
141. 60 − 7c3 d − c6 d2
(12 + c3 d)(5 − c3 d)
98. 8 − 2m − m2
(4 + m)(2 − m)
102. 48 − 8x − x2
(12 + x)(4 − x)
106. 80 − 16w − w2
(20 + w)(4 − w)
110. 64 − 30y − y 2
(32 + y)(2 − y)
114. 100 − 15k − k 2
(20 + k)(5 − k)
118. 200 − 10r − r2
(20 + r)(10 − r)
122. 600 − 10a − a2
(30 + a)(20 − a)
126. w6 − 14w3 − 15
(w3
− 15)(w3
+ 1)
130. r6 − 22r3 − 75
(r3
− 25)(r3
+ 3)
134. 140 − 4a4 − a8
(14 + a4 )(10 − a4 )
138. 30 − 13xy − x2 y 2
(15 + xy)(2 − xy)
142. r4 s8 − 4r2 s4 − 32
(r2 s4 − 8)(r2 s4 + 4)
83. w2 − 36w − 160
(w − 40)(w + 4)
87. n2 − 95n − 500
(n − 100)(n + 5)
91. b2 − 85b − 1500
(b − 100)(b + 15)
95. 21 − 20w − w2
(21 + w)(1 − w)
99. 24 − 10p − p2
(12 + p)(2 − p)
103. 40 − 6w − w2
84. d2 − 54d − 360
(d − 60)(d + 6)
88. t2 − 10t − 600
(t − 30)(t + 20)
92. a2 − 88a − 1200
(a − 100)(a + 12)
96. 36 − 35r − r2
(36 + r)(1 − r)
100. 42 − a − a2
(7 + a)(6 − a)
104. 90 − 13c − c2
(10 + w)(4 − w)
(18 + c)(5 − c)
107. 80 − 2r − r2
108. 72 − x − x2
(10 + r)(8 − r)
(9 + x)(8 − x)
111. 72 − 21y − y 2
(24 + y)(3 − y)
115. 132 − p − p2
(12 + p)(11 − p)
119. 200 − 17x − x2
(25 + x)(8 − x)
123. 250 − 15k − k 2
(25 + k)(10 − k)
127. 12 − a2 − a4
(4 + a2 )(3 − a2 )
131. 95 − 14k 5 − k 10
(19 + k 5 )(5 − k 5 )
135. c12 − 26c6 − 120
(c6 − 30)(c6 + 4)
139. d4 h2 − 5d2 h − 36
112. 96 − 4a − a2
(12 + a)(8 − a)
116. 144 − 18w − w2
(24 + w)(6 − w)
120. 160 − 36m − m2
(40 + m)(4 − m)
124. 500 − 95r − r2
(100 + r)(5 − r)
128. 28 − 3m3 − m6
(7 + m3 )(4 − m3 )
132. 56 − 26w2 − w4
(28 + w2 )(2 − w2 )
136. m10 − 10m5 − 96
(m5 − 16)(m5 + 6)
140. 52 − 24mp2 − m2 p4
(d2 h − 9)(d2 h + 4)
(26 + mp2 )(2 − mp2 )
143. 20 − 8xy 2 z − x2 y 4 z 2
144. k 4 n2 r8 − 13k 2 nr4 − 48
(10 + xy 2 z)(2 − xy 2 z)
ALG catalog ver. 2.6 – page 180 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(k 2 nr4 − 16)(k 2 nr4 + 3)
IH
Topic:
Factoring trinomials of the form: ax2 + bx + c.
Directions:
11—Factor. 12—Factor, if possible. 13—Factor completely.
14—Factor, then check answer by multiplying.
1.
5m2 + 6m + 1
2.
12x2 + 4x − 1
6.
(6x − 1)(2x + 1)
9.
6a2 − a − 1
7.
(3z + 11)(z − 1)
21. 5y 2 − 14y − 3
29. 5a2 + 12a + 4
(5a + 2)(a + 2)
33. 30m2 + 7m − 1
(10m − 1)(3m + 1)
37. 14x2 + 15x + 1
26. 3b2 + 12b + 7
16. 3a2 − 4a − 7
(3a − 7)(a + 1)
20. 13u2 − 27u + 2
(13u − 1)(u − 2)
23. 2a2 − 25a − 13
(3p − 1)(p + 7)
prime
(5r + 7)(r + 1)
(5y + 1)(y + 3)
22. 3p2 + 20p − 7
(5y + 1)(y − 3)
12. 5r2 + 12r + 7
19. 5y 2 + 16y + 3
(2c − 1)(c − 11)
24. 11x2 + 32x − 3
(2a + 1)(a − 13)
prime
30. 4r2 + 11r + 7
(4r + 7)(r + 1)
34. 21x2 − 4x − 1
(7x + 1)(3x − 1)
38. 24m2 − 25m + 1
27. 7m2 − 9m − 1
8c2 + 6c + 1
(4c + 1)(2c + 1)
(13x − 5)(x + 1)
18. 2c2 − 23c + 11
(7d + 1)(d + 5)
8.
15. 13x2 + 8x − 5
(11h + 2)(h − 1)
17. 7d2 + 36d + 5
10m2 − 7m + 1
(7y − 2)(y − 1)
14. 11h2 − 9h − 2
13w2 + 12w − 1
(13w − 1)(w + 1)
11. 7y 2 − 9y + 2
(2n + 5)(n + 1)
13. 3z 2 + 8z − 11
4.
(5m − 1)(2m − 1)
10. 2n2 + 7n + 5
(3x − 7)(x − 1)
5y 2 − 4y − 1
(5y + 1)(y − 1)
(3a + 1)(2a − 1)
3x2 − 10x + 7
25. 2y 2 − 8y + 5
3.
(11k − 1)(k − 1)
(5m + 1)(m + 1)
5.
11k 2 − 12k + 1
(11x − 1)(x + 3)
prime
31. 2p2 − 7p + 6
(2p − 3)(p − 2)
35. 10y 2 + 9y − 1
(10y − 1)(y + 1)
39. 28n2 + 16n + 1
28. 11c2 + c − 13
32. 6k 2 − 13k + 2
(6k − 1)(k − 2)
36. 15a2 − 14a − 1
(15a + 1)(a − 1)
40. 21y 2 − 10y + 1
(14x + 1)(x + 1)
(24m − 1)(m − 1)
(14n + 1)(2n + 1)
(7y − 1)(3y − 1)
41. 5s2 − 13s − 6
42. 6y 2 − 11y − 7
43. 2x2 + 17x − 9
44. 9c2 + 18c − 7
(5s + 2)(s − 3)
45. 3w2 − 10w + 8
(3y − 7)(2y + 1)
46. 8u2 − 14u + 5
(2x − 1)(x + 9)
47. 7m2 − 27m + 26
(3w − 4)(w − 2)
(4u − 5)(2u − 1)
(7m − 13)(m − 2)
49. 15y 2 − y − 2
50. 2x2 − x − 21
51. 2p2 − 9p − 35
(5y − 2)(3y + 1)
(2x − 7)(x + 3)
53. 4d2 + 9d + 2
54. 4y 2 + 8y + 3
(4d + 1)(d + 2)
57. 7a2 + 9a − 10
(7a − 5)(a + 2)
61. 4y 2 + 20y − 11
(2y + 11)(2y − 1)
65. 9x2 − 12x − 5
(3x − 5)(3x + 1)
69. 3k 2 + 28k + 49
(3k + 7)(k + 7)
73. 25a2 − 15a + 2
(5a − 2)(5a − 1)
(2y + 1)(2y + 3)
58. 10n2 + 23n − 5
(5n − 1)(2n + 5)
62. 25a2 + 10a − 3
(5a + 3)(5a − 1)
66. 15c2 − 22c − 5
(5c + 1)(3c − 5)
70. 3a2 + 14a + 15
(3a + 5)(a + 3)
74. 4c2 − 24c + 11
(2c − 11)(2c − 1)
(2p + 5)(p − 7)
55. 10a2 + 11a + 3
(2a + 1)(5a + 3)
(3c + 7)(3c − 1)
48. 22q 2 − 35q + 3
(11q − 1)(2q − 3)
52. 26s2 − 21s − 5
(26s + 5)(s − 1)
56. 10c2 + 21c + 11
(10c + 11)(c + 1)
59. 8x2 + x − 7
60. 5w2 + w − 4
(8x − 7)(x + 1)
(5w − 4)(w + 1)
63. 21p2 + 8p − 5
(7p + 5)(3p − 1)
67. 21w2 − 26w − 11
(7w − 11)(3w + 1)
71. 26z 2 + 41z + 3
(13z + 1)(2z + 3)
75. 13y 2 − 41y + 6
(13y − 2)(y − 3)
ALG catalog ver. 2.6 – page 181 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
prime
64. 4h2 + 27h − 7
(4h − 1)(h + 7)
68. 5z 2 − 16z − 21
(5z − 21)(z + 1)
72. 5x2 + 57x + 22
(5x + 2)(x + 11)
76. 10p2 − 37p + 7
(2p − 7)(5p − 1)
IH
77. 10p2 − 27p + 5
78. 25m2 − 20m + 3
(5p − 1)(2p − 5)
(5m − 1)(5m − 3)
81. 5y 2 + 23y + 12
86. 18n2 − 5n − 7
(5x + 6)(x − 3)
89. 3c2 − 32c + 20
93. 24r2 + 7r − 5
(7m + 10)(m − 3)
(2x − 3)(x − 12)
105. 48r2 + 35r + 2
(3x + 4)(x + 12)
prime
113. 6y 2 + 19y + 10
(3y + 2)(2y + 5)
117. 4c2 + 9c − 9
(4c − 3)(c + 3)
121. 6x2 − 25x + 14
(3x − 2)(2x − 7)
125. 15k 2 − 26k − 8
(15k + 4)(k − 2)
129. 33c2 + c − 4
(11c + 4)(3c − 1)
133. 25v 2 − 5v − 6
(5v + 2)(5v − 3)
137. 8w2 − 42w + 27
(4w − 3)(2w − 9)
141. 12x2 + 40x + 25
(6x + 5)(2x + 5)
145. 20y 2 − 21y + 4
(4y − 1)(5y − 4)
149. 12y 2 − 19y − 21
(4y + 3)(3y − 7)
153. 5a2 + 33a + 40
(5a + 8)(a + 5)
157. 6x2 + 71x − 50
(3x − 2)(2x + 25)
110. 3t2 − 26t + 8
prime
(5p + 8)(2p + 1)
118. 9x2 + 16x − 4
(9x − 2)(x + 2)
122. 14y 2 − 23y + 8
(7y − 8)(2y − 1)
126. 6a2 − a − 15
(3a − 5)(2a + 3)
130. 4p2 + 12p − 55
(2p + 11)(2p − 5)
134. 6y 2 − 13y − 15
(6y + 5)(y − 3)
138. 35p2 − 31p + 6
(7p − 2)(5p − 3)
142. 18a2 + 51a + 8
(6a + 1)(3a + 8)
146. 12p2 − 44p + 35
(6p − 7)(2p − 5)
150. 21x2 − 4x − 12
154. 30c2 + 31c + 7
(10c + 7)(3c + 1)
158. 40a2 + 59a − 8
(8a − 1)(5a + 8)
103. 5v 2 + v − 42
104. 42p2 + 11p − 3
(6p − 1)(7p + 3)
107. 36a2 + 27a + 5
(12a + 5)(3a + 1)
114. 10p2 + 21p + 8
(7x − 6)(3x + 2)
(9 − 2n)(4 + n)
(5v − 14)(v + 3)
106. 3x2 + 40x + 48
(16r + 1)(3r + 2)
100. 36 + n − 2n2
(7y − 12)(y + 3)
102. 2x2 − 27x + 36
(5a − 9)(a − 4)
(2u + 15)(u − 3)
99. 7y 2 + 9y − 36
(5v + 1)(6v − 5)
101. 5a2 − 29a + 36
96. 2u2 + 9u − 45
(9b + 2)(5b − 1)
98. 30v 2 − 19v − 5
111. 14a2 + a − 5
prime
(8x + 9)(x + 1)
119. 4a2 + 4a − 15
(2a + 5)(2a − 3)
123. 35s2 − 39s + 10
(7s − 5)(5s − 2)
127. 10w2 − 7w − 6
(5w − 6)(2w + 1)
131. 22a2 + 27a − 9
(11a − 3)(2a + 3)
135. 15m2 − 13m − 6
(5m − 6)(3m + 1)
139. 25z 2 + 50z + 9
(5z + 9)(5z + 1)
143. 9w2 + 36w + 20
(3w + 10)(3w + 2)
147. 9x2 − 23x + 14
(9x − 14)(x − 1)
151. 18m2 + 3m − 10
(6m + 5)(3m − 2)
155. 21x2 + 43x + 20
(7x + 5)(3x + 4)
159. 20n2 − 33n − 27
ALG catalog ver. 2.6 – page 182 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
108. 7w2 + 76w + 60
(7w + 6)(w + 10)
115. 8x2 + 17x + 9
(5n + 3)(4n − 9)
92. 24k 2 + k − 3
(3k − 1)(8k + 3)
95. 45b2 + b − 2
(7a + 12)(a − 2)
97. 7m2 − 11m − 30
(3y − 14)(y + 2)
(11x − 12)(x + 2)
94. 7a2 − 2a − 24
(3r − 1)(8r + 5)
88. 3y 2 − 8y − 28
91. 11x2 + 10x − 24
(5a − 1)(4a − 7)
84. 5c2 − 42c + 16
(5c − 2)(c − 8)
(7k − 2)(4k + 1)
90. 20a2 − 39a + 7
(3c − 2)(c − 10)
83. 16w2 − 32w + 7
87. 28k 2 − k − 2
(9n − 7)(2n + 1)
80. 3x2 − 20x + 33
(3x − 11)(x − 3)
(4w − 7)(4w − 1)
(6a + 13)(2a + 1)
85. 5x2 − 9x − 18
109. 10r2 + 3r + 2
(11r − 1)(3r − 2)
82. 12a2 + 32a + 13
(5y + 3)(y + 4)
79. 33r2 − 25r + 2
112. 11x2 − 14x − 6
prime
116. 9m2 + 21m + 10
(3m + 2)(3m + 5)
120. 15y 2 + 7y − 4
(5y + 4)(3y − 1)
124. 10d2 − 29d + 21
(2d − 3)(5d − 7)
128. 6u2 − 5u − 21
(3u − 7)(2u + 3)
132. 9y 2 + 52y − 77
(9y − 11)(y + 7)
136. 21x2 − 26x − 15
(7x + 3)(3x − 5)
140. 10m2 + 49m + 49
(5m + 7)(2m + 7)
144. 6y 2 + 29y + 20
(6y + 5)(y + 4)
148. 18a2 − 25a + 8
(9a − 8)(2a − 1)
152. 10n2 + 41n − 18
(5n − 2)(2n + 9)
156. 20y 2 + 53y + 35
(5y + 7)(4y + 5)
160. 36k 2 − 9k − 10
(12k + 5)(3k − 2)
IH
161. 28z 2 + 71z + 18
162. 28k 2 + 51k + 20
(7z + 2)(4z + 9)
165.
84d2
+ d − 15
166.
174. 11 + 23a + 2a2
178. 3 + 13y + 4y 2
185. 3 + 28y
186.
190. 18 + 5c − 2c2
(7 + 10x)(1 − 2x)
193. 24 + 26r
+ 2r2
194. 3 − 23y
201. 4 − 23y
(4 − 15y)(1 − 2y)
205. 4x2 − 3xy − y 2
179. 7 − 16n + 4n2
(7 − 2n)(1 − 2n)
183. 8 + 15x − 2x2
(8 − x)(1 + 2x)
187. 21 − 17k
+ 30y 2
198. 7 − 17b + 10b2
202.
24 + 10a − 21a2
(4 − 3a)(6 + 7a)
206. 3a2 − 16ab + 5b2
− 11p − 5p2
(9 − 5p)(4 + p)
(6 + 5m)(1 − 3m)
60 + 37x + 5x2
(12 + 5x)(5 + x)
207. 2c2 + 9cd − 11d2
(4x + y)(x − y)
(a − 5b)(3a − b)
(2c + 11d)(c − 1)
209. 8n2 + 25np + 3p2
210. 7x2 − 16xy − 15y 2
211. 12r2 − 32rw + 5w2
(8n + p)(n + 3p)
213. 11w2 + 10wx − 24x2
(11w − 12x)(w + 2x)
217. 16p2 − 26pr + 9r2
(8p − 9r)(2p − r)
221. 6k 2 + km − 40m2
(3k + 8m)(2k − 5m)
225. 5x4 − 9x2 − 2
(5x2
+ 1)(x2
− 2)
229. 6 + 5w3 − 6w6
(3 − 2w3 )(2 + 3w3 )
233. 18a8 + 3a4 − 10
(6a4
+ 5)(3a4
− 2)
237. 6p2 r2 − pr − 2
(7x + 5y)(x − 3y)
214. 30c2 + 31cd + 7d2
(10c + 7d)(3c + d)
218. 8x2 + 27xy − 20y 2
(8x − 5y)(x + 4y)
222. 18n2 − 35nx + 12x2
(5 − 4c2 )(3 − 2c2 )
238. 3 + 8ab + 4a2 b2
(3pr − 2)(2pr + 1)
(3 + 2ab)(1 + 2ab)
241. 7a2 b2 + 22abc + 3c2
242. 8w2 − 2wxy − x2 y 2
(7ab + c)(ab + 3c)
245. 4a6 + 23a3 xy 2 + 15x2 y 4
247. 6w8 x2 + 17w4 xy 3 − 14y 6
(4w + xy)(2w − xy)
(4a3 + 3xy 2 )(a3 + 5xy 2 )
(3w4 x − 2y 3 )(2w4 x + 7y 3 )
(5 + r)(1 − 7r)
180. 21 + 16c + 3c2
(3 + c)(7 + 3c)
184. 3 − 26a − 9a2
188. 3 + 2x − 33x2
192. 5 − 32p + 12p2
196. 7 + 6x − 40x2
200. 10 + 39q + 14q 2
(2 + 7q)(5 + 2q)
204. 20 − 11w − 42w2
(4 − 7w)(5 + 6w)
208. 7k 2 + 24km + 17m2
(7k + 17m)(k + m)
212. 28a2 + 40ab − 3b2
216. 40y 2 − 46yz + 13z 2
(2m − 15n)(m + 3n)
219. 21a2 − 19ab − 12b2
(7a + 3b)(3a − 4b)
223. 36a2 + 45ac + 14c2
227. 8m6 + 2m3 − 3
234. 15 − 22c2 + 8c4
176. 5 − 34r − 7r2
215. 2m2 − 9mn − 45n2
226. 7 − 10y 3 + 3y 6
(4p5 + 3)(p5 − 5)
(13 + 5x)(1 − x)
(14a − b)(2a + 3b)
(12a + 7c)(3a + 2c)
230. 4p10 − 17p5 − 15
172. 13 + 8x − 5x2
(6r − w)(2r − 5w)
(9n − 4x)(2n − 3x)
(7 − 3y 3 )(1 − y 3 )
(12r − 5)(8r + 9)
(7 + 20x)(1 − 2x)
199. 6 − 13m − 15m2
203.
168. 96r2 + 68r − 45
(5 − 2p)(1 − 6p)
(7 + a)(4 + 3a)
(7 − 10b)(1 − b)
(2x − 7)(9x − 10)
(3 + 11x)(1 − 3x)
191. 28 + 25a + 3a2
195. 36
164. 18x2 − 83x + 70
(3 + a)(1 − 9a)
+ 2k 2
(7 − k)(3 − 2k)
(3 − 5y)(1 − 6y)
(7 − 3x)(3 + 5x)
+ 30y 2
(13 − p)(1 + 3p)
(9 − 2c)(2 + c)
(12 + r)(2 + 2r)
197. 21 + 26x − 15x2
175. 13 + 38p − 3p2
(5 − 2a)(5 + a)
(3 + 7y)(1 + 7y)
189. 7 − 4x − 20x2
(2 − 11y)(1 − y)
(2 − 3m)(1 + 5m)
25 − 5a − 2a2
− 33x − 108
171. 2 − 13y + 11y 2
(3 + y)(1 + 4y)
182. 2 + 7m − 15m2
56x2
(8x + 9)(7x − 12)
(11 + a)(1 + 2a)
(2 + c)(5 − 11c)
+ 49y 2
167.
(3 − 7a)(1 + a)
(6 − k)(1 − 2k)
181. 10 − 17c − 11c2
− y − 21
170. 3 − 4a − 7a2
(7 − x)(1 − 3x)
177. 6 − 13k + 2k 2
60y 2
(12y + 7)(5y − 3)
(5 + 2m)(1 + m)
173. 7 − 22x + 3x2
(3r − 10)(7r − 3)
(7k + 4)(4k + 5)
(12d − 5)(7d + 3)
169. 5 + 7m + 2m2
163. 21r2 − 79r + 30
(4m3
+ 3)(2m3
− 1)
231. 3 − 32t4 + 20t8
(3 − 2t4 )(1 − 10t4 )
235. 35x8 + 64x4 + 20
(7x4
+ 10)(5x4
+ 2)
239. 10w2 y 4 − 19wy 2 + 7
(5wy 2 − 7)(2wy 2 − 1)
243. 2k 2 m2 + kmr − 15r2
(2km − 5r)(km + 3r)
246. 12c10 − 17c5 d3 y − 5d6 y 2
248. 21a2 b4 − 37ab2 d + 12d2
ALG catalog ver. 2.6 – page 183 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(20y − 13z)(2y − z)
220. 15b2 + 51bc + 18c2
(5b + 2c)(3b + 9c)
224. 20b2 − 9by − 20y 2
(5b + 4y)(4b − 5y)
228. 6 − 7k 2 + 2k 4
(2 − k 2 )(3 − 2k 2 )
232. 25d6 + 25d3 + 6
(5d3 + 3)(5d3 + 2)
236. 24 − p5 − 10p10
(8 + 5p5 )(3 − 2p5 )
240. 2 + 9c3 d − 18c6 d2
(2 − 3c3 d)(1 + 6c3 d)
244. 3r2 − 25pqr + 8p2 q 2
(3r − pq)(r − 8pq)
(4c5 + d3 y)(3c5 − 5d3 y)
(7ab2 − 3d)(3ab2 − 4d)
II
Topic:
Mixed practice and review (factoring). See also categories ID–IH.
Directions:
11—Factor. 12—Factor, if possible. 13—Factor completely.
14—Factor, then check answer by multiplying.
1.
x2 + 3x − 10
2.
(x + 5)(x − 2)
5.
9.
p2 − 6p + 8
(p − 4)(p − 2)
15 − 16y + y 2
6.
(21 − b)(1 + b)
t2 + 2t − 24
10. c2 − 3c − 28
(t + 6)(t − 4)
25. r2 + 20r + 30
prime
29. 42 − 17w + w2
26. x2 − x − 24
37. 50 + 27x + x2
prime
41. a2 + 18a − 63
(12 − h)(9 + h)
(p − 28)(p − 2)
69. 105 + 38k + k 2
(35 + k)(3 + k)
73. y 2 − 28y + 132
(y − 22)(y − 6)
77. 112 + 54w − w2
66. n2 − 15n + 73
70. 108 − 21h + h2
(h − 12)(h − 9)
74. n2 + 23n + 112
(n + 16)(n + 7)
78. 130 − 63y − y 2
(65 + y)(2 − y)
48. x2 − 9x − 90
(d + 14)(d − 6)
(x − 15)(x + 6)
51. 92 + 27y + y 2
52. 84 − 25w + w2
(21 − w)(4 − w)
(23 + y)(4 + y)
55. k 2 − 8k − 105
56. x2 + 9x − 112
(k − 15)(k + 7)
(x + 16)(x − 7)
59. 84 + 17x − x2
60. 56 − 10d − d2
(21 − x)(4 + x)
(14 + d)(4 − d)
63. a2 − 21a + 90
(14 − x)(6 − x)
prime
(23 − k)(3 + k)
47. d2 + 8d − 84
62. 84 − 20x + x2
(12 + p)(7 + p)
44. 69 + 20k − k 2
(14 + x)(3 − x)
(y + 13)(y − 4)
61. 84 + 19p + p2
(c + 18)(c + 2)
43. 42 − 11x − x2
58. y 2 + 9y − 52
(c − 17)(c + 4)
36. 130 − 3a − a2
40. c2 + 20c + 36
(14 + n)(8 − n)
57. c2 − 13c − 68
(p − 17)(p − 4)
39. p2 − 30p + 56
54. 112 − 6n − n2
64. y 2 + 22y + 57
(a − 15)(a − 6)
prime
67. 85 − 20c − c2
71. x2 + 24x + 108
(x + 18)(x + 6)
75. 132 + 23x + x2
(12 + x)(11 + x)
79. x2 − 52x − 108
(x − 54)(x + 2)
ALG catalog ver. 2.6 – page 184 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
prime
32. p2 − 21p + 68
38. 52 − 28k + k 2
(a + 22)(a + 3)
53. 108 + 3h − h2
28. 36 − 18p + p2
(13 + a)(10 − a)
50. a2 + 25a + 66
(k − 24)(k − 3)
prime
(14 − w)(10 + w)
(12 − m)(7 + m)
49. k 2 − 27k + 72
27. 18 + 8m − m2
(y + 21)(y − 2)
(m + 11)(m − 10)
46. 84 + 5m − m2
(13 + w)(6 − w)
24. y 2 + 19y − 42
35. 140 + 4w − w2
(y − 18)(y + 3)
45. 78 − 7w − w2
(7 + a)(6 − a)
(k + 13)(k + 4)
42. y 2 − 15y − 54
(a + 21)(a − 3)
20. 42 − a − a2
31. k 2 + 17k + 52
(26 − k)(2 − k)
(25 + x)(2 + x)
(k + 9)(k + 7)
(a − 34)(a + 2)
34. m2 + m − 110
(y − 10)(y + 9)
16. k 2 + 16k + 63
23. a2 − 32a − 68
(21 + c)(3 + c)
33. y 2 − y − 90
(10 + x)(5 − x)
(9 − p)(p + 7)
30. 63 + 24c + c2
(14 − w)(3 − w)
12. 50 − 5x − x2
19. 63 + 2p − p2
(27 + d)(2 − d)
a2 + 15a − 16
(a + 16)(a − 1)
(a − 7)(a − 4)
22. 54 − 25d − d2
(18 − x)(2 + x)
8.
15. a2 − 11a + 28
(y + 8)(y − 7)
21. 36 + 16x − x2
x2 + 21x + 20
(6 − r)(5 + r)
18. y 2 + y − 56
(c − 9)(c + 6)
15 + 8k + k 2
(5 + k)(3 + k)
11. 30 + r − r2
(10 − w)(5 − w)
17. c2 − 3c − 54
4.
(x + 20)(x + 1)
14. 50 − 15w + w2
(10 + u)(7 + u)
(56 − w)(2 + w)
7.
(c − 7)(c + 4)
13. 70 + 17u + u2
12 − 4m − m2
(6 + m)(2 − m)
21 + 20b − b2
(15 − y)(1 − y)
65. y 2 + 6y − 60
3.
(y + 19)(y + 3)
prime
68. 90 + 26x + x2
72. w2 − 22w + 105
(w − 15)(w − 7)
76. 112 − 22c + c2
(14 − c)(8 − c)
80. g 2 + 64g − 132
(g + 66)(g − 2)
prime
II
81. d2 − 27d + 110
82. x2 + 32x + 112
(d − 22)(d − 5)
86. 105 + 16a − a2
(26 + x)(5 − x)
(8n − 5)(n − 1)
117. 30 + 19x − 11x2
(30 − 11x)(1 + x)
133. 8r2 + 23r + 14
(8r + 7)(r + 2)
137. 15 + 2u − 24u2
(3 + 4u)(5 − 6u)
141. 20k 2 − 17k − 24
(5k − 8)(4k + 3)
145. 42 − 37x + 8x2
(21 − 8x)(2 − x)
149. x6 − 15x3 + 36
(x3 − 3)(x3 − 12)
153. 33 + 8a6 − a12
(11 − a6 )(3 + a6 )
157. m4 − 21m2 − 100
+ 4)(m2
130. 2w2 − 5w − 10
134. 6d2 − 7d − 10
(6d + 5)(d − 2)
138. 40 − 18x − 9x2
(4 − 3x)(10 + 3x)
142. 12y 2 + 49y + 30
(3y + 10)(4y + 3)
146. 8 + 25y − 33y 2
(8 + 33y)(1 − y)
150. y 10 + y 5 − 30
(y 5 − 5)(y 5 + 6)
154. 85 − 22p2 + p4
(17 − p2 )(5 − p2 )
158. a6 + 36a3 + 180
+ 30)(a3
(7 + 11k)(1 − 2k)
120. 20y 2 − 16y + 3
(10y − 3)(2y − 1)
124. 12 + 35x − 13x2
(4 + 13x)(3 − x)
127. 10y 2 + 11y − 6
128. 6a2 + 19a + 10
(5y − 2)(2y + 3)
prime
131. 14 − 15a + 3a2
135. 35 − 47w + 6w2
(7 + w)(5 + 6w)
139. 9x2 − 45x + 50
(3x − 5)(3x − 10)
143. 4 + 53a − 42a2
(4 − 3a)(1 + 14a)
147. 26u2 − 11u − 15
(26u + 15)(u − 1)
151. 36 − 5c4 − c8
(9 + c4 )(4 − c4 )
155. w6 + 16w3 + 55
(w3
+ 5)(w3
+ 11)
159. 150 − 35k 7 + k 14
+ 6)
(15 − k 7 )(10 − k 7 )
161. 4y 4 + 3y 2 − 1
162. 6c12 − 7c6 + 2
163. 1 − 9k 4 + 14k 8
(4y 2 − 1)(y 2 + 1)
(3c6 − 2)(2c6 − 1)
(1 − 7k 4 )(1 − 2k 4 )
− 25)
(a3
(5 + 3a)(2 − a)
(3 + 4n)(1 + 6n)
(9 − 4r)(1 − 2r)
prime
116. 7 − 3k − 22k 2
123. 3 + 22n + 24n2
126. 9 − 22r + 8r2
(4 + 3p)(1 − 2p)
115. 10 + a − 3a2
(2p + 3)(p − 6)
(2h + 5)(h − 8)
125. 4 − 5p − 6p2
(10r − 1)(r − 1)
119. 2p2 − 9p − 18
122. 2h2 − 11h − 40
(12a − 7)(3a + 1)
(m2
118. 5 + 18m + 16m2
(5 + 8m)(1 + 2m)
121. 36a2 − 33a + 7
112. 10r2 − 11r + 1
(6x + 1)(x − 1)
114. 8n2 − 14n + 5
(2c + 7)(c + 3)
(5 + m)(1 + 7m)
111. 6x2 − 5x + 1
(1 + 3y)(1 + 5y)
113. 2c2 + 13c + 21
108. 5 + 36m + 7m2
(2 − a)(1 + 11a)
110. 1 + 8y + 15y 2
(1 + 4k)(1 − 2k)
(x + 29)(x − 4)
107. 2 + 21a − 11a2
(7p + 2)(p − 1)
109. 1 + 2k − 8k 2
104. x2 + 25x − 116
(g − 28)(g + 5)
106. 7p2 − 5p − 2
(3y − 5)(y − 1)
(14 − a)(10 − a)
103. g 2 − 23g − 140
(20 + m)(7 − m)
105. 3y 2 − 8y + 5
100. 140 − 24a + a2
(11 + x)(10 + x)
102. 140 − 13m − m2
(44 − z)(3 + z)
(x − 30)(x + 4)
99. 110 + 21x + x2
(k + 56)(k + 2)
101. 132 + 41z − z 2
96. x2 − 26x − 120
(y + 28)(y − 4)
98. k 2 + 58k + 112
(n − 72)(n − 2)
(n + 33)(n + 4)
95. y 2 + 24y − 112
(27 − c)(4 + c)
97. n2 − 74n + 144
92. 132 + 37n + n2
(44 − k)(3 − k)
94. 108 + 23c − c2
(35 + k)(4 − k)
(y − 18)(y + 6)
91. 132 − 47k + k 2
(x − 24)(x − 5)
93. 140 − 31k − k 2
88. y 2 − 12y − 108
(w + 22)(w − 6)
90. x2 − 29x + 120
(g + 26)(g + 5)
(20 − a)(7 − a)
87. w2 + 16w − 132
(21 − a)(5 + a)
89. g 2 + 31g + 130
84. 140 − 27a + a2
(25 + y)(4 + y)
(x + 28)(x + 4)
85. 130 − 21x − x2
129. 4y 2 + 28y + 6
83. 100 + 29y + y 2
ALG catalog ver. 2.6 – page 185 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(2a + 5)(3a + 2)
prime
132. 5 + x − 12x2
prime
136. 21 + 2k − 8k 2
(7 − 4k)(3 + 2k)
140. 25c2 + 25c + 4
(5c + 4)(5c + 1)
144. 6 − 43n + 20n2
(3 − 20n)(2 − n)
148. 16w2 + 43w + 22
(16w + 11)(w + 2)
152. 24 + 11d2 + d4
(8 + d2 )(3 + d2 )
156. r20 − 8r10 − 65
(r10 − 13)(r10 + 5)
160. 250 + 15n4 − n8
(25 − n4 )(10 + n4 )
164. 15x6 − 8x3 + y 2
(3x3 − 1)(5x3 − 1)
II
165. 21 − n5 − 2n10
166. 8 + 2a2 − 15a4
(7 + 2n5 )(3 − n5 )
169. c2 + 15cw + 14w2
(4 − 5a2 )(2 + 3a2 )
170. n2 − 7nr − 8r2
(n + r)(n − 8r)
(c + w)(c + 14w)
173. x2 y 2 − 26xy + 48
174. a2 b2 + 23ab + 60
(xy − 2)(xy − 24)
177. b2 + 21bx + 80x2
(ab + 3)(ab + 20)
178. c2 − 33cd − 70d2
(b + 16x)(b + 5x)
(c − 35d)(c + 2d)
181. 48 − 2m2 p − m4 p2
182. 90 + 27wx2 − w2 x4
(8 + m2 p)(6 − m2 p)
185. c2 + 12cw3 + 35w6
(30 − wx2 )(3 + wx2 )
186. k 8 − 13k 4 y 2 + 30y 4
(c + 5w3 )(c + 7w3 )
189. 6b2 − 5bc − 6c2
(k4
171. a2 t2 + 15at − 16
172. h2 k 2 − 7hk + 6
(at − 1)(at + 16)
175. c2 − 3cd − 40d2
(hk − 1)(hk − 6)
176. k 2 + 11km − 60m2
(c + 5d)(c − 8d)
179. 76 + 23pr + p2 r2
(k − 4m)(k + 15m)
180. 88 − 3ay − a2 y 2
(11 + ay)(8 − ay)
(19 + pr)(4 + pr)
183. d4 − 13d2 k + 40k 2
184. r2 + 17rs2 + 72s4
(d2 − 8k)(d2 − 5k)
187. a4 b6 + 7a2 b3 − 18
(a2 b3
− 2)(a2 b3
(r + 9s2 )(r + 8s2 )
188. k 6 t2 − 12k 3 t + 27
(k 3 t − 3)(k 3 t − 9)
+ 9)
(5x − 3y)(4x + 5y)
194. 2r4 − 11r2 p + 15p2
195. 14k 4 + 23k 2 x2 + 8x4
196. 10h6 − 3h3 n − 18n2
(2r2
− 5p)(r2
− 3p)
198. r2 − 2rp2 w − 15p4 w2
202. 5r2 + 19rpt2 + 18p2 t4
+ 2x)
(5r
+ 9pt2 )(r
+ 2pt2 )
206. (y + 2)2 + 8(y + 2) − 20
(x − 6)(x − 2)
209. 3(r − 2)2 + 7(r − 2) + 2
(4x4 + 5)(3x4 + 1)
+ 7)
(7m − n)(3m + 4n)
201. 15a4 y 2 + a2 xy − 2x2
205. (x − 1)2 − 6(x − 1) + 5
168. 12x8 + 19x4 + 5
(3a − 5b)(3a − b)
(r + 3p2 w)(r − 5p2 w)
− x)(5a2 y
+ 2)(6y 5
192. 20x2 + 13xy − 15y 2
(b2 c − 3d)(b2 c − 4d)
(3a2 y
(y 5
191. 21m2 + 25mn − 4n2
(7c + 3w3 )(5c − w3 )
197. b4 c2 − 7b2 cd + 12d2
− 3y 2 )
190. 9a2 − 18ab + 5b2
(3b + 2c)(2b − 3c)
193. 35c2 + 8cw3 − 3w6
− 10y 2 )(k 4
167. 6y 10 + 19y 5 + 14
y(y + 12)
210. 2(x + 1)2 − (x + 1) − 6
r(3r − 5)
(2x + 5)(x − 1)
213. 18(h − k)2 − 11m(h − k) + m2
215. 3(x − y)2 + 7z(x − y) − 6z 2
(9h − 9k − m)(2h − 2k − m)
(3x − 3y − 2z)(x − y + 3z)
(2k 2
+ x2 )(7k 2
+ 8x2 )
199. h8 + 4h4 kn2 − 21k 2 n4
(5h3 + 6n)(2h2 − 3n)
200. u2 x6 + 11ux3 y 2 + 18y 4
(h4 + 7kn2 )(h4 − 3kn2 )
(ux3 + 2y 2 )(ux3 + 9y 2 )
203. 12b2 w2 − 13bcw − 14c2
204. 21a2 − 19adn − 12d2 n2
(3bw + 2c)(4bw − 7c)
(7a + 3dn)(3a − 4dn)
207. (a − 3)2 + 10(a − 3) + 9
208. (w + 4)2 − 5(w + 4) − 24
(a + 6)(a − 2)
211. 4(c + 2)2 + 5(c + 2) − 6
(w + 7)(w − 4)
212. 6(k − 1)2 − 11(k − 1) + 3
(4c + 5)(c + 4)
214. 30a2 − 7a(b + c) − (b + c)2
216. 2d2 + 9d(h + p) + 10(h + p)2
ALG catalog ver. 2.6 – page 186 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(3k − 4)(2k − 5)
(10a + b + c)(3a − b − c)
(2d + 5h + 5p)(d + 2h + 2p)
IJ
Topic:
Factoring differences of squares.
Directions:
11—Factor. 12—Factor, if possible. 13—Factor completely.
14—Factor, then check answer by multiplying.
1.
x2 − 1
(x − 1)(x + 1)
2.
a2 − 9
5.
c2 − 16
(c − 4)(c + 4)
6.
k 2 − 49
9.
x2 − 36
(x − 6)(x + 6)
10. d2 − 64
13. m2 − 121
3.
m2 − 4
(m − 2)(m + 2)
4.
r2 − 25
(r − 5)(r + 5)
(k − 7)(k + 7)
7.
y 2 − 81
(y − 9)(y + 9)
8.
a2 − 64
(a − 8)(a + 8)
(d − 8)(d + 8)
11. p2 − 100
(a − 3)(a + 3)
14. y 2 − 169
(y − 13)(y + 13)
(m − 11)(m + 11)
17. r2 − 256
(r − 16)(r + 16)
(p − 10)(p + 10)
15. x2 − 225
12. a2 − 144
(a − 12)(a + 12)
16. a2 − 196
(a − 14)(a + 14)
20. y 2 − 900
(y − 30)(y + 30)
(x − 15)(x + 15)
18. w2 − 400
19. k 2 − 625
(k − 25)(k + 25)
(w − 20)(w + 20)
21. 9 − a2
(3 + a)(3 − a)
25. 49 − m2
(7 + m)(7 − m)
29. 169 − w2
22. 1 − x2
(1 + x)(1 − x)
26. 16 − r2
(4 + r)(4 − r)
30. 121 − a2
(11 + a)(11 − a)
(13 + w)(13 − w)
33. 289 − y 2
(17 + y)(17 − y)
23. 25 − c2
(5 + c)(5 − c)
24. 4 − a2
27. 64 − y 2
(8 + y)(8 − y)
28. 81 − d2
31. 144 − x2
(2 + a)(2 − a)
(9 + d)(9 − d)
32. 100 − c2
(10 + c)(10 − c)
36. 225 − a2
(15 + a)(15 − a)
40. 625 − y 2
(25 + y)(25 − y)
(12 + x)(12 − x)
34. 324 − r2
(18 + r)(18 − r)
35. 196 − n2
(14 + n)(14 − n)
37. 400 − k 2
(20 + k)(20 − k)
38. 256 − m2
39. 900 − x2
(16 + m)(16 − m)
41. −25 + w2
(w − 5)(w + 5)
42. −9 + y 2
(y − 3)(y + 3)
(30 + x)(30 − x)
43. −a2 + d2
(d − a)(d + a)
44. −m2 + r2
(r − m)(r + m)
45. 64 + h2
prime
49. 25p2 − 81
46. a2 + 100
prime
47. x2 + y 2
50. 9r2 − 16
(3r − 4)(3r + 4)
51. 4a2 − 49
prime
(2a − 7)(2a + 7)
(5p − 9)(5p + 9)
53. 16 − 49x2
57. 121 − 196m2
(11 − 14m)(11 + 14m)
61. 144y 2 − 169
(12y − 13)(12y + 13)
65. −144 + 49k 2
(7k − 12)(7k + 12)
69. x2 − y 2
(x − y)(x + y)
55. 81 − 4d2
(9 − 2d)(9 + 2d)
(5c − b)(5c + b)
58. 144 − 49k 2
(12 − 7k)(12 + 7k)
62. 400k 2 − 121
(20k − 11)(20k + 11)
66. −4p2 + 25q 2
(5q − 2p)(5q + 2p)
70. a2 − 9b2
(a − 3b)(a + 3b)
74. 64x2 − y 2
(8x − y)(8x + y)
77. 121c2 − d2
(11c − d)(11c + d)
52. 81c2 − 25
56. 9 − 64r2
(3 − 8r)(3 + 8r)
(8 − 5w)(8 + 5w)
59. 100 − 169a2
(10 − 13a)(10 + 13a)
63. 196c2 − 225
(14c − 15)(14c + 15)
67. −n4 + 36r2
(6r
− n2 )(6r
+ n2 )
71. c2 − 36d2
(c − 6d)(c + 6d)
73. 25c2 − b2
prime
(9c − 5)(9c + 5)
54. 64 − 25w2
(4 − 7x)(4 + 7x)
48. a2 + b2
78. 225x2 − y 2
(15x − y)(15x + y)
75. 4m2 − n2
(2m − n)(2m + n)
79. 169a2 − b2
(13a − b)(13a + b)
ALG catalog ver. 2.6 – page 187 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
60. 64 − 225x2
(8 − 15x)(8 + 15x)
64. 625a2 − 256
(25a − 16)(25a + 16)
68. −81t2 + 169w2
(13w − 9t)(13w + 9t)
72. m2 − 81n2
(m − 9n)(m + 9n)
76. 16p2 − r2
(4p − r)(4p + r)
80. 100k 2 − m2
(10k − m)(10k + m)
IJ
81. x2 − 25w2
82. y 2 − 49z 2
(x − 5w)(x + 5w)
83. c2 − 100d2
(y − 7z)(y + 7z)
85. 64x2 − 9y 2
(c − 10d)(c + 10d)
86. 25a2 − 81b2
(8x − 3y)(8x + 3y)
87. 4r2 − 49p2
(5a − 9b)(5a + 9b)
89. 36y 2 − 25w2
97. 36 − a2 b2
95. 400m2 − 121p2
(7r − 15n)(7r + 15n)
98. 64 − x2 y 2
(6 − ab)(6 + ab)
101. 9a2 + b2
99. m2 n2 − 25
102. c2 + 100d2
105. w2 x2 − y 2
106. a2 − b2 c2
prime
(a − bc)(a + bc)
110. 1 − 169x2
(11r − 1)(11r + 1)
107. a2 b2 − c2 d2
(x2 − 7)(x2 + 7)
125. 1 − 25b8
122. 25 − a4
119. 49c2 d2 − 64k 2
(5 − a2 )(5 + a2 )
126. 4w6 − 1
(2w3
129. 81 − 25m20
(3k 7
+ 1)
130. 121a30 − 64
(9 − 5m10 )(9 + 5m10 )
133. 144k 2 − 225p6
(11a15
− 8)(11a15
(b − 12 )(b + 12 )
( dc − 6)( dc + 6)
1
− x2
9
142. 100 −
+ 8)
( 31 − x)( 13 + x)
r2
s2
y
x
10 )( 2
146.
+
− 0.7z 2 )(0.5w
( a8 − 5b )( a8 + 5b )
139. 1 −
a2
25
(1 − a5 )(1 + a5 )
9
25
147.
1 2
9
w − y2
4
16
151. a6 − 6.25
(a3
(0.2 − y)(0.2 + y)
154. 1.69a8 − 0.81b2
+ 0.7z 2 )
(1.3a4
− 0.9b)(1.3a4
124. c8 − 36
(c4 − 6)(c4 + 6)
(1 − 7p5 )(1 + 7p5 )
(d2 h3 − 4)(d2 h3 + 4)
136. 256 − 121a4 c8
(16 − 11a2 c4 )(16 + 11a2 c4 )
140.
w2
−1
16
144.
4
− h2
49
148.
9 2
1
b − d2
49
81
( w4 − 1)( w4 + 1)
( 72 − h)( 72 + h)
(m − 35 )(m + 35 )
150. 0.04 − y 2
153. 0.25w2 − 0.49z 4
(15ab − 4b)(15ab + 4b)
132. d4 h6 − 16
( 21 w − 34 y)( 12 w + 34 y)
(x − 0.3)(x + 0.3)
120. 225a2 b2 − 16c2
(30t3 − 13w2 )(30t3 + 13w2 )
143. m2 −
y
10 )
149. x2 − 0.09
(0.5w
a2
b2
−
64 25
(8wx − 5y)(8wx + 5y)
+ 1)
135. 900t6 − 169w4
(10 − rs )(10 + rs )
x2
y2
−
4
100
116. 64w2 x2 − 25y 2
(10 − x4 y)(10 + x4 y)
(8r4 s − 20)(8r4 s + 20)
138.
(1 − 9kp)(1 + 9kp)
128. 1 − 49p10
− 1)(3k7
131. 100 − x8 y 2
134. 64r8 s2 − 400
(12k − 15p3 )(12k + 15p3 )
(3 − y 3 )(3 + y 3 )
127. 9k 14 − 1
− 1)(2w3
112. 1 − 81k 2 p2
(7cd − 8k)(7cd + 8k)
123. 9 − y 6
prime
(mnp − 1)(mnp + 1)
(10a − bc)(10a + bc)
(6m − 7rp)(6m + 7rp)
(1 − 5b4 )(1 + 5b4 )
( x2 −
104. 25 + a2 b2
108. m2 n2 p2 − 1
115. 100a2 − b2 c2
118. 36m2 − 49r2 p2
(9k − 12mn)(9k + 12mn)
145.
(ab − 7)(ab + 7)
(6wx − 1)(6wx + 1)
9(3 − 2cd)(3 + 2cd)
117. 81k 2 − 144m2 n2
c2
− 36
d2
prime
111. 36w2 x2 − 1
114. 81 − 36c2 d2
(4wx − 7)(4wx + 7)
141.
(13b − 30c)(13b + 30c)
100. a2 b2 − 49
103. 64x2 + y 2
(1 − 13x)(1 + 13x)
113. 16w2 x2 − 49
1
4
96. 169b2 − 900c2
(ab − cd)(ab + cd)
109. 121r2 − 1
137. b2 −
(12c − 13d)(12c + 13d)
(mn − 5)(mn + 5)
(wx − y)(wx + y)
121. x4 − 49
92. 144c2 − 169d2
(20m − 11p)(20m + 11p)
(8 − xy)(8 + xy)
prime
(3m − 10k)(3m + 10k)
(9x − 12y)(9x + 12y)
94. 49r2 − 225n2
(10a − 9b)(10a + 9b)
88. 9m2 − 100k 2
91. 81x2 − 144y 2
(4b − 11c)(4b + 11c)
93. 100a2 − 81b2
(m − 12n)(m + 12n)
(2r − 7p)(2r + 7p)
90. 16b2 − 121c2
(6y − 5w)(6y + 5w)
84. m2 − 144n2
− 2.5)(a3
152. 2.25 − c4
+ 2.5)
155. 0.64 − d6 k 2
+ 0.9b)
(0.8 − d3 k)(0.8 + d3 k)
ALG catalog ver. 2.6 – page 188 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
( 73 b − 19 d)( 73 b + 19 d)
(1.5 − c2 )(1.5 + c2 )
156. m2 p4 − 1.21
(mp2 − 1.1)(mp2 + 1.1)
IJ
157. a4 − 1
(a2
158. 1 − x4
+ 1)(a − 1)(a + 1)
161. 81 − y 4
162. u4 − 81
(9 + y 2 )(3 − y)(3 + y)
165. a4 − b4
(a2
+ b2 )(a + b)(a − b)
169. y 8 − 1
(y 4
+ 1)(y 2
(1 + x2 )(1 + x)(1 − x)
(u2 + 9)(u + 3)(u − 3)
173. (a − 5)2 − 36
(a + 1)(a − 11)
177. (x + y)2 − z 2
(x + y + z)(x + y − z)
181. 4n4 − (n − 1)4
(n2 + 2n − 1)(3n2 − 2n + 1)
(y 2
160. 16 − w4
+ 4)(y + 2)(y − 2)
163. p4 − 625
(p2 + 25)(p + 5)(p − 5)
167. 81m4 − k 4
(x2
(9m2
+ 4y 2 )(x + 2y)(x − 2y)
(16 + a4 )(4 + a2 )(2 + a)(2 − a)
174. 49 − (x + 4)2
+ 9)(x5
178. 1 − (a + b)2
179. (r − s)2 − t2
(y 2 + 4y + 1)(7y 2 + 4y + 1)
− 3)
175. (c + 1)2 − 121
(c − 10)(c + 12)
182. (2y + 1)4 − 9y 4
(r − s + t)(r − s − t)
183. (a + b)2 − (a − c)2
(b + c)(2a + b − c)
ALG catalog ver. 2.6 – page 189 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(4c2 + 9d2 )(2c + 3d)(2c − 3d)
172. w4 − x8
+ 3)(x5
(3 − x)(11 + x)
(1 + a + b)(1 − a − b)
(25 + m2 )(5 + m)(5 − m)
168. 16c4 − 81d4
+ k 2 )(3m + k)(3m − k)
171. x20 − 81
(x10
(4 + w2 )(2 + w)(2 − w)
164. 625 − m4
166. x4 − 16y 4
170. 256 − a8
+ 1)(y + 1)(y − 1)
159. y 4 − 16
(w2 + x4 )(w + x2 )(w − x2 )
176. 100 − (p − 8)2
(18 − p)(2 + p)
180. 9p2 − (q + 3r)2
(3p + q + 3r)(3p − q − 3r)
184. (k + 4)2 − (k − m)2
(m + 4)(2k − m + 4)
IK
Topic:
Factoring perfect square trinomials (binomial squares).
Directions:
11—Factor. 12—Factor, if possible. 13—Factor completely.
14—Factor, then check answer by multiplying.
1.
x2 + 2x + 1
5.
m2 + 14m + 49
(x + 1)2
2.
y 2 + 4y + 4
6.
k 2 + 10k + 25
(y + 2)2
(k + 5)2
3.
a2 + 8a + 16
7.
x2 + 12x + 36
(a + 4)2
(x + 6)2
4.
p2 + 6p + 9
8.
a2 + 16a + 64
(p + 3)2
(a + 8)2
(m + 7)2
9.
y 2 + 20y + 100
(y
10. c2 + 18c + 81
(a + 13)2
15. w2 + 28w + 196
+ 15)2
+ 14)2
(w
18. p2 + 60p + 900
(x + 16)2
(m − 2)2
25. w2 − 10w + 25
16. c2 + 40c + 400
(c + 20)2
19. a2 + 100a + 2500
(p + 30)2
21. m2 − 4m + 4
(a + 50)2
22. x2 − 2x + 1
(x − 1)2
26. p2 − 14p + 49
(p − 7)2
− 5)2
23. k 2 − 6k + 9
20. y 2 + 80y + 1600
(y + 40)2
(k − 3)2
27. m2 − 16m + 64
24. c2 − 8c + 16
(c − 4)2
28. y 2 − 12y + 36
(y − 6)2
(m − 8)2
29. x2 − 18x + 81
(x − 9)2
30. a2 − 20a + 100
31. c2 − 24c + 144
(a − 10)2
34. w2 − 26w + 169
− 15)2
− 13)2
(w
37. a2 − 60a + 900
(m − 20)2
42. 9 + 6a + a2
40. y 2 − 100y + 2500
(c − 40)2
(3 + a)2
43. 1 + 2y + y 2
36. x2 − 28x + 196
(x − 14)2
39. c2 − 80c + 1600
(p − 16)2
(2 + x)2
(x − 11)2
35. m2 − 40m + 400
38. p2 − 32p + 256
(a − 30)2
32. x2 − 22x + 121
(c − 12)2
33. y 2 − 30y + 225
41. 4 + 4x + x2
12. x2 + 24x + 144
(x + 12)2
14. y 2 + 30y + 225
(y
17. x2 + 32x + 256
(y
11. p2 + 22p + 121
(p + 11)2
13. a2 + 26a + 169
(w
(c + 9)2
+ 10)2
(y − 50)2
(1 + y)2
44. 25 + 10m + m2
(5 + m)2
45. 16 + 8a + a2
(4 + a)2
46. 64 + 16x + x2
(8 + x)2
47. 49 + 14w + w2
48. 81 + 18c + c2
(9 + c)2
(7 + w)2
49. 36 + 12k + k 2
(6 + k)2
50. 144 + 24r + r2
51. 100 + 20a + a2
(12 + r)2
53. 121 + 22w + w2
(11 + w)2
(14 + a)2
61. 9 − 6w + w2
54. 400 + 40c + c2
58. 1600 + 80 + m2
65. 64 − 16m + m2
62. 4 − 4y + y 2
56. 900 + 60y + y 2
(13 + r)2
(30 + y)2
59. 256 + 32x + x2
(40 + m)2
(3 − w)2
(15 + p)2
55. 169 + 26r + r2
(20 + c)2
57. 196 + 28a + a2
52. 225 + 30p + p2
(10 + a)2
60. 2500 + 100p + p2
(16 + x)2
(y − 2)2
66. 16 − 8c + c2
(4 − c)2
(50 + p)2
63. 25 − 10x + x2
(5 − x)2
64. 1 − 2a + a2
67. 81 − 18x + x2
(9 − x)2
68. 49 − 14y + y 2
(1 − a)2
(8 − m)2
69. 144 − 24p + p2
70. 36 − 12n + n2
(12 − p)2
73. 400 − 40m + m2
(20 − m)2
77. 1600 − 80n + n2
(40 − n)2
(6 − n)2
71. 225 − 30a + a2
(15 − a)2
74. 121 − 22y + y 2
(11 − y)2
78. 196 − 28x + x2
(14 − x)2
75. 900 − 60x + x2
(30 − x)2
79. 2500 − 100p + p2
(50 − p)2
ALG catalog ver. 2.6 – page 190 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
72. 100 − 20w + w2
(10 − w)2
76. 169 − 26a + a2
(13 − a)2
80. 256 − 32y + y 2
(16 − y)2
(7 − y)2
IK
81. y 2 + 3y + 9
prime
85. x2 + 2xy + y 2
(x + y)2
82. 49 + 7a + a2
prime
86. a2 − 2ab + b2
(a − b)2
83. x2 − 11x + 121
prime
87. c2 − 4cd + 4d2
(2m − n)2
(2r
93. x2 + 6xy + 9y 2
(w
97. 16q 2 − 8qr + r2
(a + 6b)2
(11r − p)2
− 9k)2
(3c + 2d)2
113. 16x2 − 56xy + 49y 2
(4x − 7y)2
− 3r)2
(7c + 4k)2
117. 25a2 + 60ay + 36y 2
(5a + 6y)2
118. 36x2 − 60xy + 25y 2
(10k − 3m)2
(3c + 10d)2
125. 4r2 + 44rx + 121x2
+ 11x)2
126. 121x2 − 44xy + 4y 2
(12a − 5y)2
(5m + 12k)2
133. 49b2 − 21by + 9y 2
prime
134. 25a2 + 55ax + 121x2
(8a + 7b)2
(3c − 15x)2
138. 49c2 − 112cd + 64d2
145. 81c2 + 180cd + 100d2
(9c + 10d)2
+ 3w)2
149. 121x2 − 88xy + 16y 2
(11x − 4y)2
− 9x)2
119. 4a2 − 28ac + 49c2
150. 16c2 + 88cx + 121x2
123. 49m2 + 84mn + 36n2
124. 36p2 − 84ps + 49s2
(6p − 7s)2
127. 64a2 − 48ab + 9b2
128. 9k 2 + 48kp + 64p2
(3k + 8p)2
131. 49p2 + 140pr + 100r2
132. 100x2 − 140xy + 49y 2
(10x − 7y)2
135. 25x2 − 50xy + 100y 2
139. 169g 2 − 52gk + 4k 2
− 2k)2
136. 9a2 + ab + 81b2
prime
(5a + 7b)2
(1 + x4 )2
159. u6 + 2u3 + 1
157. 1 − 2w3 + w6
(1 − w3 )2
158. 1 + 2x4 + x8
161. a10 + 2a5 + 1
(a5 + 1)2
162. m14 − 2m7 + 1
163. 1 − 2x6 + x12
− 1)2
ALG catalog ver. 2.6 – page 191 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
148. 4r2 + 60rp + 225p2
(2r + 15p)2
151. 25a2 + 70ab + 49b2
155. 1 − 2a5 + a10
144. 400a2 − 120ab + 9b2
(20a − 3b)2
147. 225m2 − 60mn + 4n2
(x2 − 1)2
140. 4m2 + 40mn + 100n2
(2m + 10n)2
143. 9x2 + 120xy + 400y 2
154. x4 − 2x2 + 1
(m7
120. 49d2 + 28dy + 4y 2
(7d + 2y)2
(15m − 2n)2
(4c + 11x)2
(y 2 + 1)2
116. 4w2 − 36wx + 81x2
(2w − 9x)2
(3x + 20y)2
146. 100w2 − 180wx + 81x2
(10w
112. 4a2 + 20ak + 25k 2
(2a + 5k)2
115. 81p2 + 36pr + 4r2
(13g
142. 225r2 + 90rw + 9w2
(15r
111. 25u2 − 20uw + 4w2
prime
(7c − 8d)2
141. 9c2 − 90cx + 225x2
108. 400x2 + 40xy + y 2
(20x + y)2
(7p + 10r)2
prime
137. 64a2 + 112ab + 49b2
107. 169w2 − 26wy + y 2
(8a − 3b)2
130. 25m2 + 120mk + 144k 2
104. m2 − 24mn + 144n2
(m − 12n)2
(7m + 6n)2
(11x − 2y)2
129. 144a2 − 120ay + 25y 2
103. x2 + 20xy + 100y 2
(2a − 7c)2
122. 9c2 + 60cd + 100d2
100. 49x2 + 14xy + y 2
(7x + y)2
(9p + 2r)2
(6x − 5y)2
121. 100k 2 − 60km + 9m2
99. 64c2 − 16cd + d2
(5u − 2w)2
114. 49c2 + 56ck + 16k 2
96. n2 + 8nk + 16k 2
(n + 4k)2
(13w − y)2
110. 4w2 − 12wr + 9r2
(2w
95. a2 − 8ay + 16y 2
(x + 10y)2
106. 225a2 + 30ac + c2
92. 9a2 − 6ac + c2
(3a − c)2
(8c − d)2
(15 + c)2
109. 9c2 + 12cd + 4d2
+ m)2
(a − 4y)2
102. g 2 − 18gk + 81k 2
(g
105. 121r2 − 22rp + p2
153. y 4 + 2y 2 + 1
− 3x)2
(5b + w)2
101. a2 + 12ab + 36b2
(2r
(3k
98. 25b2 + 10bw + w2
− r)2
(x + 2y)2
91. 9k 2 + 6km + m2
+ p)2
94. w2 − 6wx + 9x2
(x + 3y)2
(4q
90. 4r2 + 4rp + p2
prime
88. x2 + 4xy + 4y 2
(c − 2d)2
89. 4m2 − 4mn + n2
84. 100 − 50n + n2
152. 49x2 − 70xw + 25w2
(7x − 5w)2
(1 − a5 )2
(u3 + 1)2
(1 − x6 )2
156. 1 + 2c2 + c4
(1 + c2 )2
160. y 8 − 2y 4 + 1
(y 4 − 1)2
164. 1 + 2c10 + c20
(1 + c10 )2
IK
165. w4 + 10w2 + 25
(w2
166. k 4 − 6k 2 + 9
(k 2 − 3)2
167. x4 − 14x2 + 49
+ 5)2
(x2
169. 4 − 4y 2 + y 4
170. 64 + 16x2 + x4
(2 − y 2 )2
(w4
185. 9x8 + 30x4 + 25
(x2
(wx3
194. a8 + 2a4 b2 c3 + b4 c6
197. 9a4 − 48a2 b2 + 64b4
201. a2 + a +
1
4
(k 3 m5
202. x2 − 5x +
25
4
9
25
206. 9c2 + 8c +
16
9
(a −
4 2 16
p − p + 16
9
3
214.
( 32 p − 4)2
y2
7y
+
+ 49
64
4
(x − 52 )2
203. w2 − 3w +
207.
218.
200. 4w4 + 60w2 y 4 + 225y 8
(2w2 + 15y 4 )2
9
4
204. k 2 + 7k +
(w − 32 )2
1 2
p + 4p + 100
25
3
9
211. y 2 − y +
2
16
208.
49
4
(k + 72 )2
25 2
r − 15r + 36
16
( 45 r − 6)2
2
1
212. w2 + w +
3
9
(y − 34 )2
(w + 31 )2
9 2
10 )
25 2 40
y + y + 64
36
3
215. 9n2 +
x2
5x
−
+ 25
16
2
24
16
n+
5
25
216. 81x2 −
(3n + 45 )2
( 56 y + 8)2
( y8 + 7)2
(w2 x4 − y)2
( 15 p + 10)2
9
81
210. a2 − a +
5
100
(x + 56 )2
196. w4 x8 − 2w2 x4 y + y 2
− n4 )2
199. 25c6 + 60c3 x + 36x2
(3c + 43 )2
5
25
209. x2 + x +
3
36
(x6 − y 4 )2
(5c3 + 6x)2
(a + 12 )2
(5m − 35 )2
192. x12 − 2x6 y 4 + y 8
+ n7 )2
195. k 6 m10 − 2k 3 m5 n4 + n8
(7x − 10y 2 )2
205. 25m2 − 6m +
(10 + 3m3 )2
191. m10 + 2m5 n7 + n14
+ b2 c3 )2
198. 49x2 − 140xy 2 + 100y 4
(3a2 − 8b2 )2
217.
(a3 + b)2
(m5
(a4
(3 − n5 )2
188. 100 + 60m3 + 9m6
(7 − 2a6 )2
190. a6 + 2a3 b + b2
− y 2 )2
184. 9 − 6n5 + n10
(8 + c4 )2
187. 49 − 28a6 + 4a12
− y 3 )2
193. w2 x6 − 2wx3 y 2 + y 4
213.
183. 64 + 16c4 + c8
(2y 3 − 11)2
189. x4 − 2x2 y 3 + y 6
(11 + 5x2 )2
+ 10)2
186. 4y 6 − 44y 3 + 121
(3x4 + 5)2
180. 121 + 110x2 + 25x4
(7 − 4y 2 )2
182. w8 + 20w4 + 100
− 6)2
(3d2 − 12)2
179. 49 − 56y 2 + 16y 4
(2 − 11x2 )2
181. a6 − 12a3 + 36
(6 − c2 )2
176. 9d4 − 72d2 + 144
(5x2 + 7)2
178. 4 − 44x2 + 121x4
(10 + 3k 2 )2
(a3
175. 25x4 + 70x2 + 49
(2z 2 + 9)2
177. 100 + 60k 2 + 9k 4
172. 36 − 12c2 + c4
(9 + a2 )2
174. 4z 4 + 36z 2 + 81
(3w2 − 2)2
(n2 + 10)2
171. 81 + 18a2 + a4
(8 + x2 )2
173. 9w4 − 12w2 + 4
168. n4 + 20n2 + 100
− 7)2
( x4 − 5)2
219.
( 2c
3 −
7 2
12 )
(9x −
4cw
w2
4c2
−
+
9
15
25
220.
w 2
5)
21
49
x+
2
144
b
8br
16r2
+
+
49
21
9
( 7b +
4r 2
3 )
Fractions and decimals
221. d2 − 0.8d + 0.16
222. h2 + 1.6h + 0.64
(d − 0.4)2
(h + 0.8)2
225. 9t2 + 6.6t + 1.21
226. 4w2 − 2.8w + 0.49
(3t + 1.1)2
229. 0.36y 2 − 0.84y + 0.49
(2w
− 0.7)2
(0.6y − 0.7)2
231. 0.04n2 + 0.02n + 0.0025
(0.2n + 0.05)2
233. (x − 2)2 + 10(x − 2) + 25
235. 9 − 6(r + 2) + (r + 2)2
(x + 3)2
(1 − r)2
237. 16 − 8(a + b) + (a + b)2
239. (c − d)2 + 16(c − d) + 64
(4 − a − b)2
(c − d + 8)2
223. p2 + 0.2p + 0.01
224. k 2 + 2.4k + 1.44
(p + 0.1)2
(k + 1.2)2
227. 100u2 + 26u + 1.69
228. 25a2 − 3a + 0.09
(10u + 1.3)2
230. 0.16c2 + 0.72c + 0.81
(5a − 0.3)2
(0.4c + 0.9)2
232. 0.09x2 − 0.03x + 0.0025
234. (y + 4)2 − 14(y + 4) + 49
(0.3x − 0.05)2
(y − 3)2
236. 36 + 12(a − 10) + (a − 10)2
238. 1 − 2(m − r) + (m − r)2
240. (x + y)2 − 18(x + y) + 81
ALG catalog ver. 2.6 – page 192 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(a − 4)2
(1 − m + r)2
(x + y − 9)2
IL
Topic:
Combined methods (factoring).
Directions:
11—Factor. 12—Factor, if possible. 13—Factor completely.
14—Factor, then check answer by multiplying.
1.
2y 2 − 18
2(y − 3)(y + 3)
2.
5a2 − 125
3.
5(a + 5)(a − 5)
5.
81p − p3
p(9 − p)(9 + p)
6.
49w3 − w5
108c2 − 3
4.
3(6c + 1)(6c − 1)
7.
x3 − 25x
12(m − 2)(m + 2)
x(x − 5)(x + 5)
8.
w3 (7 − w)(7 + w)
9.
3a2 − 12b2
10. 4x2 − 36y 2
3(a − 2b)(a + 2b)
14. 50r2 s − 8s
5y(x − 5y)(x + 5y)
18. 14k − 56k 5 m2
19q(2p − q 2 )(2p + q 2 )
22. x4 yz 2 − x2 y 3
x2 y(xz
ab(bc − a)(bc + a)
25. 99x3 y 2 − 44xy 4
12h2 k(k
29. 20mp4 − 20m
33. 8xy 9 − 128x5 y
37.
4 2 1
k −
5
5
1
5 (2k
41.
a2 b3 (1 + a4 b4 )(1 + a2 b2 )(1 − ab)(1 + ab)
38.
25
4 (3w
20. 48w3 z − 75wz 5
3wz(4w − 5z 2 )(4w + 5z 2 )
36u(2u − x)(2u + x)
23. km5 − h2 km3
24. n2 pr3 − n4 p3 r
km3 (m − h)(m + h)
n2 pr(r − np)(r + np)
27. 5y 2 z 4 − 80y 4 z 2
5y 2 z 2 (z
28. 64a2 b5 − 36a2 b3
42.
196 2 100 2
a −
b
3
3
4
3 (7a − 5b)(7a + 5b)
− x)(3w + x)
31. 2a5 − 162a
32. 3xy 2 − 3xy 6
2a(a2 + 9)(a − 3)(a + 3)
5m2 (m4 + p4 )(m2 + p2 )(m − p)(m + p)
36. 81c14 d2 − c2 d2
c2 d2 (9c6 + 1)(3c3 − 1)(3c3 + 1)
2 2 8
y −
3
3
40.
39.
2
3 (y
− 2)(y + 2)
43.
9 36 2 4
− d h
7
7
9
7 (1 − 2dh)(1 + 2dh)
46. 0.04xy 5 − 0.01x3 y
47. 0.25p − 0.81m2 p3
p(0.5 − 0.9mp)(0.5 + 0.9mp)
48. 1.21h3 − 0.64hk 2
53. 2x2 + 20x + 42
2(x + 3)(x + 7)
57. 60 + 10y − 10y 2
10(3 − y)(2 + y)
50.
4 2 4 2
πr − πR
3
3
4
3 π(r
− R)(r + R)
54. 6x2 − 30x + 24
6(x − 1)(x − 4)
58. 84 − 2r − 2r2
2(6 − r)(7 + r)
61. 5k 2 + 50k + 125
62. 72p2 − 48p + 8
+ 5)2
8(3p − 1)2
5(k
65. a3 − 11a2 + 28a
a(a − 7)(a − 4)
66. c4 − 8c3 − 33c2
c2 (c − 11)(c + 3)
9 25 2
− a
2
2
1
2 (3 − 5a)(3 + 5a)
a(0.4b2 − 0.3c)(0.4b2 + 0.3c)
2πh(R − r)(R + r)
3xy 2 (1 + y 2 )(1 − y)(1 + y)
34. 5m10 − 5m2 p8
45. 0.16ab4 − 0.09ac2
49. 2πR2 h − 2πr2 h
4a2 b3 (4b − 3)(4b + 3)
− 4y)(z + 4y)
1
10 (3 + r)(3 − r)
− 1)(2k + 1)
225 2 25 2
w − x
4
4
9
1
− r2
10 10
3cd(c − 3d)(c + 3d)
19. 144u3 − 36ux2
2r(4 + r2 )(2 − r)(2 + r)
8xy(y 2 − 2x)(y 2 + 2x)(y 4 + 4x2 )
16. 3c3 d − 27cd3
2p(4p − 3w)(4p + 3w)
− 4)(k + 4)
30. 32r − 2r5
20m(p2 + 1)(p − 1)(p + 1)
2(3m − 2y)(3m + 2y)
15. 32p3 − 18pw2
− y)(xz + y)
26. 12h2 k 3 − 192h2 k
11xy 2 (3x − 2y)(3x + 2y)
12. 18m2 − 8y 2
4(5a − b)(5a + b)
14k(1 − 2k 2 m)(1 + 2k 2 m)
21. ab3 c2 − a3 b
35. a2 b3 − a10 b11
11. 100a2 − 4b2
2s(5r − 2)(5r + 2)
17. 76p2 q − 19q 5
k 6 − 100k 4
k 4 (k − 10)(k + 10)
4(x − 3y)(x + 3y)
13. 5x2 y − 125y 3
12m2 − 48
51. 2πV 3 − 8πV
2πV (V − 2)(V + 2)
55. 3c2 + 27c − 66
3(c − 2)(c + 11)
59. 70 + 49x + 7x2
7(2 + x)(5 + x)
63. 2 − 28m + 98m2
2(1 − 7m)2
67. a3 + 15a2 − 16a
a(a + 16)(a − 1)
ALG catalog ver. 2.6 – page 193 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
44.
16 4 2 144
m p −
5
5
16
2
2
5 (m p − 3)(m p + 3)
xy(0.2y 2 − 0.1x)(0.2y 2 + 0.1x)
h(1.1h − 0.8k)(1.1h + 0.8k)
52. 50πav 2 − 2πa
2πa(5v − 1)(5v + 1)
56. 4p2 − 28p − 72
4(p − 9)(p + 2)
60. 42 − 27c + 3c2
3(2 − c)(7 − c)
64. 12 + 24x + 12x2
12(x + 1)2
68. x5 + 14x3 + 24x
x(x2 + 12)(x2 + 2)
IL
69. 48p3 + 2p5 − p7
70. 30r − 17r2 + r3
p3 (8 − p2 )(6 + p2 )
74. 25w3 − 10w4 + w5
a(8 + a)2
w3 (5 − w)2
77. 2y 3 + 2y 2 − 12y
2y(y + 3)(y − 2)
81. 28m2 − 28m3 + 7m4
− 8)(m2
+ 1)
82. 300a + 60a2 + 3a3
7m2 (2 − m)2
3a(10 + a)2
85. 6x2 − 6xy − 72y 2
86. 5a2 + 10ab − 40b2
6(x − 4y)(x + 3y)
5(a + 4b)(a − 2b)
89. 10ax2 + 10ax − 20a
90. 105b − 6by − 3by 2
10a(x + 2)(x − 1)
3b(5 − y)(7 + y)
93. 45x2 − 30x3 + 5x4
94. 27m + 18mp + 3mp2
5x2 (x − 3)2
3m(3 + p)2
97. w4 z + 8w3 z 2 + 15w2 z 3
99. 42cd5 − c2 d3 − c3 d
w2 z(w + 3z)(w + 5z)
cd(7d2 + c)(6d2 − c)
101. 5a2 b2 + 90ab2 + 325b2
5b2 (a + 13)(a + 5)
103. 5a3 m + 30a2 m2 − 80am3
105. 60 + 4x − 8x2
5am(a + 8m)(a − 2m)
106. 30m2 + 84m + 54
4(3 − x)(5 + 2x)
6(5m + 9)(m + 1)
109. 8y 4 − 19y 3 + 6y 2
y 2 (8y
110. 5r + 21r2 − 26r3
− 3)(y − 2)
r(5 + 26r)(1 − r)
113. 6c + 22c2 + 20c3
114. 105k 5 − 27k 3 − 6k
3k(5k 2 − 2)(7k 2 + 1)
2c(3 + 5c)(1 + 2c)
117. 18x3 z 4 + 3x3 z 2 − 3x3
3x2 (2x2
+ 1)(3z 2
118. 4a2 + 16a2 y − 84a2 y 2
4a2 (1 − 3y)(1 + 7y)
− 1)
121. 45by − 60by 2 − 25by 3
5by(3 + y)(3 − 5y)
123. 50km5 − 40km3 + 6km
125. 6x2 + 8xy − 8y 2
2mk(5m2 − 1)(5m2 − 3)
2(3x − 2y)(x + 2y)
127. 30b2 − 5by 2 − 5y 4
5(2b − y 2 )(3b + y 2 )
129. 18h4 − 48h2 k + 32k 2
131. 50x2 − 120xy 3 + 72y 6
133. 9a2 + 54ab + 45b2
2(3h2 − 4k)2
2(5x − 6y 3 )2
9(a + 5b)(a + b)
135. 30m2 − 78mn + 36n2
6(5m − 3n)(m − 2n)
m(6 + m)(2 − m)
75. x6 − 6x4 + 9x2
76. p4 + 22p3 + 121p2
x2 (x2 − 3)2
78. 3m5 − 21m3 − 24m
3m(m2
72. 12m − 4m2 − m3
k 2 (3 + k)(5 + k)
r(15 − r)(2 − r)
73. 64a + 16a2 + a3
71. 15k 2 + 8k 3 + k 4
p2 (p + 11)2
79. 2x4 + 4x3 − 48x2
80. 6a3 − 12a2 − 48a
2x2 (x + 6)(x − 4)
6a(a − 4)(a + 2)
83. 5c3 + 40c2 + 80c
84. 3y 6 − 36y 4 + 108y 2
5c(c + 4)2
3y 2 (y 2 − 6)2
87. 12n2 − 24mn − 36m2
88. 100c2 + 54cd + 2d2
12(n − 3m)(n + m)
2(25 + d)(2 + d)
91. 2t3 − 30t2 + 72t
92. 3x4 + 63x3 + 60x2
3x2 (x + 20)(x + 1)
2t(t − 12)(t − 3)
95. 4h2 k 2 + 40h2 k + 100h2
4h2 (k
96. 2y 6 − 48y 4 + 288y 2
+ 5)2
2y 2 (y 2 − 12)2
98. 8x2 y + 2x2 y 2 − x2 y 3
x2 y(4 − y)(2 + y)
100. a6 b3 − 15a4 b4 + 36a2 b5
a2 b3 (a2 − 12b)(a2 − 3b)
102. 4m3 r − 12m2 r2 − 40mr3
4mr(m − 5r)(m + 2r)
104. 2x2 y − 14x2 y 2 + 20x2 y 3
2x2 y(1 − 5y)(1 − 2y)
107. 70 + 10y − 80y 2
108. 48d2 − 88d − 56
10(7 + 8y)(1 − y)
8(2d + 1)(3d − 7)
111. 24a5 + 19a3 + 2a
a(8a2
+ 1)(3a2
112. 3k 2 − 35k 4 + 22k 6
k 2 (3k 2 − 2)(k 2 − 11)
+ 2)
115. 8x + 54x2 − 14x3
116. 32y 4 + 104y 3 + 60y 2
4y 2 (2y + 5)(4y + 3)
2x(1 + 7x)(4 − x)
119. 9n2 r2 + 42n2 r + 45n2
3n2 (3r
120. 66c3 − 50b2 c3 + 4b4 c3
2c3 (11 − b2 )(3 − 2b2 )
+ 5)(r + 3)
122. 25r3 w3 + 65r3 w2 − 30r3 w
124. 30p2 x − 99p2 x2 − 21p2 x3
126. 12a4 − 40a2 b + 28b2
3p2 x(5 + x)(2 − 7x)
4(3a2 − 7b)(a2 − b)
128. 24w2 − 32wz 3 − 56z 6
130. 12bc2 + 36b2 c + 27b3
5r3 w(5w − 2)(w + 3)
8(3w − 7z 3 )(w + z 3 )
3b(2c + 3b)2
132. 75r3 + 90mr2 + 27m2 r
3r(5r + 3m)2
134. 18x2 + 36xy − 14y 2
2(3x + 7y)(3x − y)
136. 90a2 − 75ac − 90c2
15(2a − 3c)(3a + 2c)
137. 3x3 − 8x2 y − 35xy 2
x(3x + 7y)(x − 5y)
138. 12a3 + 7a2 b − 10ab2
139. 3c3 d − 4c2 d2 − 4cd3
cd(3c + 2d)(c − 2d)
140. 2k 2 m3 + 11k 3 m2 + 5k 4 m
ALG catalog ver. 2.6 – page 194 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
a(3a − 2b)(4a + 5b)
k 2 m(2m + 5k)2
IL
141. a4 − 24a2 − 25
143. y 4 + y 2 − 20
(a2 + 1)(a − 5)(a + 5)
(y 2 + 5)(y − 2)(y + 2)
145. 2p4 − 7p2 − 4
(2p2 + 1)(p − 2)(p + 2)
147. 3 − 10m2 − 8m4
(3 + 2m2 )(1 − 2m)(1 + 2m)
149. 8a5 − 198a3 − 50a
151. k 5 r − 8k 3 r3 − 9kr5
2a(4a2 + 1)(a − 5)(a + 5)
kr(k2 + r2 )(k − 3r)(k + 3r)
142. 10 − 9x2 − x4
(10 + x2 )(1 − x)(1 + x)
144. 27 + 6n2 − n4
(3 + n2 )(3 − n)(3 + n)
146. 5c4 − 3c2 d2 − 2d4
148. 18x4 + 25x2 − 3
(5c2 + 2d2 )(c − d)(c + d)
(2x2 + 3)(3x − 1)(3x + 1)
150. 12c3 − 19c5 + 4c7
c3 (3 − 4c2 )(2 − c)(2 + c)
152. 64b2 + 28a2 b4 − 2a4 b6
2b2 (2 + a2 b2 )(4 − ab)(4 + ab)
153. 36 − 13y 2 − y 4
(3 − y)(3 + y)(2 − y)(2 + y)
154. c4 − 29c2 + 100
(c − 5)(c + 5)(c − 2)(c + 2)
155. 16 − 17r2 + r4
(4 − r)(4 + r)(1 − r)(1 + r)
156. x4 − 37x2 + 36
(x − 6)(x + 6)(x − 1)(x + 1)
157. w4 − 2w2 + 1
(w − 1)2 (w + 1)2
158. y 4 − 8y 2 + 16
(y − 2)2 (y + 2)2
159. 81 − 18c2 + c4
(3 − c)2 (3 + c)2
160. 16 − 8a2 + a4
(2 − a)2 (2 + a)2
ALG catalog ver. 2.6 – page 195 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
IM
Topic:
Factoring by grouping.
Directions:
11—Factor. 12—Factor, if possible. 13—Factor completely.
14—Factor, then check answer by multiplying.
1.
(m + 3)2 − 12m
3.
4xy + (x − y)2
5.
x(y + 2) + 3(y + 2)
7.
(w − z)5u − (w − z)7
9.
h(m2 + 3m) − (m2 + 3m)
(m − 3)2
(x + y)2
(x + 3)(y + 2)
11. 3u(w − z) + (w − z)
13. a(y − 1) − b(1 − y)
(w − z)(5u − 7)
(h − 1)(m2 + 3m)
(3u + 1)(w − z)
(a − b + c)(r + s)
19. b2 (b2 + 1) + b(b2 + 1) + 4(b2 + 1)
4.
(a + b)2 − 4ab
6.
r(p2 + 5) − s(p2 + 5)
8.
k(h2 − k) + 4(h2 − k)
(k + 4)(h2 − k)
10. (rx − 5) + 2r(rx − 5)
(1 + 2r)(rx − 5)
12. (r3 + 1) − (r3 + 1)4p
(1 − 4p)(r3 + 1)
(r + 2)2
(a − b)2
(r − s)(p2 + 5)
(y − z)(x − y)
16. (x − 2y)2a + (2y − x)
(3c + 1)(2c − d)
17. a(r + s) − b(r + s) + c(r + s)
8r + (r − 2)2
14. (x − y)y + (y − x)z
(a + b)(y − 1)
15. 3c(2c − d) − (d − 2c)
2.
(b2 + b + 4)(b2 + 1)
21. k(k − 1) + km(k − 1) + m(k − 1) − 2(k − 1)
(k + km + m − 2)(k − 1)
(2a − 1)(x − 2y)
18. m2 (n − 3) + m(n − 3) − (n − 3)
20. 2s(s + t) − t(s + t) − (s + t)
(m2 + m − 1)(n − 3)
(2s − t − 1)(s + t)
22. a(x + y) − 3b(x + y) + 5c(x + y) − d(x + y)
(a − 3b + 5c − d)(x + y)
23. (u − 2w)r3 − (u − 2w)3r2 + (u − 2w)r + (u − 2w)
24. (2a + 3)by + (2a + 3)c − (2a + 3)bx − (2a + 3)y
(2a + 3)(by + c − bx − y)
(u − 2w)(r3 − 3r2 + r + 1)
25. (x − y + z)2x − (x − y + z)y + (x − y + z)3z
(x − y + z)(2x − y + 3z)
26. mr(r + p − 2) + m2 (r + p − 2) + (r + p − 2)
(mr + m2 + 1)(r + p − 2)
27. 4a(2x + y − 3) − 3b(2x + y − 3) + cd(2x + y − 3)
(4a − 3b + cd)(2x + y − 3)
28. (n2 + n − 1) + (n2 + n − 1)2n − (n2 + n − 1)n2
(n2 + n − 1)(1 + 2n − n2 )
Two-and-two grouping
29. 5x − 5y + x2 − xy
31. ac + bc + ad + bd
33. a2 − a − ac + c
(5 + x)(x − y)
(c + d)(a + b)
(a − c)(a − 1)
35. ax − bx − ay + by
(x − y)(a − b)
30. km + 7k + 2m + 14
(k + 2)(m + 7)
32. w2 − wz + 6w − 6z
(w + 6)(w − z)
34. k 2 − 5k − mk + 5m
(k − m)(k − 5)
36. pr − 3p − 8r + 24
(p − 8)(r − 3)
37. 2nr + 3p2 + 6pr + pn
(2r + p)(n + 3p)
38. 4x2 − 5y − xy + 20x
39. 50k 2 − 10kr + 5k − r
(10k + 1)(5k − r)
40. 9a2 + 3ab + 3a + b
41. 4xy + 30 − 5y − 24x
43. 8y − 3z 2 − 12z + 2yz
45. 7c3 − 28c2 + 3c − 12
(4x − 5)(y − 6)
(2y − 3z)(z + 4)
(7c2 + 3)(c − 4)
(x + 5)(4x − y)
(3a + 1)(3a + b)
42. 11a2 − 3b + 33a − ab
(a + 3)(11a − b)
44. 5mp − 18pm − 2m2 + 45p2
46. 4d5 + d3 + 8d2 + 2
(d3 + 2)(4d2 + 1)
47. 3w2 − 7wx + 3wx2 − 7x3
(w + x2 )(3w − 7x)
48. k 3 n + k 2 n3 + k + n2
49. 12ab + 5c + 48a2 b + 20ac
(12ab + 5c)(4a + 1)
50. 2m3 − 6m2 r + 9m − 27r
51. abc + 7a2 + 2b2 c + 14ab
(bc + 7a)(a + 2b)
(k 2 n + 1)(k + n2 )
52. −40ab − 3ac + 15bc + 8a2
ALG catalog ver. 2.6 – page 196 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(m + 9p)(5p − 2m)
(2m2 + 9)(m − 3r)
(a − 5b)(8a − 3c)
IM
53. 6pr2 − pw2 − 3rw + 2p2 rw
55. n2 r2 + n3 p + r3 p + np2 r
(2pr − w)(pw + 3r)
(n2 + rp)(r2 + np)
54. a2 bc − ab2 d + ac2 d − bcd2
56. x2 y 3 − xyz − xy 2 z 2 + z 3
57. 3p2 r2 + 12p2 r + 5pr2 + 20pr
pr(3p + 5)(r + 4)
58. 2a5 + 6a2 − 4a4 − 3a3
59. −8x2 y + 2xy 2 + 28x2 − 7xy
x(7 − 2y)(4x − y)
60. 12w4 + 6w3 − 4w2 − 2w
61. 2k 4 − 2k 3 n − k 3 − k 2 n
k 2 (k − n)(2k − 1)
63. a2 c2 − 6ac2 + ac3 − 6c3
c2 (a + c)(a − 6)
65. 2ab2 c + 8ab2 − 4b3 c − 16b3
2b2 (c + 4)(a − 2b)
67. 12ax2 − 48axy − 16bxy + 4bx2
4x(x − 4y)(3a − b)
(ab + cd)(ac − bd)
(xy − z 2 )(xy 2 − z)
a2 (a − 2)(2a2 − 3)
2w(3w2 − 1)(2w + 1)
62. 2x2 y − 10xy + 6x2 − 30x
2x(y + 3)(x − 5)
64. 60am − 20mx − 30ax + 10x2
10(x − 2m)(x − 3a)
66. 10r2 x − 5rwx − 10r2 w + 5rw2
68. 14nr2 + 21nr − 14r2 s − 21rs
5r(x − w)(2r − w)
7r(n − s)(2r + 3)
Three-and-one grouping
69. a2 + 4a + 4 − x2
(a + 2 + x)(a + 2 − x)
70. b2 − 8b + 16 − c2
71. x2 − 6x + 9 − y 2
(x − 3 + y)(x − 3 − y)
72. m2 + 10m + 25 − n2
(b − 4 + c)(b − 4 − c)
(m + 5 + n)(m + 5 − n)
73. x2 − 2xy + y 2 − 9w2
(x − y + 3w)(x − y − 3w)
74. r2 + 2rw + w2 − 36z 2
75. a2 + 2ab + b2 − 25n2
(a + b + 5n)(a + b − 5n)
76. c2 − 2cd + d2 − 4b2
(c − d + 2b)(c − d − 2b)
77. y 2 − a2 − 6a − 9
(y + a + 3)(y − a − 3)
78. n2 − b2 − 8b − 16
79. x2 − c2 + 4c − 4
(x + c − 2)(x − c + 2)
80. p2 − m2 + 2mn − n2
81. 4c2 − 4c + 1 − 9d2
(2c − 1 + 3d)(2c − 1 − 3d)
83. w2 − 10wy + 25y 2 − 64
85. p2 − m2 − n2 + 2mn
87. a2 − 2ac − c2 − b2
(w − 5y + 8)(w − 5y − 8)
(p + m − n)(p − m + n)
(a − c + b)(a − c − b)
89. x2 − 64 + y 2 + 2xy
(x + y + 8)(x + y − 8)
91. m2 + n2 − 81 − 2mn
(m − n + 9)(m − n − 9)
93. 3x2 y − 6xy 2 + 3y 3 − 3yz 2
3y(x − y + z)(x − y − z)
(r + w + 6z)(r + w − 6z)
(n + b + 4)(n − b − 4)
(p + m − n)(p − m + n)
82. a2 + 6ab + 9b2 − 25c2
(a + 3b + 5c)(a + 3b − 5c)
84. 25b2 + 10bx + x2 − d2
(5b + x + d)(5b + x − d)
86. w2 − 2xy − x2 − y 2
88. 2ab − a2 + n2 − b2
90. h2 + 2hk − 49 + k 2
(w + x + y)(w − x − y)
(n + a − b)(n − a + b)
(h + k + 7)(h + k − 7)
92. 100 − 4r2 − w2 + 4rw
(10 + 2r − w)(10 − 2r + w)
94. 7k 2 r2 − 28kmr2 + 28m2 r2 − 7r4
7r2 (k − 2m + r)(k − 2m − r)
95. 50n3 + 20n2 p + 2np2 − 2nt2
2n(5n + p + t)(5n + p − t)
96. 20a2 b2 − 60ab3 + 45b4 − 5b2 c2
5b2 (2a − 3b + c)(2a − 3b − c)
Three-and-two grouping, and other variations
97. a2 + 2ab + b2 + ac + bc
(a + b)(a + b + c)
98. x2 − 2xy + y 2 + wx − wy
99. c2 − 2cd + d2 − 4c + 4d
(c − d)(c − d − 4)
100. m2 + 2mr + r2 − m − r
101. a3 − a(c − d)2
103. x(x − y)2 − 9x
a(a + c − d)(a − c + d)
x(x − y + 3)(x − y − 3)
105. x2 − 16 − (x − 4)2
8(x − 4)
107. a2 − b2 − (a + b)2
−2b(a + b)
109. (m − 2)(m2 − 7) + 3(m − 2)
102. 3(b − c)2 − 12
104. 4k − k(m − r)2
(m + r)(m + r − 1)
3(b − c + 2)(b − c − 2)
k(2 + m − r)(2 − m + r)
106. 9 − w2 − (3 − w)2
108. m2 − p2 − (m − p)2
(m − 2)2 (m + 2)
(x − y)(w + x − y)
2w(3 − w)
2p(m − p)
110. (a2 + 1)(a + 3) − 2(a + 3)
ALG catalog ver. 2.6 – page 197 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(a + 3)(a + 1)(a − 1)
IM
111. (p2 − 8)(p + 1) − (p + 1)
113. y(y 2 − 1) − 2(y + 1)
(p + 1)(p + 3)(p − 3)
114. a(a2 − 4) − 3(a − 2)
(y + 1)2 (y − 2)
115. c(c2 − x2 ) − 2x2 (c − x)
112. (x − 4) + (x2 − 5)(x − 4)
(a − 2)(a − 1)(a + 3)
116. 2a2 (a − b) − b(a2 − b2 )
(c − x)2 (c + 2x)
(x − 4)(x + 2)(x − 2)
(a − b)2 (2a + b)
117. a(a + 1)(a + 2) − 3(a + 1)
(a + 1)(a − 1)(a + 3)
118. 6(m − 2) + m(m − 5)(m − 2)
119. r(r + 1)(r − 6) + 8(r + 1)
(r + 1)(r − 2)(r − 4)
120. k(k + 5)(k − 3) − 10(k + 5)
121. 25(c + 7) − c2 (c + 7)
(c + 7)(5 + c)(5 − c)
123. 4a2 (a − b) − b2 (a − b)
125. (c − d)3 + 4cd(c − d)
(a − b)(2a + b)(2a − b)
129. (y 2 − 4)2 − (y − 2)2
(1 − x + 3y)(1 + x − 3y)(x + 3y)2
133. 4a2 (2a − 1) − 4a(2a − 1) + (2a − 1)
(2a − 1)3
(k + 5)(k − 5)(k + 2)
(y − 3z)(x + 3y)(x − 3y)
(w + 6)(4w + 1)(4w − 1)
(a − 1)(a − 2)(a − 3)
128. (r + 2)3 − 9(r + 2)
(x + y)(x − y)2
(y + 1)(y + 3)(y − 2)2
131. (x + 3y)2 − (x2 − 9y 2 )2
124. 16w2 (w + 6) − (w + 6)
126. (a − 2)3 − (a − 2)
(c − d)(c + d)2
127. (x + y)3 − 4xy(x + y)
122. x2 (y − 3z) − 9y 2 (y − 3z)
(m − 2)2 (m − 3)
(r + 2)(r + 5)(r − 1)
130. (w2 − 9)2 − (w − 3)2
(w + 2)(w + 4)(w − 3)2
132. (4m2 − 1)2 − (2m + 1)2
4m(m − 1)(2m + 1)2
134. k 2 (3k + 1) − 2k(3k + 1) − 8(3k + 1)
(3k + 1)(k + 2)(k − 4)
135. c2 (c2 − 4) + 4c(c2 − 4) + 4(c2 − 4)
137. a2 + 2ab + b2 − c2 + 6c − 9
(c − 2)(c + 2)3
(a + b − c + 3)(a + b + c − 3)
136. y 2 (y 2 − 9) + 5y(y 2 − 9) + 6(y 2 − 9)
138. x2 − 2xy + y 2 − w2 − 10w − 25
(x − y − w − 5)(x − y + w + 5)
139. 4a2 + 4ab + b2 − c2 − 6cd − 9d2
(2a + b − c − 3d)(2a + b + c + 3d)
141. a2 + 2ax + x2 − a4 + 2a2 x2 − x4
(a + x)2 (1 − a + x)(1 + a − x)
143. x2 y 2 − 3x2 y + 2x2 − 4y 2 + 12y − 8
(x − 2)(x + 2)(y − 2)(y − 1)
140. x2 − 8xy + 16y 2 − 9w2 + 6wz − z 2
(x − 4y − 3w + z)(x − 4y + 3w − z)
142. c4 − 2c2 d2 + d4 − c2 + 2cd − d2
(c − d)2 (c + d − 1)(c + d + 1)
144. a2 b2 − 5ab2 + 4b2 − 9a2 + 45a − 36
(b − 3)(b + 3)(a − 4)(a − 1)
ALG catalog ver. 2.6 – page 198 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(y + 2)(y − 3)(y + 3)2
IN
Topic:
Factoring sums and differences of cubes.
Directions:
11—Factor. 12—Factor, if possible. 13—Factor completely.
14—Factor, then check answer by multiplying.
1.
x3 + 1
(x + 1)(x2 − x + 1)
2.
y 3 + 27
(y
5.
a3 − 27
6.
(a − 3)(a2
9.
x3 − 64
10. 1 + k 3
(2 + y)(4 − 2y
7.
(1 + k)(1 − k + k 2 )
14. 27 − x3
18. k 3 + 125
− 5)(r2
+ 5)(k 2
(k
+ 5r + 25)
(p − 10)(p2
(c + 10)(c2
+ 10p + 100)
− 10c + 100)
34. 125 − 8y 3
+ 4y 2 )
38. 27m3 + 64
+ xy
+ y2 )
45. m3 + 8n3
(a − b)(a2
46. 8x3 + y 3
(m + 2n)(m2 − 2mn + 4n2 )
49. a3 − 27b3
(a − 3b)(a2
50. 64p3 − q 3
+ 3ab + 9b2 )
53. 125x3 + 8y 3
(4p − q)(16p2
57. y 3 − w3 x3
− wx)(y 2
+ 4pq
58. a3 b3 − c3
+ wxy
+ w2 x2 )
24. k 3 − 216
(k − 6)(k 2 + 6k + 36)
28. 1000 − a3
(10 + x)(100 − 10x + x2 )
(ab − c)(a2 b2
32. 1 − 27p3
(4r
(1 − 3p)(1 + 3p + 9p2 )
36. 216w3 + 125
− 3)(16r2
(6w + 5)(36w2 − 30w + 25)
+ 12r + 9)
40. 27 − 64y 3
(10 + 3x)(100 − 30x + 9x2 )
(3 − 4y)(9 + 12y + 16y 2 )
44. a3 + b3
(x + y)(x2
− xy
+ y2 )
(a + b)(a2 − ab + b2 )
48. 8k 3 − w3
(r − 2p)(r2 + 2rp + 4p2 )
(3w
(2k − w)(4k 2 + 2kw + w2 )
52. m3 + 216n3
+ x)(9w2
− 3xw
+ x2 )
(m + 6n)(m2 − 6mn + 36n2 )
56. 27c3 − 64d3
(4n − 5r)(16n2 + 20nr + 25r2 )
59. k 3 + m3 n3
+ abc + c2 )
(10 − a)(100 + 10a + a2 )
(1 + 2a)(1 − 2a + 4a2 )
55. 64n3 − 125r3
(2a + 3b)(4a2 − 6ab + 9b2 )
(5 + w)(25 − 5w + w2 )
(r + 6)(r2 − 6r + 36)
51. 27w3 + x3
+ q2 )
54. 8a3 + 27b3
(5x + 2y)(25x2 − 10xy + 4y 2 )
(5 − y)(25 + 5y
47. r3 − 8p3
(2x + y)(4x2 − 2xy + y 2 )
(2 − y)(4 + 2y + y 2 )
20. 125 + w3
43. x3 + y 3
+ ab + b2 )
(3 + x)(9 − 3x + x2 )
+ y2 )
39. 1000 + 27x3
(3m + 4)(9m2 + 12m + 16)
42. a3 − b3
(x − y)(x2
(k
+ mn)(k 2
(3c − 4d)(9c2 + 12cd + 16d2 )
60. p3 r3 + q 3
− kmn + m2 n2 )
61. 27y 3 + w3 x3
(3y + wx)(9y 2 − 3wxy + w2 x2 )
62. a3 b3 + 8c3
63. 64k 3 − m3 n3
(4k − mn)(16k 2 + 4kmn + m2 n2 )
64. p3 r3 − 125q 3
(pr − 5q)(p2 r2 + 5prq + 25q 2 )
(x2 + 1)(x4 − x2 + 1)
66. 1 + y 6
67. a6 − 8
(a2 − 2)(a4 − 2a2 + 4)
68. 8 − m6
(2 − m2 )(4 + 2m2 + m4 )
69. y 6 − 1
(y + 1)(y − 1)(y 4 + y 2 + 1)
70. 1 − a6
(1 + a)(1 − a)(1 + a2 + a4 )
(2 + w)(2 − w)(16 + 4w2 + w4 )
72. x6 − 64
ALG catalog ver. 2.6 – page 199 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(pr + q)(p2 r2 − prq + q 2 )
(ab + 2c)(a2 b2 − 2abc + 4c2 )
65. x6 + 1
71. 64 − w6
(c − 2)(c2 + 2c + 4)
16. 8 − y 3
35. 64r3 − 27
(5 − 2y)(25 + 10y
(5a − 2)(25a2 + 10a + 4)
41. x3 − y 3
c3 − 8
(1 − m)(1 + m + m2 )
31. 1 + 8a3
(3x + 1)(9x2 − 3x + 1)
(3 + 4x)(9 − 12x + 16x2 )
37. 125a3 − 8
8.
12. 27 + x3
27. 1000 + x3
30. 27x3 + 1
33. 27 + 64x3
11. 64 + m3
23. r3 + 216
(6 − w)(36 + 6w + w2 )
(2y − 1)(4y 2 + 2y + 1)
(y
− 5k + 25)
26. c3 + 1000
29. 8y 3 − 1
(y − 1)(y 2 + y + 1)
19. 125 − y 3
22. 216 − w3
(6 + y)(36 − 6y + y 2 )
25. p3 − 1000
(m + 4)(m2 − 4m + 16)
− 2a + 4)
15. 1 − m3
(3 − x)(9 + 3x + x2 )
17. r3 − 125
21. 216 + y 3
y3 − 1
m3 + 64
(4 + m)(16 − 4m + m2 )
(4 − a)(16 + 4a + a2 )
(r
4.
+ 4x + 16)
+ y2 )
13. 64 − a3
a3 + 8
(a + 2)(a2
− 3y + 9)
(x − 4)(x2
+ 3a + 9)
8 + y3
3.
+ 3)(y 2
(1 + y 2 )(1 − y 2 + y 4 )
(x + 2)(x − 2)(x4 + 4x2 + 16)
IN
73. a9 + 1
(a + 1)(a2 − a + 1)(a6 − a3 + 1)
74. p9 − 1
75. 1 + y 9
(1 + y)(1 − y + y 2 )(1 − y 3 + y 6 )
76. 1 − w9
(1 − w)(1 + w + w2 )(1 + w3 + w6 )
(a3 + b)(a6 − a3 b + b2 )
77. x3 − y 6
(x − y 2 )(x2 + xy 2 + y 4 )
78. a9 + b3
79. w6 − x9
(w2 − x3 )(w2 + wx3 + x6 )
80. c12 + d3
(p − 1)(p2 + p + 1)(p6 + p3 + 1)
(c4 + d)(c8 − c4 d + d2 )
81. 8m9 + n12
(2m3 + n4 )(4m6 − 2m3 n4 + n8 )
82. c12 − 27d6
83. 64a3 + b12
(4a + b4 )(16a2 − 4ab4 + b8 )
84. x15 − 125y 3
85. w6 − x3 y 9
(w2 − xy 3 )(w4 + w2 xy 3 + x2 y 6 )
86. a3 b15 + c9
87. 8m6 − n12 p3
89. a12 − 64
(2m2 − n4 p)(4m4 − 2m2 n4 p + n8 p2 )
(a2 − 2)(a2 + 2)(a8 + 4a4 + 16)
(c4 − 3d2 )(c8 + 3c4 d2 + 9d4 )
(x5 − 5y)(x10 + 5x5 y + 25y 2
(ab5 + c3 )(a2 b10 − ab5 c3 + c6 )
88. c3 d9 + 27k 6
90. 64 − c12
(cd3 + 3k 2 )(c2 d6 − 3cd3 k 2 + 9k4)
(2 + c2 )(2 − c2 )(16 + 4c4 + c8 )
91. x12 − 1
(x − 1)(x + 1)(x2 + 1)(x4 + x2 + 1)(x4 − x2 + 1)
92. 1 − y 12
93. x4 − 8x
x(x − 2)(x2 + 2x + 4)
94. 15 + 15m3
(1 − y)(1 + y)(1 + y 2 )(1 + y 2 + y 4 )(1 − y 2 + y 4 )
15(1 + m)(1 − m + m2 )
95. 27k 2 − k 5
k 2 (3 − k)(9 + 3k + k 2 )
96. 2a3 + 128
97. 3y + 81y 4
3y(1 + 3y)(1 + 3y + 9y 2 )
98. 250t5 − 16t2
2t2 (5t − 2)(25t2 + 10t + 4)
3w(3 − 2x)(9 + 6x + 4x2 )
2(a + 4)(a2 − 4a + 16)
99. 2bc + 128bc4
2bc(1 + 4c)(1 + 4c + 16c2 )
100. 81w − 24wx3
101. a6 + 2a3 − 3
(a3 + 3)(a − 1)(a2 + a + 1)
102. x6 − 3x3 − 40
103. y 6 + 16y 3 + 15
105. p6 + 7p3 − 8
107. 8k 6 − 9k 3 + 1
(y 3 + 15)(y + 1)(y 2 − y + 1)
(p + 2)(p − 1)(p2 − 2p + 4)(p2 + p + 1)
(2k − 1)(k − 1)(4k 2 + 2k + 1)(k 2 + k + 1)
104. c6 + 6c3 − 16
(x3 + 5)(x − 2)(x2 + 2x + 4)
(c3 − 2)(c + 2)(c2 − 2c + 4)
106. x6 − 26x3 − 27
108. 27y 6 + 35y 3 + 8
ALG catalog ver. 2.6 – page 200 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(x + 1)(x − 3)(x2 − x + 1)(x2 + 3x + 9)
(3y + 2)(y + 1)(9y 2 − 6y + 4)(y 2 − y + 1)
IO
Topic:
Factoring expressions with variable exponents.
Directions:
11—Factor. 12—Factor, if possible. 13—Factor completely.
14—Factor, then check answer by multiplying.
1.
xn+2 + xn
xn (x2 + 1)
2.
x3n + x2n
x2n (xn + 1)
3.
y 2k+n + y 3k+2
4.
y x+3 (1 − y 4 )
y 2k (y n + y k+2 )
5.
4x3r+2 + 8x4r
4x3r (x2
9.
6.
+ 2xr )
5k b (3k 4b
ay b3y+1 + ay+1 b2y
18. 36 − 9w2a − w4a
(a2x − 4)(a2x + 5)
(12 + w2a )(3 − w2a )
22. 10c2d − 9cd + 2
(5 − 7y x )(1 + 2y x )
25. 3x2n − 8xn y − 3y 2
(5cd
(nx + 3)(nx − 3)
− 2)(2cd
− 1)
26. 8r4 + 15r2 pn + 7p2n
(3xn + y)(xn − 3y)
29. n2x − 9
prn (pr2n
(8r2 + 7pn )(r2 + pn )
30. 25 − r2a
(5 − ra )(5 + ra )
8.
15. 3 + 4y n + y 2n
12. 50x2k y n − 30xk+3 y 2n
10xk y n (5xk − 3x3 y n )
16. r2u − 9ru + 14
(3 + y n )(1 + y n )
19. m8n − 7m4n + 6
(ru − 7)(ru − 2)
20. 1 − 2y 4x − 3y 8x
(m4n − 6)(m4n − 1)
23. 2a2c − 9ac − 5
(2ac
+ 1)(ac
(1 − 3y 4x )(1 + y 4x )
24. 6 − 13pr + 6p2r
(3 − 2pr )(2 − 3pr )
− 5)
27. a2x − 3ax+y + 2a2y
28. 5a4 + 4a2+n − a2n
(ax − ay )(ax − 2ay )
31. 1 − x10a
(5a2 − an )(a2 + an )
32. y 8n − 36
(1 − x5a )(1 + x5a )
33. x2y − x6
34. m2 − n4r
(xy − x3 )(xy + x3 )
37. a2x + 2a + 1
(ax + 1)2
35. a2x − b2x
41. x6a − 4x3a + 4
(ax − bx )(ax + bx )
38. 25 − 10k n + k 2n
39. 1 − 2x4n + x8n
(xa
46. r2x + rx+2 + r4
− xb )2
(rx
49. xm+2 − xm
53. x2a+1 + 2xa+1 − 3x
x(xa
+ 3)(xa
− 5)(ax
y(y n
67. y 8x − 1
+ 2)(y n
rx (rx
+ 4)
+ 1)(rx
− 6)
62. x4n − 2x2n − 8
(ax − 1)(ax + 1)(a2x + 3)
65. b4x − c4
+ 2)
58. r3x − 5r2x − 6rx
− 2)
61. a4x + 2a2x − 3
− 2)(y n
54. y 2n+1 + 6y n+1 + 8y
− 1)
57. a3x − 7a2x + 10ax
ax (ax
3y(y n
(xn − 2)(xn + 2)(x2n + 2)
(bx − c)(bx + c)(b2x + c2 )
(y x − 1)(y x + 1)(y 2x + 1)(y 4x + 1)
(a3k − 7)2
43. 9y 2x + 30y x + 25
44. 16k 2 − 8kmn + m2n
(4k − mn )2
47. x2a + 2xa ax + a2x
(xa
+ r2 )2
50. 3y 2n+1 − 12y
xm (x − 1)(x + 1)
40. a6k − 14a3k + 49
(3y x + 5)2
(10 + p2r )2
45. x2a − 2xa+b + x2b
(xy − y x )(xy + y x )
(1 − x4n )2
42. 100 + 20p2r + p4r
(x3a − 2)2
(y 4n − 6)(y 4n + 6)
36. x2y − y 2x
(m − n2r )(m + n2r )
(5 − k n )2
48. k 6 − 2k 3 nk + n2k
+ ax )2
(k 3 − nk )2
51. 45ax − 5a3x
52. m3a − ma+2
5ax (3 + ax )(3 − ax )
55. p2n+1 + 10pn+1 + 25p
p(pn
ma (ma − m)(ma + m)
56. u2x+1 − 14ux+1 + 49u
u(ux − 7)2
+ 5)2
59. a3k − 2a2k + ak
ak (ak
60. x3y + 4x2y + 4xy
xy (xy + 2)2
− 1)2
63. a4x − 2a2x + 1
64. x4n − 18x2n + 81
(ax − 1)2 (ax + 1)2
66. 16 − a4n
68. 1 − r8u
ALG catalog ver. 2.6 – page 201 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
xa y − x2a y a+2
xa y(1 − xa y a+1 )
+ 1)
12a2 n2 (2nx − 3ax )
− 3)
21. 5 + 3y x − 14y 2x
p2 r3n + prn
pk rn+1 (pr − pk )
(5 − ax )(2 + ax )
17. a4x + a2x − 20
7.
11. 24a2 nx+2 − 36ax+2 n2
14. 10 + 3ax − a2x
13. x2n − xn − 6
+ 2)(xn
− 7k 5 )
10. pk+1 rn+2 − p2k rn+1
ay b2y (by+1 + a)
(xn
15k 5b − 35k b+5
y x+3 − y x+7
(xn − 3)2 (xn + 3)2
(2 − an )(2 + an )(4 + a2n )
(1 − ru )(1 + ru )(1 + r2u )(1 + r4u )
JA
Topic:
Using zero-product rule.
Directions:
15—Solve.
1.
5w = 0
0
5.
k(k − 5) = 0
9.
−3m(13 + m) = 0
124—Solve over IR.
0, 5
2.
10r3 = 0
6.
−p(p − 9) = 0
3.
−9h = 0
0, 9
7.
z(z + 3) = 0
0, −7
11. −4a(11 − a) = 0
0
10. 4x(7 + x) = 0
0
0, −3
0, 11
4.
3a2 = 0
8.
−y(y + 14) = 0
0
12. 2p(9 − p) = 0
0, −14
0, 9
0, −13
13. (x − 3)(x + 5) = 0
14. (a − 7)(a + 4) = 0
3, −5
15. (c + 2)(c − 9) = 0
7, −4
17. (k − 7)(k − 1) = 0
18. (y − 4)(y − 8) = 0
7, 1
16. (m + 5)(m − 7) = 0
−2, 9
4, 8
−5, 7
19. (w + 10)(w + 3) = 0
20. (g + 8)(g + 11) = 0
−10, −3
21. (u + 5)(u + 5) = 0
22. (p + 9)(p + 9) = 0
−5
−9
−8, −11
23. (h − 12)(h − 12) = 0
24. (k − 6)(k − 6) = 0
6
12
25. (b − 12)(b + 12) = 0
26. (a − 7)(a + 7) = 0
±7
27. (b + 1)(b − 1) = 0
±1
28. (a + 10)(a − 10) = 0
±12
±10
29. (h + 4)2 = 0
30. (k + 7)2 = 0
−4
33. (12 + h)(8 + h) = 0
−1, −13
37. (7 − w)(11 − w) = 0
41. (y + 30)(1 − y) = 0
53. 4y(1 + 3y) = 0
0,
46. −5m2 (m + 2) = 0
7
4
50. p(5 − 9p) = 0
54. 3x(8x + 1) = 0
0, − 13
−1,
61. (1 − 6q)(2 + 5q) = 0
3
2
66. (5u + 8)(5u + 8) = 0
±16
47. −7x(x − 5)2 = 0
48. 5r(r + 4)3 = 0
0, 5
51. b(25b + 1) = 0
1
0, − 25
55. p2 (3 − 8p) = 0
0,
52. 2c(1 + 16c) = 0
0,
4
9
60. (5 + p)(5 − 8p) = 0
4
9
−5,
5
8
64. (6a − 9)(4a − 7) = 0
63. (3b + 2)(3b + 1) = 0
− 23 , − 13
67. (4w + 3)2 = 0
1
0, − 16
56. 4m3 (9m − 4) = 0
8
3
59. (4 − 9r)(6 + r) = 0
−6,
0, −4
9 7
6, 4
68. (6m − 7)3 = 0
− 34
7
6
− 85
5
4
70. (5k − 18)(7k − 4) = 0
69. (3u + 25)(8u + 3) = 0
− 23 , − 38
18 4
5 , 7
5, ±7
79. −3y 2 (y + 1)(y + 4)(y − 7) = 0
1
12 , 7
72. (3m + 5)(3m − 5) = 0
± 27
± 53
0, 5, −8
76. −2h2 (h − 6)(h − 8) = 0
0, −1, −9
0,
71. (7d − 2)(7d + 2) = 0
74. g(g − 5)(g + 8) = 0
0, −2, 3
77. (m − 5)(m + 7)(m − 7) = 0
81. q(12q − 1)(7 − q) = 0
0, − 18
− 58 , 12
65. (4x − 5)(4x − 5) = 0
75. 5c(c + 1)(c + 9) = 0
5
9
62. (5 + 8g)(1 − 2g) = 0
2
1
6,−5
73. −p(p + 2)(p − 3) = 0
0,
0, 2
58. (y + 1)(2y − 3) = 0
5
3
7, 2
44. (z − 16)(16 + z) = 0
±25
0, −7
57. (3k − 5)(k + 8) = 0
−8,
43. (y − 25)(25 + y) = 0
−4, 13
45. 10y 2 (y − 7) = 0
40. (y − 7)(2 − y) = 0
5, 3
42. (z + 4)(13 − z) = 0
−30, 1
7, −3
39. (x − 5)(3 − x) = 0
22, 3
8
36. (w − 7)(3 + w) = 0
4, −6
38. (22 − x)(3 − x) = 0
7, 11
32. (p − 8)2 = 0
15
35. (u − 4)(6 + u) = 0
34. (1 + c)(13 + c) = 0
−12, −8
49. k(4k − 7) = 0
31. (m − 15)2 = 0
−7
0, −1, −4, 7
0, 6, 8
78. (w + 6)(w − 6)(w + 9) = 0
80. 5x(x + 4)(x − 9)(x − 2) = 0
82. −g(8 − g)(10g − 1) = 0
ALG catalog ver. 2.6 – page 202 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
±6, −9
0, −4, 9, 2
0, 8, tf 110
JA
83. −3w(5w − 12)(w − 15) = 0
0,
85. −5k(4k − 5)(10k − 9) = 0
87. 7h2 (1 − 5h)(11 + 6h) = 0
0, 54 ,
91. (3a − 4)(3a + 4)(a + 3) = 0
−3, 4
95. (4y − 1)2 (4y + 1)2 = 0
97. (x2 − 1)(x2 + 16) = 0
99. (25p2 − 1)(p2 + 25) = 0
9
10
0, 15 , − 11
6
89. (4x − 3)(15x − 1)(2x + 7) = 0
93. (c + 3)2 (c − 4)3 = 0
12
5 , 15
± 14
±1
± 15
7
3 1
4 , 15 , − 2
± 43 , −3
84. 6z(z − 11)(3z − 8) = 0
0, 11,
8
3
86. 7u(3u − 2)(5u − 7) = 0
0, 23 ,
7
5
88. −2k 2 (5 + 13k)(5 − 6k) = 0
5
0, − 13
,
90. (5y − 4)(8y + 7)(11y − 1) = 0
92. (4c − 9)(7c − 2)(7c + 2) = 0
94. (r − 5)4 (r + 6)2 = 0
98. (y 2 + 9)(y 2 − 4) = 0
100. (9r2 + 1)(9r2 − 1)
ALG catalog ver. 2.6 – page 203 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
7 1
4
5 , − 8 , 11
9, ± 27
5, −6
96. (y + 10)3 (10y − 1)2 = 0
±2
± 13
−10,
5
6
1
10
JB
Topic:
Simple quadratic equations (factorable). See also category JC.
Directions:
15—Solve.
1.
x2 − 9x = 0
5.
0 = 10y + y 2
9.
m2 + m = 0
0, 9
13. 0 = 2y 2 − y
0, −10
4.
0 = 2k − k 2
0, 2
0, −7
7.
11x + x2 = 0
0, −11
8.
a2 + 6a = 0
0, −6
14. 0 = w + 8w2
0, − 18
15. 3a2 + a = 0
0, − 43
29. 0 = 35a − 21a2
y2
10
0, 14
1
2
0,
25. 6w2 + 8w = 0
45. y =
0 = w2 + 7w
0 = a2 − 14a
11. 0 = x2 − x
0,
18. 3m2 − 4m = 0
22. 0 = 2y + 6y 2
1
2
34. w2 = −w
38. 9c = c2
0, −14
46. −
0, 10
49. c2 + 4c − 5 = 0
0,
0, − 13
23. 3c2 + 15c = 0
0, −5
−5, 1
k2
=k
4
0,
4
7
0, − 52
47. x = −
0, −4
50. m2 − 3m − 10 = 0
61. 0 = w2 − 6w + 10
65. 0 = r2 + 13r − 30
Ø
18, 1
73. r2 + 8r − 48 = 0
−12, 4
77. 0 = x2 + 28x + 75
−25, −3
81. 0 = 3y 2 + 24y + 45
−3, −5
85. 5y 2 − 20y − 60 = 0
6, −2
89. 3x2 − 21x − 90 = 0
10, −3
13
5
0, 1
0, −6
0,
1
15
0, 8
52. 0 = y 2 − 18y + 45
−10, 3
60. 0 = k 2 + k − 20
4, −5
4, −3
Ø
−6, 5
63. y 2 − y + 15 = 0
Ø
67. 0 = a2 − 8a − 33
11, −3
70. m2 + 9m + 18 = 0
−6, −3
74. 0 = a2 − 19a − 20
20, −1
78. 0 = y 2 − 20y + 64
16, 4
82. 0 = 2k 2 − 24k + 70
7, 5
86. 0 = 4x2 + 32x − 80
2, −10
90. 0 = 5w2 + 45w − 180
−12, 3
y2
=y
8
0,
56. c2 + 7c − 30 = 0
59. m2 − m − 12 = 0
−15, 2
69. 0 = n2 − 19n + 18
32. 15x2 − 39x = 0
15, 3
−10, 7
62. n2 + 5n + 10 = 0
66. 0 = y 2 + y − 30
0, − 45
48.
0, −5
55. r2 + 3r − 70 = 0
−12, −2
28. 0 = 16c + 20c2
44. 15w2 = w
0, − 13
0, − 67
0, 3
40. −6m = m2
0, 20
1
5
24. 12x − 4x2 = 0
36. −y = −y 2
0, −1
−3, −7
58. a2 + 14a + 24 = 0
27, 1
x2
5
2
5
51. 0 = x2 + 10x + 21
10, −4
57. 0 = y 2 − 28y + 27
0, − 72
43. −n = 3n2
1
0, − 12
54. n2 − 6n − 40 = 0
10, −5
31. 42r + 12r2 = 0
39. y 2 = 20y
0, 9
−2, 5
53. a2 − 5a − 50 = 0
0,
0, −1
0,
20. 0 = 7k 2 + 6k
10
3
27. 0 = 25y 2 − 10y
35. −5a2 = 5a
0, −1
42. −12a2 = a
1
2
16. y − 5y 2 = 0
0, − 13
19. 0 = 10r − 3r2
30. 0 = 8y 2 + 20y
5
3
0,
12. 0 = 10c + 10c2
0, 1
4
3
0,
26. 12m − 21m2 = 0
0, 1
0,
6.
124—Solve over IR.
3.
0, 5
0, 1
21. 0 = 10a2 − 5a
41. r = 2r2
5y − y 2 = 0
10. 3y − 3y 2 = 0
0, − 25
37. p2 = −14p
2.
26—Solve by any method.
0, −1
17. 2x + 5x2 = 0
33. 8k = 8k 2
23—Solve by factoring.
71. c2 + 18c + 32 = 0
64. 0 = x2 + 3x + 20
Ø
68. 0 = p2 − 6p − 55
11, −5
72. 0 = y 2 − 12y + 32
8, 4
−16, −2
75. 0 = r2 − 11r − 60
15, −4
79. a2 + 21a + 90 = 0
−15, −6
83. 6a2 + 48a − 54 = 0
−1, 9
87. 0 = 2x2 + 22x + 48
−8, −3
91. 2y 2 + 4y − 96 = 0
−8, 6
ALG catalog ver. 2.6 – page 204 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
76. k 2 + 16k − 36 = 0
−18, 2
80. y 2 − 19y + 88 = 0
11, 8
84. 8p2 − 8p − 48 = 0
−2, 3
88. 3w2 − 45w + 42 = 0
14, 1
92. 0 = 4x2 − 28x − 240
12, −5
JB
93. 0 = 6k 2 + 84k + 270
94. 2x2 + 34x + 132 = 0
−9, −5
95. 3c2 − 72c + 240 = 0
−11, −6
97. 0 = 4x2 + 8x − 480
20, 4
98. 0 = 2c2 − 16c − 168
−12, 10
129. 81 = c2
126. 9 − c2 = 0
±5
130. 144 = r2
±9
137. 63 − 7p2 = 0
±3
138. 24y 2 − 24 = 0
145. a2 + 25 = 0
146. 0 = y 2 + 100
132. w2 = 100
±20
139. 0 = −15m2 + 60
±1
143. 14 − 14a2 = 0
±7
147. −c2 = 64
Ø
150. 36 − 12m + m2 = 0
−2
128. 0 = p2 − 36
±4
±7
135. x2 − 400 = 0
±11
142. 0 = 196 − 4m2
Ø
149. w2 + 4w + 4 = 0
131. a2 = 49
±12
134. 0 = 121 − k 2
30, −10
127. 0 = 16 − m2
±3
±14
±11
124. 0 = 300 − 20y − y 2
−50, 10
133. 0 = 196 − y 2
141. 0 = −2w2 + 242
−25, −8
123. 0 = 500 + 40k − k 2
−20, 12
125. a2 − 25 = 0
120. 0 = c2 + 33c + 200
30, 6
122. 240 + 8a − a2 = 0
20, −7
12, −8
119. y 2 − 36y + 180 = 0
16, 10
121. 140 − 13x − x2 = 0
116. 0 = r2 − 4r − 96
−15, 7
118. 0 = a2 − 26a + 160
−10, −15
27, −1
115. 0 = a2 + 8a − 105
−20, 4
117. x2 + 25x + 150 = 0
112. w2 − 26w − 27 = 0
−38, 2
114. 80 + 16n − n2 = 0
−40, 3
−10, −9
111. 0 = k 2 + 36k − 76
−18, 2
113. 120 + 37k − k 2 = 0
108. 0 = m2 + 19m + 90
−20, −6
110. 36 + 16y − y 2 = 0
46, −1
16, −5
107. w2 + 26w + 120 = 0
56, 2
109. 0 = 46 − 45a − a2
104. 0 = x2 − 11x − 80
20, −4
106. 0 = y 2 − 58y + 112
30, 2
−9, 8
103. a2 − 16a − 80 = 0
−18, 3
105. r2 − 32r + 60 = 0
100. 6p2 + 6p − 452 = 0
32, −2
102. c2 + 15c − 54 = 0
−17, 3
14, 3
99. 3y 2 − 90y − 192 = 0
14, −6
101. 0 = m2 + 14m − 51
96. 0 = 4p2 − 68p + 168
6
±2
±10
136. c2 − 169 = 0
±13
140. 0 = 48 − 3r2
±4
144. 7x2 − 175 = 0
±1
148. −49 = k 2
Ø
151. 0 = 25 + 10r + r2
±6
−5
±5
Ø
152. 0 = k 2 − 20k + 100
10
153. 0 = 9 − 6a + a2
154. 0 = a2 + 24a + 144
3
155. n2 − 14n + 49 = 0
7
156. 81 + 18c + c2 = 0
−9
−12
157. 0 = 128 + 32w + 2w2
158. 0 = 8y 2 − 16y + 8
1
−8
160. 3x2 − 24x + 48 = 0
4
−11
161. 0 = x2 − 40x + 400
162. k 2 − 18k + 81 = 0
9
163. r2 + 26r + 169 = 0
164. 0 = y 2 + 16y + 64
−8
−13
20
165. p2 + 5p + 25 = 0
166. 0 = u2 − 2u + 4
Ø
169. 7w2 − 8w + 1 = 0
1,
1
7
173. 0 = 2a2 + 7a + 3
−3, − 12
177. 6p2 − 11p − 7 = 0
7
3
181. 0 = 3 + 13m + 4m2
−3, − 14
Ø
170. 0 = 11k 2 + 10k − 1
−1,
− 12 ,
159. 0 = 242 + 44y + 2y 2
1
11
174. 3r2 + 20r − 7 = 0
−7,
1
3
178. 8x2 − 14x + 5 = 0
1 5
2, 4
182. 0 = 2 + 7y − 15y 2
2
1
3,−5
167. 0 = 9 + 3x + x2
Ø
171. 0 = 2a2 + 7a + 5
168. 49 − 7a + a2 = 0
Ø
172. 3x2 − x − 2 = 0
1, − 23
−1, − 52
175. 2c2 − 23c + 11 = 0
11,
1
2
179. 0 = 21h2 − 4h − 1
1
1
3,−7
183. 11 + 23x + 2x2 = 0
−11, − 12
ALG catalog ver. 2.6 – page 205 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
176. 0 = 5x2 − 64x − 13
13, − 15
180. 0 = 24y 2 − 25y + 1
1,
1
24
184. 3 − 4x − 7x2 = 0
−1,
3
7
JB
185. 10x2 + 23x − 5 = 0
− 52 , 15
189. 0 = 3t2 + 14t + 15
−3, − 53
193. 0 = 20c2 + 11c − 3
1
3
5,−4
197. 2y 2 − 27y + 36 = 0
12,
3
2
201. 0 = 10r2 − 81r + 8
8,
1
10
205. 3 − 23p + 30p2 = 0
1 3
6, 5
209. 35a2 − 31a + 6 = 0
3 2
5, 7
186. 0 = 14m2 + m − 3
3
1
7,−2
190. 25y 2 − 20y + 3 = 0
194. 0 = 18a2 − 5a − 7
7
9
198. 3p2 + 40p + 48 = 0
−12, − 43
202. 9c2 + 16c − 4 = 0
−2,
2
9
206. 0 = 25 − 5w − 2w2
−5,
5
1
3,−5
191. 4r2 − 24r + 11 = 0
1 11
2, 2
1 3
5, 5
− 12 ,
187. 0 = 15c2 − 22c − 5
5
2
210. 0 = 18y 2 + 51y + 8
− 83 , − 16
195. 12y 2 + 7y − 5 = 0
−1,
5
12
199. 0 = 40c2 − 11c − 2
1
2
5,−8
203. 14p2 − 23p + 8 = 0
1 8
2, 7
207. 0 = 7 − 17a + 10a2
1,
7
10
211. 12x2 − 44x + 35 = 0
5 7
2, 6
ALG catalog ver. 2.6 – page 206 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
188. 8u2 + 11u + 3 = 0
− 38 , −1
192. 0 = 12m2 + 32m + 13
− 12 , − 13
6
196. 7x2 − 11x − 30 = 0
3, − 10
7
200. 0 = 42h2 + h − 5
5
1
3 , − 14
204. 0 = 6m2 − 13m − 15
3, − 56
208. 18 + 5x − 2x2 = 0
−2,
9
2
212. 0 = 21r2 − 4r − 12
− 23 ,
6
7
JC
Topic:
Advanced quadratic equations (factorable). See also category JB.
Directions:
15—Solve.
23—Solve by factoring.
26—Solve by any method.
1.
s2 + 32 = 153
±11
2.
u2 = 2u2 − 49
5.
x2 + 3x = 10
−5, 2
6.
−y 2 = 15 − 16y
±7
15, 1
124—Solve over IR.
3.
4a2 + 25 = 5a2
±5
4.
−11 = −m2 + 70
7.
12 − m2 = 4m
−6, 2
8.
x2 + 20 = −21x
±9
−20, −1
9.
t2 = −2t + 24
10. 30 − r2 = −r
−6, 4
−5, 6
11. 70 = −17y − y 2
12. a2 = 11a − 28
7, 4
−10, −7
13. r2 − 6r = −9
14. −x2 + 10x = 25
3
17. 3y 2 − 8y = −5
1,
18. 2 + 21a = 11a2
5
3
5
1
2, − 11
15. d2 + 16 = −8d
16. 12z + 36 = −z 2
−4
19. 7w2 + 3 = −10w
20. 16k = 5k 2 + 3
−6
3,
1
5
−1, − 37
21. −8k 2 = −1 − 2k
22. 5x = 6x2 − 1
1, − 16
23. c − 3c2 = −10
24. 13r − 2 = 15r2
2, − 53
1 2
5, 3
− 14 , 12
25. c2 − 3c − 50 = 4
26. 10 = 46 + 16x − x2
−6, 9
−2, 18
29. 4 − 3w = 46 − 20w + w2 30. k 2 + 30k = 13k − 52
−13, −4
14, 3
33. −2c2 − 8 = 56 − 3c2
34. p2 + 10p = 10p + 144
±8
±12
37. 13r2 − 28 = 4r2 − 12
38. 32t2 − 46 = 7t2 − 45
± 43
± 15
41. −2c2 − 3c = 10c + 21
42. 10 − a = 3a2 − 2a
−3, − 72
2, − 53
45. 44 + 27x + 2x2 = x2 − 6
49. t2 − 10t + 15 = 10t − 85
28. a2 − 27a − 68 = 5a
−7, 8
−2, 34
31. 2y − 15 = y 2 + y − 105
32. x − 120 = 3x − x2
10, −9
12, −10
35. 3h + 10 = h2 + 3h + 1
36. 4a2 + 48 = 5a2 − 52
±3
±10
39. 25x2 − 13x + 4 = 7x
40. −16a2 − 6a = 18a + 9
2
5
− 34
43. 5x2 − 30 = 19x − 6x2
−1,
44. 2p2 − 24 = 9p − 6
30
11
6, − 32
46. 3p2 − 5p = 4p2 − 35p + 56
−2, −25
47. −4a2 + 3 = −3a2 + 18a − 60
27. 56 + 3p − p2 = 2p
−21, 3
28, 2
48. 21 − 11x − 2x2 = −x2 − 21
−14, 3
50. 3y 2 + 4y + 49 = 2y 2 − 10y
10
51. −u2 + 8u = −2u2 + 32u − 144
53. −k 2 + 7k + 3 = −20k + 75
12
3, 24
52. 5z − 47 = z 2 + 21z + 17
−7
−8
54. 64 + 3h − 2h2 = −h2 − 44
−9, 12
55. 6c2 − 5c = 5c2 + 3c + 105
−7, 15
56. 100 − 5x = 16 − 22x + x2
−4, 21
57. 4p2 + 16p − 49 = 16p + 32
± 92
58. −24t2 − t + 36 = 25t2 − t
± 67
60. −45d − 74 = 4d2 − 9d + 7
− 92
59. 5y 2 − 6y − 1 = −4y 2 − 2
1
3
61. 3x2 − 21x + 21 = x2 − 4x
7,
3
2
63. −4p2 − 44p − 12 = −7p2 + 3
65. 22 − p = 6p2 + 4p + 18
4
1
2,−3
67. 36a2 − 25a + 2 = 8a − 5
69. 9k 2 + 5k + 1 = −25k − 24
7 1
12 , 3
− 53
15, − 13
62. 12 − 16y = 13y 2 + 44y − 13
−5,
64. 6 − 11m = 8m2 + 19m + 13
− 14 , − 72
66. 3n2 + 12 = 9 − 22n − 21n2
68. 7y 2 − 1 = −3y 2 − 11y + 5
− 34 , − 16
− 32 ,
70. 72a − 2 = −100a2 + 12a − 11
ALG catalog ver. 2.6 – page 207 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
5
13
2
5
3
− 10
JC
71. 66p2 − 14 = −15p2 + 18p − 15
73. 35 − 3w + w2 = −50w − 5w2
75. 10 + 4u2 + 8u = 25u2 + 7u
77. x(x + 4) = 12
1
9
−7, − 56
11
4
74. −2r2 − 2r = 6r2 + 21r + 14
−2, − 78
76. 12u + 5 = −26u2 + u + 20
5
2
7,−3
78. −8 = x(x − 6)
2, −6
72. −16t2 + 90t − 22 = 2t + 99
2, 4
79. 0 = k(8 + k) + 15
3, 5
−1,
15
26
80. 21 − b(b − 20) = 0
21, −1
81. x(7 − 9x) − 10x = 0
82. 0 = 8y − 3y(y + 6)
0, − 13
0, − 10
3
85. 5(x − 3) = 2x(x − 3)
3,
0,
86. x(x − 5) + 5(5 − x) = 0
5
5
2
89. 0 = 4x(4x − 1) − 4x + 1
91. x(x + 3) − 5x = x + 10
83. −4x(x − 5) = 11x
84. 2x = x(7x + 11)
87. 0 = 2x(3x + 4) − (x − 2) 88. 2(10 − x) = (x − 3)(x)
5, −4
−1, − 23
90. 4(x + 6) + 3x = 2(x2 − 3)
1
4
92. 3a − 15 + 2(a2 − 5a) = 0
5, −2
6, − 52
5, − 32
93. 3(x2 + x + 1) − x(x − 2) = x2 + x
−1, −3
94. 6x2 = x(9x + 13) − 2(x2 − x + 8)
95. 8(2x2 + 3x − 2) = 4x(3x + 7) − 16
1, 0
96. −3x(2x − 5) + 5(x2 − 3x + 6) = 5
97. (r − 2)(r − 2) = 36
98. 49 = (p + 5)(p + 5)
8, −4
101. 121 = (m + 6)2
2, −12
5, −17
8, −2
105. (x − 3)(x − 4) = 2
2, −6
109. (2x − 3)(2x + 3) = 7
± 53
113. (x + 1)(2x + 1) = 3
−2,
117. (4a − 1)2 + 2 = 8a
−1,
2
3
107. −4 = (x + 2)(x + 7)
111. 24 = (4x + 1)(4x − 1)
115. (2m − 1)(m − 3) = 18
119. (p + 3)2 − p = 15
122. −y − 14 = (y + 4)(3y − 2)
−2, − 32
−4, −5
112. 5 = (11x − 2)(11x + 2)
3
± 11
116. (4n − 3)(2n + 1) = 7
−1,
1, − 12
−3, − 23
2
124. 4y + 23 = (3y + 5)(y + 3)
125. (x − 3)(x + 3) = 2x2 − 18
±3
126. 4a2 + 10a − 24 = (3a − 5)(a + 4)
129. (2n − 6)2 = 10(2n − 6) − 25
−2, −4
131. 33 + 5(2y − 7) = 9y + 3(y − 2)2
2,
133. 5(x2 + 1) − (4x − 1)(x − 1) = 0
−1, −4
2
3
1, −4
2, 14
−1
132. 10n − (2n − 3)2 = (n + 4)(n − 4) + 15
7
3
135. 11r + (r + 2)(2 − r) = 33 − (4r − 3)2
−4,
128. x2 + 9x − 43 = (2x + 3)(x − 5)
130. 0 = (2x − 1)2 − 3x(x − 2)
11
2
5
4
120. 3x2 + x = (x + 1)2
123. (y + 8)(y − 4) = 2y 2 − 28
127. (3x − 2)(2x + 5) = 5x2 + 5x − 18
6
2
5,−5
108. (x + 8)(x + 1) = −12
1, −6
0, − 53
121. (x + 2)(x − 6) = 5x2 + 10x
104. (5w − 2)2 = 16
5, − 32
118. 25 − 8x2 = (2x + 5)2
1 3
4, 4
±8
± 54
114. −2 = (x − 1)(3x + 4)
1
2
±5
100. 55 = (y − 3)(y + 3)
−3, −6
110. (3x + 7)(3x − 7) = −24
±2
103. 0 = (3w + 2)2 − 9
1, −16
1
5
3,−3
106. 7 = (x − 1)(x + 5)
2, 5
99. (x − 8)(x + 8) = 17
±9
102. (x − 3)2 − 25 = 0
0, − 97
9
4
134. 6 − (3x − 4)(x + 3) = −2x(2x + 8)
5
4
3,−5
136. (x − 7)(2x + 5) − (x − 4)2 = 9x − 27
ALG catalog ver. 2.6 – page 208 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
4,
2
5
−2, −9
−2, 12
JD
Topic:
Higher order equations (factorable).
Directions:
15—Solve.
1.
15n = −n3
5.
3k 4 + 9k 3 = 0
−3, 0
9.
−8p4 = −12p3
3
2,0
0
13. n3 − 9n = 0
17. 4x4 = 16x2
r4 + 8r2 = 0
6.
−7x4 = 28x3
0
−4, 0
10. 0 = 20c3 − 45c4
14. y 3 = 49y
±3, 0
22. 9p2 = 49p4
± 15 , 0
25. a3 + 15a2 − 16a = 0
4
9,0
±5, 0
± 37 , 0
26. c4 − 3c3 − 28c2 = 0
−16, 1, 0
7, −4, 0
29. 0 = w4 − 18w3 + 72w2
−2, −18, 0
33. 0 = 7h3 − 7h2 − 42h
38. 0 = s3 + 14s2 + 49s
−7, 0
3, 0
41. 7a3 − 5a2 − 2a = 0
−5, − 17 , 0
45. 5x3 − 14x2 + 8x = 0
−1,
49. 0 = 20y 4 − 16y 3 + 3y 2
1
10 , 0
54. −y 4 = 19y 3 − 42y 2
−8, 7, 0
61. y 4 − 16 = 0
± 94 , 0
27. 0 = k 4 + 16k 3 + 63k 2
24. 144w3 = w
62. −x4 = −81
65. 0 = a4 − 26a2 + 25
±2, ±3
69. 81k 4 − 18k 2 + 1 = 0
70. 0 = 16n4 − 8n2 + 1
± 13
75. t3 + 2t2 − 9t − 18 = 0
7, ±2
−2, ±3
77. 0 = a3 + 5a2 + 3a + 15
1
−5
1
± 12
,0
−10, 5, 0
31. p3 − 13p2 − 68p = 0
32. m4 + m3 − 110m2 = 0
−11, 10, 0
35. 3d3 + 33d2 + 72d = 0
36. 0 = 6y 4 + 12y 3 − 48y 2
−4, 2, 0
39. 0 = t4 − 10t3 + 25t2
40. z 3 + 20z 2 + 100z = 0
−10, 0
43. 2b4 − 5b3 + 3b2 = 0
44. 0 = 3n3 + 8n2 − 11n
3
2,0
1, − 11
3 ,0
47. 0 = 14p3 + p2 − 3p
48. 0 = 6x4 + 7x3 + 2x2
− 23 , − 12 , 0
51. 0 = 7w − 3w2 − 22w3
52. 5n2 + 18n3 + 16n4 = 0
− 12 , − 58 , 0
55. 42a2 = a3 + a4
−7, 6, 0
56. d3 = 25d2 + 54d
59. g 3 − 23g 2 = 140g
60. 132y + 41y 2 = y 3
63. 1 − 81h4 = 0
−3, 44, 0
64. 16m4 = 1
± 13
67. 0 = x4 − 20x2 + 64
71. y 4 − 2y 2 + 1 = 0
±1, ±3
72. 0 = u4 − 8u2 + 16
±1
74. 0 = r3 + 8r2 − r − 8
−8, ±1
76. 0 = d3 − 3d2 − 25d + 75
3, ±5
78. m3 − 10m2 + 6m − 60 = 0
80. r3 + 2r2 + 9r + 18 = 0
ALG catalog ver. 2.6 – page 209 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
± 12
68. w4 − 10w2 + 9 = 0
±2, ±4
± 12
73. x3 − 7x2 − 4x + 28 = 0
79. 0 = y 3 − y 2 + 4y − 4
±3
66. c4 − 13c2 + 36 = 0
±1, ±5
±8, 0
28. 0 = x3 − 5x2 − 50x
−5, 28, 0
2, 72, 0
±2
±6, 0
27, −2, 0
58. −144n2 = n4 − 74n3
−10, −11, 0
− 16
3 ,0
20. 5u4 − 320u2 = 0
±4, 0
−21, 2, 0
57. 21x3 + 110x2 = −x4
7, 0
16. −r4 = −36r2
±10, 0
7
1
2 , − 11 , 0
− 13 , − 15 , 0
53. y 3 + y 2 = 56y
42h3 = 6h4
0
12. 6s4 + 32s3 = 0
− 56 , 0
23. 16u4 − 81u2 = 0
8.
− 12 , 37 , 0
50. y + 8y 2 + 15y 3 = 0
1 3
2 , 10 , 0
19. −3s3 = −48s
1,
46. 10r4 + 9r3 − r2 = 0
2, 45 , 0
11. 18y 3 = −15y 2
5, 0
0 = t4 + 12t2
5, 0
42. 0 = 5h2 + 36h3 + 7h4
1, − 27 , 0
0 = 11x4 − 55x3
4.
−8, −3, 0
1, 9, 0
37. r4 − 6r3 + 9r2 = 0
7.
0
−4, 17, 0
34. 5x4 − 50x3 + 45x2 = 0
−2, 3, 0
s3 = −6s
−9, −7, 0
30. 0 = c3 + 20c2 + 36c
6, 12, 0
3.
15. 0 = c4 − 100c2
±7, 0
18. 0 = −7t3 + 175t
±2, 0
21. 0 = y − 25y 3
2.
−2
10
±2
JE
Topic:
Solving for other variables (factorable equations). See also categories GH (first degree equations) and LM
(rational equations).
Directions:
15—Solve.
1.
A = P + Prt; for P
2.
A
1 + rt
5.
bx = c + ax; for x
6.
10
C +1
13. A =
2`w + 2wh + 2`h; for h
2V − IR
; for I
2I
r` − a
; for r
r−`
29. x2 − 3ax = 0; for x
nE − rI
; for n
nI
r` − a
; for `
r−`
30. ay 2 − y = 0; for y
2D
2t − `
V − 3c
3c + 3
S − 2bc
2b + 2c
Rr
; for R
R−r
20. I =
ad
; for a
a+c−d
27. ay + 1 = y(b − c); for y
24. F =
0, 3a
31. at2 + bt = 0; for t
0, −
33. y 2 − 6ay + 5a2 = 0; for y 34. 2x2 + bx − b2 = 0; for x
b
2
ad
; for d
a+c−d
Fa + Fc
F +a
28. d(n + 1) = c(d − 1); for d
c
c−n−1
1
−
a−b+c
1
a
nE
; for n
R + nr
IR
E − Ir
Fd − Fc
F −d
0,
2at − a`
; for a
2
12. D =
15. S = 2(ab + bc + ac); for a 16. V = 3(c + cd + d); for d
23. F =
c+y
a+b
−b,
180(n − 2)
; for n
n
kr
K −r
26. ax − y = c − bx; for x
a, 5a
bz − cd = az; for z
8.
cd
b−a
19. K =
a + sr
s+r
y−c
a−x
r
1−r
360
180 − a
rI
E − RI
22. s =
an = cm − bn; for n
11. a =
A − 2wh
2w + 2h
s` − a
s−`
25. ab + c = xb + y; for b
T1 − T 2
; for T1
T1
18. R =
S − Sr = r; for S
4.
cm
a+b
14. A =
2`w + 2wh + 2`h; for `
2V
2r + R
21. s =
7.
T2
1−E
A − 2`w
2w + 2`
17. r =
c + ay = dy; for y
10. E =
Ft = mv1 − mv2 ; for m
Ft
v1 − v2
c
d−a
10 − r
; for r
r
C=
3.
E
r+R
c
b−a
9.
E = Ir + IR; for I
32. w2 + 2bw = 0; for w
0, −2b
b
a
35. c2 − 2cd − 3d2 = 0; for c 36. p2 − 5pw + 4w2 =
0; for w p, p
−d, 3d
4
37. k 2 − ak − k + a = 0; for k 38. n2 + an − bn − ab =
0; for n −a, b
a, 1
39. y 2 − cy − dy + cd =
0; for y c, d
40. a2 + ac + a + c = 0; for a
41. 16a2 − b2 = 0; for a
43. 0 = x2 − 36y 2 ; for y
44. −9p2 + q 2 = 0; for q
±
42. 0 = 4x2 − c2 ; for c
45. 0 = y 2 + 2ay + a2 ; for y
46. c2 − 2cd + d2 = 0; for c
−a
49. x2 − 8rx = 20r2 ; for r
x
±
6
±2x
b
4
d
53. x2 + 4x − 3xy − 12y = 0; for x
a
a
,−
8
3
−4, 3y
55. 2my + 5m − 8y 2 − 20y = 0; for y
±3p
47. x2 − 4xy + 4y 2 = 0; for x 48. 0 = a2 + 6ab + 9b2 ; for b
2y
50. a2 = 24t2 + 5at; for t
x
x
,−
10
2
−c, −1
5
m
,−
4
2
51. 5cy = y 2 − 14c2 ; for y
−
52. d2 + 12x2 = 8dx; for d
7c, −2c
54. 3s2 − 6s + st − 2t = 0; for s
2x, 6x
2, −
56. 4n2 t − nt + 12n − 3 = 0; for n
ALG catalog ver. 2.6 – page 210 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
a
3
t
3
1
3
,−
4
t
JF
Topic:
Word problems involving factoring. See also category NH (quadratic formula).
Directions:
0—(No explicit directions.) 16—Solve and check.
40—Write an equation and solve.
39—Translate and solve.
1.
Find a positive number whose square is 12 more
than the number itself. 4
2.
Find a negative number whose square is 20 more
than the number itself. −4
3.
Find a negative number whose square is 2 more than
the number itself. −1
4.
Find a positive number whose square is 30 more
than the number itself. 6
5.
The sum of two numbers is 19. The sum of their
squares is 193. Find the numbers. 7, 12
6.
The sum of two numbers is 14. The sum of their
squares is 100. Find the numbers. 6, 8
7.
The sum of two numbers is 3. The sum of their
squares is 89. What are the numbers? 8, −5
8.
The sum of two numbers is 16. The sum of their
squares is 416. What are the numbers? 20, −4
9.
The difference of two numbers is 6. The sum of their
squares is 116. Find the numbers. 4, 10
10. The difference of two numbers is 12. The sum of
their squares is 170. Find the numbers. 1, 13
11. A positive number is one-half of another number.
The sum of the numbers is 28 less than the square of
the smaller number. What are the numbers? 7, 14
12. A positive number is one-third of another number.
The sum of the numbers is 5 less than the square of
the larger number. What are the numbers? 1, 3
13. When the square of a number is subtracted from
11 times the number, the difference is 18. Find the
number. 9
14. When the square of a number is subtracted from
8 times the number, the difference is 12. Find the
number. 6
15. Seven less than the square of a positive number is the
same as 6 times the number. What is the number?
16. 22 less than the square of a positive number is the
same as 9 times the number. What is the number?
7
11
17. If twice the square of a positive number is decreased
by 5 times the number, the difference is 12. Find the
number. 4
18. If one-half the square of a positive number is
decreased by twice the number, the difference is 16.
Find the number. 8
19. Five times the square of a negative number is
48 more than the number. What is the number?
20. Eight times a negative number is 20 less than the
square of the number. What is the number? −2
−3
Consecutive integers
21. Find two consecutive integers whose product is 306.
17, 18 or −18, −17
23. The product of two consecutive integers is 210. Find
two pairs of numbers that satisfy this condition.
14, 15 or −15, −14
22. Find two consecutive integers whose product is 182.
13, 14 or −14, −13
24. The product of two consecutive integers is 380. Find
two pairs of numbers that satisfy this condition.
19, 20 or −20, −19
25. Find two consecutive even integers whose product
is 288. 16, 18 or −18, −16
26. Find two consecutive odd integers whose product
is 143. 11, 13 or −13, −11
27. Find two consecutive odd integers whose product
is 168. not possible
28. Find two consecutive even integers whose product
is 255. not possible
29. Find two consecutive odd integers such that the sum
of their squares is 130. 7, 9 or −9, −7
30. Find two consecutive even integers such that the
sum of their squares is 244. 10, 12 or −12, −10
31. The square of the sum of two consecutive integers
is 529. What are the integers? 11, 12 or −12, −11
32. The square of the sum of two consecutive integers
is 1681. What are the integers? 20, 21 or −21, −20
33. Find two consecutive odd integers such that the sum
of their squares is 164. not possible
34. Find two consecutive even integers such that the
square of their sum is 225. not possible
35. Find two consecutive positive integers such that the
difference of their squares is 25. 12, 13
36. Find two consecutive negative integers such that the
difference of their squares is 29. −14, −15
ALG catalog ver. 2.6 – page 211 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
JF
37. Find two consecutive odd integers such that the
square of the first, increased by the second, is 32.
5, 7
38. Find two consecutive even integers such that the
square of the second, decreased by the first, is 92.
8, 10
39. When the first of two consecutive integers is added
to the square of the second, the result is 5. What
are the integers? −4, −3 or 1, 2
40. When the first of two consecutive integers is added
to the square of the second, the result is −1. What
are the integers? −2, −1 or −1, 0
41. The product of two consecutive integers is 8 more
than twice their sum. What are the integers?
42. The sum of two consecutive integers is 19 less than
half their product. What are the integers?
5, 6 or −2, −1
8, 9 or −5, −4
43. Find two consecutive even integers such that the
square of the smaller is 4 more than 10 times the
larger. 12, 14 or 0, −2
44. Find two consecutive odd integers such that the
square of the larger is 3 less than 12 times the
smaller. 7, 9 or 1, 3
45. Twice the square of an integer is 20 more than the
product of the integer and the next consecutive
integer. What are the integers? 5, 6 or −4, −3
46. Half the square of an integer is 15 more than the
sum of the the integer and the next consecutive
integer. What are the integers? 8, 9 or −4, −3
47. Find three consecutive integers such that the square
of the third, added to the first, is 130.
48. Find three consecutive integers such that the square
of the second, added to the third, is 133.
9, 10, 11 or −14, −13, −12
10, 11, 12 or −13, −12, −11
49. Find three consecutive odd integers such that the
first times the second is 8 more than 5 times the
third. 7, 9, 11
50. Find three consecutive even integers such that the
first times the third is 4 less than 12 times the
second. 10, 12, 14
51. Find four consecutive positive integers such that the
sum of the squares of the second and third is 85.
52. Find four consecutive negative integers such that the
square of the sum of the first and fourth is 289.
5, 6, 7, 8
−10, −9, −8, −7
53. Find four consecutive even integers such that the
product of the first and second is 10 less than the
sum of the third and fourth. 0, 2, 4, 6
54. Find four consecutive odd integers such that the sum
of the first three integers is 10 less than the square
of the fourth. −5, −3, −1, 1
55. Find three consecutive integers such that the product
of the first and third is 1 less than the square of the
middle integer. any three consec. integers
56. Find four consecutive integers such that the product
of the first and fourth is 2 less than product of the
middle integers. any four consec. integers
Area, perimeter
57. A rectangular driveway is 12 meters longer than it is
wide. Its area is 1260 square meters. Find its length
and width. 42, 30 m
58. A rectangular garden is 3 feet longer than it is wide.
Its area is 54 square feet. Find its length and width.
59. The area of a rectangular carpet is 165 sq ft. The
width is 4 ft less than the length. What are the
dimensions? 11 × 15 ft
60. The area of a rectangular parking lot is 1800 sq m.
The width is 14 m less than the length. What are
the dimensions? 36 × 50 m
61. The length of a rug is twice its width. Its area is
12.5 m2 . Find the dimensions of the rug. 2.5 × 5 m
62. The length of a corridor is 4 times its width. Its
area is 121 ft2 . Find the dimensions of the corridor.
9, 6 ft
5.5 × 22 ft
63. The width of a rectangle is one-third of the length.
If the area is 108 cm2 , what is the width and length?
6, 18 cm
64. The width of a rectangle is one-half of the length. If
the area is 128 sq ft, what is the width and length?
8, 16 ft
65. The length of a rectangle is 2 inches more than
three times the width. Find the length and width if
the area is 85 in2 . 17, 5 in.
66. The length of a rectangle is 5 cm less than twice
the width. Find the length and width if the area is
88 sq cm. 11, 8 cm
67. The width of a rectangular walkway is 12 ft less than
half the length. What are the dimensions of the
walkway if the area is 1080 ft2 ? 18 × 60 ft
68. The width of a rectangular playground is 4 meters
more than one-third of the length. What are
the dimensions of the playground if the area is
3060 sq m ? 34 × 90 m
ALG catalog ver. 2.6 – page 212 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
JF
69. The perimeter of a building is 112 m. It covers an
area of 720 sq m. Find the dimensions of the building
(assume it is rectangular). 20 × 36 m
70. The perimeter of a garden is 34 ft. It covers an area
of 60 ft2 . Find the dimensions of the garden (assume
it is rectangular). 5 × 12 ft
71. A rectangular piece of wood has an area of 315 sq in.
and a perimeter of 81 in. Find its length and width.
72. A rectangular counter top has an area of 6 m2 and a
perimeter of 11 m. Find its length and width.
30, 10.5 in.
4, 1.5 m
73. The base of a triangle is 4 in. more than the height.
The area is 6 sq in. Find the base and height. 6, 2
74. The base of a triangle is 5 m less than the height.
The area is 33 m2 . Find the base and height. 6, 11
75. The height of a triangle is 3 cm more than twice the
base. The area is 45 cm2 . Find the base and height.
76. The height of a triangle is 1 foot less than twice the
base. The area is 95 sq ft. Find the base and height.
6, 13
10, 19
77. The length and width of 3 × 5 in. rectangle are both
increased by the same amount in order to form a
rectangle with an area of 48 sq in. By how much
were the length and width increased? 3 in.
78. The length and width of 4 × 9 cm rectangle are both
increased by the same amount in order to form a
rectangle with an area of 66 cm2 . By how much were
the length and width increased? 2 cm
79. The length and width of 10 × 12 ft rectangle are both
decreased by the same amount in order to form a
rectangle with an area of 35 sq ft. By how much were
the length and width decreased? 5 ft
80. The length and width of 8 × 20 m rectangle are both
decreased by the same amount in order to form a
rectangle with an area of 28 m2 . By how much were
the length and width decreased? 6 ft
81. A room is 2 ft longer than it is wide, and the ceiling
is 9 ft high. If the total area of the walls and ceiling
is 516 sq ft, find the dimensions of the room.
82. A room is 2 m longer than it is wide, and the ceiling
is 3 m high. If the total area of the walls and ceiling
is 84 m2 , find the dimensions of the room. 6 × 4 × 3 m
12 × 10 × 9 ft
83. A room is twice a long as it is wide, and the ceiling
is 8.5 ft high. If the total area of the walls and
ceiling is 621 sq ft, find the dimensions of the room.
18 × 9 × 8.5 ft
84. A room is twice a long as it is wide, and the ceiling
is 3.5 m high. If the total area of the walls and
ceiling is 198 m2 , find the dimensions of the room.
12 × 6 × 3.5 m
Borders
85. A photograph is 11 × 14 in. A frame of uniform
width is placed around the photograph. The area of
the frame is 150 in2 . Find the width of the frame.
86. A painting is 25 × 40 cm. A frame of uniform width
is placed around the painting. The area of the frame
is 504 cm2 . Find the width of the frame. 3.5 cm
2.5 in.
87. The outer dimensions of a picture frame are
20 × 12 cm. The area of the picture that is exposed
is 84 sq cm. What is the width of the frame? 3 cm
88. The outer dimensions of a picture frame are
14 × 18 in. The area of the picture that is exposed is
192 in2 . What is the width of the frame? 1 in.
89. A city lot is 16 m wide and 34 m long. How wide a
strip must be cut off one end and one side to make
the area of the lot 360 m2 ? 4 m
90. A piece of wood is 16 in. wide and 30 in. long. How
wide a strip must be cut off one end and one side to
make the area of the wood 275 sq in.? 5 in.
91. A sidewalk of uniform width is built around three
sides of a rectangular lot (one of the shorter sides is
left alone). The dimensions of the lot are 30 × 10 ft.
The total area of the lot and sidewalk is 528 ft2 .
What is the width of the sidewalk? 3 ft
92. A sidewalk of uniform width is built around three
sides of a rectangular building (one of the longer
sides is left alone). The dimensions of the building
are 40 × 28 m. The total area of the building and
sidewalk is 1320 m2 . What is the width of the
sidewalk? 2 m
93. A painting is 5 inches longer than it is wide. If a
2 inch border is added to all sides, then the total
area will be 414 sq in. What are the dimensions of
the painting itself? 14 × 19 in.
94. A photograph is 3 cm longer than it is wide. If a
4 cm border is added to all sides, then the total area
will be 460 sq cm. What are the dimensions of the
photograph itself? 12 × 15 cm
95. When a border of uniform width is placed around
an 8 × 10 in. photograph, the area is increased by
243 sq in. What is the width of the border? 4.5 in.
96. When a border of uniform width is placed around a
9 × 12 m carpet, the area is increased by 100 sq m.
What is the width of the border? 2 m
ALG catalog ver. 2.6 – page 213 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
JF
97. A rectangular swimming pool is 10 m wide and 15 m
long. The pool is surrounded by a deck of uniform
width, whose area is equal to the area of the pool.
How wide is the deck? 2.5 m
98. A rectangular garden is 12 ft wide and 15 ft long.
The garden is surrounded by a dirt path of uniform
width, whose area is half the area of the garden.
How wide is the path? 1.5 m
99. A table cloth is 30 inches long and 20 inches wide.
The cloth is going to be enlarged by adding a border
of uniform width to all sides. How wide should the
border be, in order to double the area of the cloth?
100. A warehouse platform is 10 meters long and 3 meters
wide. The platform is going to be enlarged by
adding a strip of uniform width to all sides. How
wide should the strip be, in order to double the area
of the platform? 1 m
5 in.
101. A rectangular patio is surrounded on three sides by a
fence (the remaining side is up against the house). If
the area of the patio is 150 ft2 , and the total length
of fence is 35 ft, what is the length and width of the
patio? 20, 7.5 ft
102. A rectangular patio is surrounded on three sides by
a fence (the remaining side is up against the house).
If the area of the patio is 45 m2 , and the total length
of fence is 19 m, what is the length and width of the
patio? 10, 4.5 m
FENCE1.PCX
FENCE1.PCX
103. A rectangular flower bed, whose dimensions are
4 × 11 m, has one of its longer sides against a house.
The remaining three sides are to be increased by a
strip of uniform width, so that the area of the garden
is increased by 75%. How wide should that strip be?
1.5 ft
104. A rectangular flower bed, whose dimensions are
6 × 15 ft, has one of its longer sides against a house.
The remaining three sides are to be increased by a
strip of uniform width, so that the area of the garden
is increased by 50%. How wide should that strip be?
1.5 ft
GARDEN1.PCX
GARDEN1.PCX
Volume
105. The length of a rectangular piece of sheet metal is
3 cm less than twice the width. A 4 cm square is cut
from each corner of the metal, and the sides are bent
up so that the metal forms an open box. If the
volume of the box is 208 cm3 , what are the original
dimensions of the metal? 12 × 21 cm
106. The length of a rectangular piece of sheet metal is
5 inches less than twice the width. A 2 inch square
is cut from each corner of the metal, and the sides
are bent up so that the metal forms an open box.
If the volume of the box is 132 in3 , what are the
original dimensions of the metal? 10 × 15 in.
MAKBOX2.PCX
MAKBOX2.PCX
107. A box is formed by cutting an 4 cm square from
each corner of a square piece of cardboard, and then
folding the sides. If the volume of the resulting box
is 64 cm3 , what is the original size of the cardboard?
12 × 12 cm
108. A box is formed by cutting an 3 inch square from
each corner of a square piece of cardboard, and then
folding the sides. If the volume of the resulting box
is 48 in3 , what is the original size of the cardboard?
10 × 10 in.
ALG catalog ver. 2.6 – page 214 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
JF
109. A rectangular piece of sheet metal is twice as long
as it is wide. Squares measuring 5 inches on a side
are cut from each corner of the metal, and the sides
are folded to form a box. If the volume of the box
is 2040 in3 , what are the original dimensions of the
sheet metal? 22 × 44 in.
111. A rectangular piece of cardboard is 6 cm longer
than it is wide. When an 2 cm square is cut out of
each corner, and the sides folded up, the resulting
box has a volume of 135 cm3 . What are the original
dimensions of the cardboard? 13 × 19 cm
110. A rectangular piece of sheet metal is three times
as long as it is wide. Squares measuring 2 cm on
a side are cut from each corner, and the sides are
folded to form a metal pan. If the volume of the pan
is 312 cm3 , what are the original dimensions of the
sheet metal? 10 × 30 cm
112. A rectangular piece of cardboard is 2 inches longer
than it is wide. When a 5 inch square is cut out of
each corner, and the sides folded up, the resulting
box has a volume of 70 in3 . What are the original
dimensions of the cardboard? 15 × 17 in.
Complementary & supplementary angles
113. Find two complementary angles which meet this
condition: the first angle measure is 18 ◦ more than
the square of the second angle measure. 82 ◦, 8 ◦
114. Find two complementary angles which meet this
condition: the first angle measure is 20 ◦ less than
the square of the second angle measure. 80 ◦, 10 ◦
115. Find two supplementary angles such that the square
of the first angle measure is 80 ◦ more than twice the
second angle measure. 20 ◦, 160 ◦
116. Find two supplementary angles such that the square
of the first angle measure is 5 ◦ more than four times
the second angle measure. 25 ◦, 155 ◦
117. The two angles are complementary. If a = x2 + 6x
and b = 10x + 10, solve for x and find the measure of
each angle. 4; 40, 50 ◦
118. If a = x2 and b = 10x + 15, solve for x and find the
measure of each angle. 5; 25, 65 ◦
COMPL02.PCX
119. The two angles are supplementary. If c = x2 + 12x
and d = 7x + 30, solve for x and find the measure of
each angle. 6; 108, 72 ◦
COMPL02.PCX
120. If c = x2 + 6x and d = x + 10, solve for x and find
the measure of each angle. 10; 160, 20 ◦
SUPPL02.PCX
ALG catalog ver. 2.6 – page 215 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
SUPPL02.PCX
KA
Topic:
Proportions.
Directions:
15—Solve.
1.
a
1
=
18
2
5.
w
32
=
40
25
9.
15
3
=
10
k
13.
45
12
=
r
120
17.
33—Solve each proportion.
2.
b
1
=
4
32
3.
y
3
=
8
88
33
4.
x
3
=
65
5
39
6.
x
27
=
36
20
7.
c
21
=
8
24
7
8.
a
48
=
16
7
21
10.
4
16
=
n
12
11.
12
4
=
18
r
6
12.
5
15
=
p
12
4
32
14.
48
12
=
x
112
28
15.
105
15
=
140
a
20
16.
65
15
=
104
c
24
28
y
=
4
36
252
18.
a
36
=
9
45
180
19.
h
33
=
121
11
363
20.
b
52
=
169
13
676
21.
42
25
=
c
75
126
22.
25
63
=
50
y
126
23.
4
52
=
36
d
24.
56
7
=
m
42
25.
w
2
=
6
15
4
5
26.
20
5
=
3
h
27.
8
p
=
40
2
2
5
28.
4
32
=
r
5
29.
9
6
=
4
x
30.
r
3
=
6
8
31.
5
3
=
n
8
40
3
32.
7
h
=
3
5
33.
15
p
=
28
7
34.
5
35
=
c
17
35.
a
5
=
26
12
65
6
36.
27
15
=
y
7
63
5
37.
y
0.2
=
30
3
38.
6
12
=
0.5
x
39.
56
7
=
n
0.5
4
40.
0.3
a
=
2
20
3
41.
3
36
=
0.25
c
3
42.
r
0.25
=
32
2
4
43.
0.75
x
=
9
48
4
44.
56
7
=
y
0.75
45.
18
6
=
x
0.8
2.4
46.
0.6
y
=
4
18
2.7
47.
r
0.7
=
12
4
2.1
48.
4
16
=
0.8
w
3.2
49.
a
58
=
6
0.9
8.7
50.
0.8
4
=
n
62
12.4
51.
4
0.7
=
64
y
11.2
52.
x
63
=
0.7
3
14.7
53.
k
3/4
=
5
20
3
54.
3
16
=
3/4
y
4
55.
x
1/4
=
12
3
1
56.
15
5
=
m
2/3
2
57.
24
6
=
x
5/8
5
2
58.
a
7/8
=
24
6
7
2
59.
5
28
=
3/7
r
12
5
60.
5/7
y
=
4
35
25
4
61.
p
2/3
=
6
1/2
62.
2/3
1/3
=
x
9
63.
w
2/5
=
1/4
15
24
64.
k
16
=
3/8
2/3
65.
12
a
=
9/4
5/8
66.
c
3/8
=
16
5/9
67.
1/3
3/4
=
x
21
68.
2/3
12
=
3/4
y
69.
3
3x
=
4
16
4
70.
4b
2
=
18
3
3
71.
2
3
=
24
4y
72.
2
2
=
3a
15
73.
8
16
=
7
2c
7
74.
24
8
=
3y
5
5
75.
3
3a
=
8
16
76.
2d
3
=
28
7
77.
21
7
=
16n
4
78.
2
3
=
16
40x
79.
25w
5
=
12
6
80.
18
9y
=
7
2
9
20
2
8
3
15
4
2
8
2
3
3
4
8
15
3
3
4
9
4
17
7
1
18
54
5
3
5
ALG catalog ver. 2.6 – page 216 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
468
28
3
9
2
2
5
336
5
8
35
3
6
9
27
2
5
6
4
7
KB
Topic:
Writing and simplifying ratios.
Directions:
46—Write each ratio in simplest form.
1.
30 to 42
5:7
2.
51 to 18
5.
7 to 84
1 : 12
6.
90 to 6
9.
132 to 48
11 : 4
13. 50 to 5000
17.
1
2
to 5
3
4
to
5
8
33. 0.05 to 0.3
1 : 10
1:6
37. 20 cents to 1 dollar
2 : 15
4.
120 to 45
7.
11 to 99
1:9
8.
64 to 4
11. 90 to 162
14. 2 to 3000
1 : 1500
15. 150,000 to 75
26.
6:5
10 to 75
3 : 10
1
5
6
7
to
3
14
19. 7 to 3 12
15 : 1
22. 12 to 2 41
3:4
29. 3.2 to 32
15 : 1
3.
10. 75 to 250
18. 3 to
1 : 10
21. 7 12 to 10
25.
1 : 100
17 : 6
16 : 3
4:1
23.
4
5
27.
3
10
to
12. 168 to 72
5:9
2000 : 1
16. 300 to 90,000
20. 16 to 2 32
2:1
4:3
9
20
7:3
2:3
24.
2
3
28.
1
18
to
10
3
to
1 : 300
6:1
1:5
1
14
7:9
100 : 1
31. 1.5 to 450
1 : 300
32. 64 to 0.08
800 : 1
34. 2.4 to 0.6
4:1
35. 0.7 to 1.54
5 : 11
36. 8.4 to 10.5
4:5
38. 15 cents to 5 dollars
39. 1.25 dollars to 45 cents
3 : 100
40. 95 cents to 1 dollar
25 : 9
42. 50 cents to 3.5 dollars
1 : 1000
19 : 20
43. 200 cents to 1 dollar
1:7
45. 18 hours to 2 days
3
5
16 : 1
30. 75 to 0.75
1:5
41. 1 cents to 10 dollars
to
8:3
44. 8 dollars to 30 cents
2:1
80 : 3
46. 3 minutes to
112 seconds 45 : 28
47. 100 minutes to
24 hours 5 : 72
48. 1 hour to 1 week
49. 4 21 minutes to
312 seconds 45 : 52
50. 6 12 days to 44 hours
51. 24 seconds to 1 hour
52. 2.75 hours to
132 minutes 5 : 4
53. 100 m to 60 km
54. 2.5 kg to 750 g
3:8
57. 750 cm to 5 m
1 : 600
3:2
61. 0.3 km to 1500 cm
39 : 11
1 : 150
10 : 3
58. 32 cm to 0.32 m
1:1
62. 5 km to 4500 mm
20 : 1
55. 48 g to 1.8 kg
15 : 7
73.
2 m2
to
80 cm2
20 : 3
1 : 250
to
25 : 4
5 mm2
200000 : 1
77. 1 cm3 to 1 m3
78. 1 cm3 to 1000 mm3
1 : 1000000
1 : 10
1:1
86. 1 foot to 80 inches
3 : 20
89. 2 miles to 10000 feet
132 : 125
90. 0.5 miles to 1320 feet
2:1
93. 21000 inches to 2 miles
25 : 144
97. 16 oz. to 5 lbs.
82. 10 feet to 24 yards
5 : 36
85. 14 feet to 168 inches
94. 16 yards to 100 inches
144 : 25
1:5
98. 3.5 lbs. to 2.4 oz.
70 : 3
60. 5 m to 150 cm
10 : 3
64. 10 mm to 10 km
1 : 1000000
67. 60 m to 270 mm
68. 6600 mm to 2.4 m
11 : 4
71. 2.4 L to 240 mL
75.
1000 m2
to
10 : 1
1 km2
72. 0.5 L to 900 mL
76.
1 cm2
to
1 mm2
5:9
100 : 1
1 : 1000
79. 2 cm3 to 50 mm3
40 : 1
1:1
81. 3 feet to 10 yards
1000 : 1
2000 : 9
70. 5 L to 800 mL
74.
56. 25 km to 25 m
1:1
66. 3.6 mm to 4.5 cm
1 m2
50 : 1
63. 0.7 km to 70000 cm
2 : 25
69. 1 L to 150 mL
2 : 75
59. 18 m to 36 cm
10000 : 9
65. 3 cm to 14 mm
1 : 168
80. 5000 cm3 to 3 m3
1 : 600
83. 18 yards to 8 feet
27 : 4
87. 6 inches to 60 feet
1 : 120
91. 1 mile to 26400 feet
1:5
84. 6 yards to 18 feet
1:1
88. 5 feet to 36 inches
5:3
92. 1 mile to 1 foot
1 : 5280
95. 250 yards to
250 inches 36 : 1
96. 24 inches to 24 miles
99. 1500 lbs. to 2 tons
100. 8 tons to 640 lbs.
3:8
ALG catalog ver. 2.6 – page 217 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1 : 63360
25 : 1
KB
101. 6 fluid oz. to 1 cup
3:4
105. 60 cups to 1 gallon
15 : 4
109. 1 square foot to
1 square inch 144 : 1
102. 3 quarts to 6 gallons
1:8
106. 16 fluid oz. to 1 quart
1:2
110. 40 square inches to
2 square feet 5 : 36
103. 4 fluid oz. to 5 pints
1 : 20
107. 4.5 cups to
13.5 fluid oz.
104. 1 gallon to 68 fluid oz.
32 : 17
108. 1.5 quarts to 3 cups
8:3
111. 100 square feet to
1 square mile
2:1
112. 5 square yards to
25 square feet 9 : 5
1 : 278784
113. 8 cubic inches to
1 cubic foot 1 : 216
114. 12 cubic feet to
1 cubic yard 4 : 9
115. 1 cubic yard to
27 cubic feet 1 : 1
ALG catalog ver. 2.6 – page 218 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
116. 1 cubic foot to
80 cubic inches
108 : 5
KC
Topic:
Ratios derived from equations.
Directions:
72—Find the ratio of x to y.
1.
9x = 3y
5.
48x = 80y
9.
x
y
=
63
14
1/3
5/3
2/9
13. 36y − 45x = 0
17. 72y = 36x
7y = 28x
6.
105y = 56x
10.
y
x
=
8
20
1/4
15/8
5/2
14. 0 = 56y − 48x
4/5
18. 21x = 105y
2/1
21. 9x − 4y = y
2.
24x = 18y
3/4
4.
42y = 77x
6/11
7.
17x = 85y
5/1
8.
96y = 12x
8/1
11.
x
y
=
121
55
5/11
12.
y
x
=
85
10
15. 0 = 65x − 26y
7/6
19. 5x − 5y = 0
5/1
22. 2x = 26x − 3y
5/9
3.
16. 12x − 30y = 0
2/5
20. 0 = x − y
1/1
23. 3x − 7y = x + y
1/8
2/17
4/1
5/2
1/1
24. 9y − 8x = 12x − 3y
3/5
25. 4(x + y) = 6y
26. 3x = −9(x − y)
1/2
3/4
27. 8x − 5y = 5(x + y)
28. 7(y − x) = 21x − y
10/3
29. −3(x − y) = −7(x − y)
30. 2(x − y) = 6(y − x)
1/1
31. 5(x + 3y) = 17(x − y)
1/1
33.
3x
2y
=
5
7
37.
5y
25x
=
4
16
41.
3x
2y
=
2
3
45.
3x + 2y
2
=
5y
3
2/7
32. 2(y − 5x) = −2(x + 7y)
8/3
2/1
10/21
34.
7
5
y= x
3
4
28/15
35.
2
8
x= y
9
3
12/1
36.
11
5
y= x
20
2
4/5
38.
4y
2x
=
9
3
2/3
39.
2y
2x
=
7
7
1/1
40.
10
30
x=
y
13
39
1/1
42.
4x
5y
=
5
4
25/16
43.
5
10
=
7y
21x
44.
1
1
=
18x
45y
5/2
46.
7
x − 2y
=
12
2y
47.
7y + 15
5x + 10
=
3
2
48.
x+8
5y + 4
=
6
3
52.
3x − y
= −9
3y − x
56.
12y
9x
=
5 − 4y
2 − 3x
4/9
4/9
19/6
2/3
11/50
10/1
14/15
49.
y
=2
10x − 7y
53.
1
4
=
5x − 2y
x + 6y
50. −1 =
3/4
54.
16x
4x − 5y
51. 8 =
1/4
2
7
=
6y + x
9x − y
4/1
55.
14x + 3y
x + 3y
7/2
3x
9y
=
2x + 1
6y + 1
3/1
13/3
8/15
14/19
57. 49x2 = y 2
58. 225y 2 = x2
1/7
61. 36x2 − 121y 2 = 0
11/6
59. 16y 2 = 25x2
15/1
62. 0 = 81y 2 − 64x2
9/8
60. 4x2 = 9y 2
4/5
63. 0 = 16x2 − 169y 2
13/4
3/2
64. 144y 2 − 225x2 = 0
4/5
65.
y
4x
=
9y
4x
3/4
66.
y
18x
=
2x
y
1/6
67.
x + 3y
12x + y
=
2y
8x
68.
4y
9x
=
x + 4y
9x + y
1/2
69. 2y 2 = x2
√
2/1
70. 3x2 = y 2
√
3/3
71. 25x2 = 5y 2
ALG catalog ver. 2.6 – page 219 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
5/5
72. 12y 2 = 4x2
√
3/1
2/3
KD
Topic:
Word problems involving ratios.
Directions:
0—(No explicit directions.)
1.
The directions for a liquid detergent read: “Use 5 mL
of detergent for every 2 liters of water.” What is the
ratio of detergent to water in simplest form? 1 : 200
2.
When the bakery makes cookie dough, it adds
4 grams of salt to every 15 kg of flour. What is the
ratio of salt to flour in simplest form? 1 : 3750
3.
When a company mixes concrete, it uses one ton of
gravel for each 300 pounds of cement. What is the
ratio of gravel to cement in simplest form? 20 : 3
4.
To get ready for a triathlon, Martina swims 500 yards
for every 4 miles of running. What is the ratio of
her swimming to running distance in simplest form?
25 : 88
5.
In a certain fuel mixture, the ratio of gas to oil
is 15 : 2. What fraction of the total is oil? 2
6.
In a drink mix, the ratio of fruit juice to soda is 3 : 8.
What fraction of the total is fruit juice? 3
17
11
7.
In a rectangle, the ratio of length to width is 4 : 1.
What is the ratio of length to perimeter? 4 : 5
8.
9.
An epoxe glue contains resin and hardener. If resin
is 38 of the total mixture, what is the ratio of resin
to hardener? 3 : 5
10. A pesticide is a mixture of water and chemicals.
9
of the total mixture is water, what is the ratio
If 10
of water to chemicals? 9 : 1
In a rectangle, the ratio of length to width is 7 : 6.
What is the ratio of width to perimeter? 6 : 13
3
adhesive.
11. A paint is made up of 15 pigment and 10
The rest is a drying agent (mostly paint thinner).
What is the ratio of pigment to adhesive to drying
agent? 2 : 3 : 5
12. A certain drink is made up of 14 lemon juice and
1
12 sweetener (mainly corn syrup). The rest is water.
What is the ratio of lemon juice to sweetener to
water? 3 : 1 : 8
13. In a recent poll, the ratio of opposition to support
for the tax was 5 : 4. If 558 people were polled, how
many expressed opposition? 310
14. 1056 people were surveyed about the new law. The
ratio of supporters to opponents was 7 : 4. How many
people expressed support? 672
15. The ratio of apricot to plum trees in an orchard
is 9 : 5. If there are 392 trees altogether, how many
of each kind are there? 252, 140
16. The ratio of women to men enrolled at a private
college is 7 : 9. If the total enrollment is 1504, find
the number of women and men. 658, 846
17. The sides of a triangle are in a ratio of 3 : 6 : 4. The
perimeter of the triangle is 65 cm. What is the
length of each side? 15, 30, 20 cm
18. The sides of a triangle are in a ratio of 2 : 5 : 8. The
perimeter of the triangle is 105 mm. What is the
length of each side? 14, 35, 56 mm
19. The length and width of a rectangle are in a 5 : 3
ratio. Find the length if the perimeter is 48. 15
20. The length and width of a rectangle are in a 7 : 2
ratio. Find the width if the perimeter is 72. 8
21. The angles in a triangle are in a ratio of 2 : 3 : 5.
Find the measure of each angle. 36, 54, 90
22. The angles in a triangle are in a ratio of 5 : 7 : 8.
Find the measure of each angle. 45, 63, 72
23. The angles in a triangle are in a ratio of 1 : 1 : 4.
What is the measure of each angle? 30, 30, 120
24. The angles in a triangle are in a ratio of 3 : 4 : 5.
What is the measure of each angle? 45, 60, 75
25. The angles in a quadrilateral are in a ratio of
4 : 4 : 5 : 7. What is the measure of each angle?
26. The angles in a quadrilateral are in a ratio of
3 : 4 : 5 : 6. What is the measure of each angle?
52, 52, 65, 91
27. The angles in a quadrilateral are in a ratio of
6 : 7 : 8 : 9. Find the measure of each angle.
72, 84, 96, 108
60, 80, 100, 120
28. The angles in a quadrilateral are in a ratio of
2 : 2 : 3 : 3. Find the measure of each angle.
72, 72, 108, 108
ALG catalog ver. 2.6 – page 220 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
KE
Topic:
Word problems involving proportions.
Directions:
0—(No explicit directions.)
40—Write an equation and solve.
1.
A woman drives 200 km in 5 hours. At the same
rate, how far would she travel in 8 hours? 320 km
2.
A race car goes 385 miles in 2 hours. At that rate,
how far will it go in 6 hours? 570 mi
3.
A car used 7 gallons of gasoline to go a distance of
210 miles. At that rate, how much gasoline would
the car use to go 495 miles? 16.5 gal
4.
A truck went 180 km on 32 liters of diesel fuel. At
that rate, how far could the truck go on a full tank
of 164 liters? 922.5 km
5.
A drink is made up of 10 parts sparkling water to
3 parts lemon juice. How much juice should be
mixed with 80 mL of water? 24 mL
6.
A salad dressing is made up of 2 parts vinegar to
3 parts oil. How much vinegar should be used with
50 mL of oil? 33 1 mL
A cake recipe calls for a 3 : 1 mixture of flour to
sugar. If 8 cups of flour are used, how much sugar
should be added? 2 2 cup
8.
A hospital must be staffed with 5 nurses per
17 patients. How many nurses are required for
255 patients? 75
10. A school is staffed with at least 3 teachers per
65 students. How many teachers are at the school if
there are 1430 students? 66
11. In a county election, the incumbent received 5 votes
for every 3 received by the challenger. If the
incumbent received 6210 votes, how many did the
challenger get? 3726
12. In a class election, Silvia received 7 votes for every
6 received by her opponent. If Silvia received
161 votes, how many did her opponent get? 138
13. At a rate of $2.75 per square yard, how much will
9 square yards of carpet cost? $24.75
14. At rate of $3.50 per square foot, how much will
5 square feet of wood panelling cost? $17.50
15. Jen buys 24 candy bars for $6.40. At the same price
per candy bar, how much would she have to pay for
15 of them? $4
16. Gustavo buys 32 liters of gas for $10.00. At the same
rate, how much would he spend on 40 liters of gas?
17. Nine books cost $12. At the same price per book,
how much would five of them cost? $6.67 (rounded)
18. Nine postcards cost $1. At the same price per card,
how much would 20 of them cost? $2.22 (rounded)
19. At a small college, there are at least 2 professors for
every 25 students. If 518 students attend the college,
what is the least number of professors there?
20. A public swimming pool is required to have at
least one lifeguard for every 50 visitors in the pool
area. How many lifeguards are required if there are
275 visitors? 6 (rounded)
7.
3
9.
42 (rounded)
3
An old motorcycle engine requires a 20 : 1 mixture of
gas to oil. How much oil should be used with 2 liters
of gas? 0.1 liter
$12.50
21. In a marketing survey, 108 people, or 4 out of 5,
preferred Chewy-O’s. How many people were
included in the survey? 135
22. On a vocabulary quiz, Luis answered only 27, or 3
out of 5, questions correctly. How many questions
were on the quiz? 45
23. On a test with 40 questions, you answer 34 of them
correctly. If a perfect score is 100 points, and the
questions are of equal value, how many points will
you receive? 85
24. It is estimated that 2 out of 25 people in a city
use public transportation on weekends. How many
people does that represent, if the population of the
city is 124 thousand? 9920
25. 130 meters of copper wire weighs 18 kilograms.
What would 546 meters of the same kind of wire
weigh? 75.6 kg
26. If 30 feet of a fencing material weighs 21 pounds,
what does 75 feet of the same material weigh?
27. If it takes 8 meters of fabric to make 5 pairs of
curtains, how many meters are required to make
22 pairs? 35.2
28. 42 yards of material are used to upholster 8 sets of
living room furniture. How many yards are needed
for 14 sets? 73.5
52.5 lbs
ALG catalog ver. 2.6 – page 221 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
KE
29. Thomas is saving for a motorcycle. After 14 weeks,
he has already saved $560. At that rate, how long
will it take him to save an additional $740?
30. Ms. Marovich is a computer consultant. She earned
$171 for 6 hours of work. At that rate of pay, how
much does she earn in a 40 hour work week? $1140
18 12 weeks
31. A basketball player scored 192 points in 16 games.
At that rate, how many points will she score in the
remaining 6 games of the season? 72
32. A baseball player got 12 hits in his first 38 times at
bat. To keep the same batting average, how many
hits should he have after 133 “at bats”? 42
33. The scale on an architectural drawing is 1 inch
to 30 feet. What does 4.5 inches on the drawing
represent in “real-life”? 135 ft
34. A map has a scale of 160 km to 1 cm. What is the
actual distance between two cities that are 6.2 cm
apart on the map? 992 km
35. If 2 34 inches on a map represents 110 miles, what
distance does 1 21 inches on the map represent? 60 mi
36. On a road map, 3 cm represents 20 km. If two towns
are really 130 km apart, how far apart are they on
the map? 19.5 cm
37. A cake recipe calls for 21 teaspoon of baking powder
for 34 cup of flour. If the recipe is changed to include
6 cups of flour, how much baking powder is needed?
38. A pancake recipe requires 21 cup of wheat flour for
every 1 13 cup of white flour. How much wheat flour
should be mixed with 5 cups of white flour? 2 cups
4
39. A recipe for 2 kilograms of chocolate chip cookies
calls for 38 kg butter. How much butter would needed
to make a 6 kilogram batch of cookies? 1 1 kg
8
40. A punch recipe calls for 1 34 liters of orange juice for
every 4 liters of ginger ale. If 2 31 liters of orange
juice are used, how much ginger ale should be added?
5 13 liter
41. A wallet-size photo is 2 × 3 inches. If the shorter
side is enlarged to 5 inches, what will be the length
of the other side? 7 1 in.
42. A photograph that is 6.5 cm wide and 8 cm tall must
be enlarged so that is is 20 cm tall. How wide will it
be? 16.25 cm
43. A 2 12 ft by 3 21 ft poster is going to be reduced so that
the longer side is only 14 inches. What will be the
length of the shorter side? 10 in.
44. The image on a movie screen is 12 meters wide and
9 meters high. If the height of the image on film is
only 1.2 cm, what is its width on film? 1.6 cm
45. A “cubit” is an ancient Greek measure for distance.
If 1 cubit is about 53 cm, how many centimeters are
in 8.5 cubits? ≈ 450.5
46. A “siliqua” is an ancient Roman measure for weight.
If 1 gram is about 5.26 siliqua, how many grams are
in 1 siliqua? ≈ 0.19
47. Pharmacists used to measure weight by “grains”
One grain is about the same as 67 mg. What is the
equivalence, in grains, to 3.5 mg? ≈ 0.054
48. As a measure of energy consumption, one horsepower
is about the same as 0.75 kilowatts. What is the
equivalence, in kilowatts, to an 85 horsepower rating?
2
≈ 63.75
ALG catalog ver. 2.6 – page 222 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
LA
Topic:
Reciprocals.
Directions:
45—Write the reciprocal.
1
5
1.
5
5.
−
9.
−0.2
1
4
−4
−5
5
8
13. 1.6
2.
−3
6.
1
2
2
10. 0.04
25
14. −2.3
−
17. x
1
x
18. −y
21.
a
b
b
a
22. −
25.
1 1
+
2 4
29. 2
33.
1
2
4
3
2
5
37. −
1
n4
41. n−8
26.
xy
y−x
−n4
n8
−
1 1
−
3 6
2
3
−
34.
1 1
+
a
c
38.
1
x2
10
23
1
7.
−
1
3
7
−
1
y
23. −cd
6
27.
3
11
31. 5 +
ac
a+c
35.
−x5
100
17
−
1 1
+
6 2
43. y −3
ALG catalog ver. 2.6 – page 223 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1
cd
−1
8.
9
5
1
4
12. −0.25
−4
16. −0.06
50
3
28.
4
21
−
−r7
y3
1
x
−x
1
wz
1 1
−
4 3
32. 4 −
20
3
−1
5
9
24. wz
3
2
3 3
−
5 4
1
r7
4.
20. −
y
k
m
39. −
7
3
2
15. 0.17
19.
x2
42. −x−5
3.
11. 0.5
1
y
−
m
k
30. −3
1
1
−
x y
1
3
−
1
2
36.
2 2
+
3 7
40.
1
u6
−12
2
7
21
20
u6
44. −a−2
−a2
LB
Topic:
Least common multiples.
Directions:
70—Find the least common multiple.
1.
6, 8
5.
4, 5, 6
9.
27, 36, 30
24
60
13. 7x2 , 12
540
84x2
17. 10k 2 n3 , 45k 4 n
2.
4, 18
6.
3, 4, 5
60
3.
9, 27
7.
12, 16, 24
4.
16, 64
64
48
8.
5, 6, 8
120
27
10. 35, 21, 15
105
11. 18, 32, 24
144
12. 36, 40, 22
3690
14. 14a, 42a3
42a3
15. 3n2 , 18n5
18n5
16. 22r4 , 10r
110r4
18. 4x8 y 2 , 4y 3
90k 4 n3
21. 12cx, 18c8 x3 , 6c7
4x8 y 3
22. 33h2 k 5 , 15h4 , 11k 2
36c8 x3
19. 30s, 12s3 t4
60s3 t4
23. 6a, 8a3 y 3 , 24a6 y
26. w, w2 + 3w
12x − 18
30. a − b, b
2w(w − 7)
33. y 2 + 6y, y 2
37. 6c, 12c2 + 9c
w2 + 3w
4(x + 4)
38. 50b3 − 10b, 5b
6c(4c + 3)
41. xy 3 , xy 2 − x2 y 2
27. 5, 10a − 25b
31. 4p + 1, p
b(a − b)
34. 4, 2x + 8
y 2 (y + 6)
10b(5b2
35. n3 , n3 − n2
60r2 s4
39. 4m3 − 16m2 , 10m
2c2 d(2c − d)
45. 5a − 10b, a − b
46. y + 6, y 2 + 6y
y(y + 6)
50. 3m2 n − mn, 21m − 7
10mr(m − 2r)
47. 8x − 20, 12x − 30
48. 3p + 12, 6p + 24
53. y 2 + 6y + 9, y + 3
54. 2 − x, 10 − 3x − x2
+ 3)2
(2r − 3)(2r + 3)
59. x − 6, x2 − 10x + 24, x − 4
61. c2 − 4cd, c2 − 16d2
c(c − 4d)(c + 4d)
63. u3 − 4u2 + 4u, 6u − 12
65. n2 − 1, n2 − n − 2
10a(a + 3b)
55. a + 6, a2 − 36
56. 4 − 9k 2 , 2 − 3k
(n − 1)(n + 1)(n − 2)
69. k − 6, k 2 − 36, k + 6
71. a2 − 4, 5a + 10, a − 2
(z + 1)(z − 3)(z + 2)
5(a − 2)(a + 2)
4c(c + 1)(c − 1)
77. r2 − 7r + 12, r2 − r − 6
− 16
3n(n + 2)(n + 4)
(r − 3)(r − 4)(r + 2)
(a + 4)(a − 4)(a − 1)
81. 3x3 + 9x2 + 6x, x3 − 4x, 3x2
83. 8d, 2d3 − 6d2 − 36d, 3d2 + 9d
(2 − 3k)(2 + 3k)
58. 16a2 − b2 , 4a + b, 4a − b
(4a − b)(4a + b)
60. x2 + x − 30, x + 6, x − 5
(x + 6)(x − 5)
62. 10a + 2b, 50a2 − 2b2
2(5a − b)(5a + b)
66. r2 + 8r + 15, r2 − 9
3(x − 3y)(x + 5y)
(r + 3)(r + 5)(r − 3)
68. m2 + m − 2, m2 + 7m + 10
70. y 2 + 3y − 10, y − 2, y + 5
(k − 6)(k + 6)
75. 3n + 6, n2 + 4n, n2 + 6n + 8
(a − 6)(a + 6)
64. x2 + 2xy − 15y 2 , 3x − 9y
6u(u − 2)2
67. z 2 − 2z − 3, z 2 − z − 6
73. 4c + 4, c2 − c, c2 − 1
(x − 6)(x − 4)
52. 5a + 15b, 2a2 + 6ab
3y 2 (y + 1)
(5 + x)(2 − x)
57. 2r − 3, 4r2 − 9, 2r + 3
6(p + 4)
51. 3y 4 + 3y 2 , y 2 + y
7mn(3m − 1)
8x(x + 5)
− 5a + 4, a2
44. 10m − 20r, 5mr
12(2x − 5)
49. 8x + 40, x2 + 5x
3x2 (x + 1)(x + 2)(x − 2)
24d(d + 3)(d − 6)
10(3a − 2)
2x2 (x2 − 5)
3abc(b − 5)
5(a − 2b)
8(5x + 7)
40. 2x2 , x4 − 5x2
20m2 (m − 4)
− 1)
y4 − y2
36. 6a − 4, 10
n3 (n − 1)
xy 3 (1 − x)
79.
24. 10rs2 , 4s4 , 6rs4
32. 8, 5x + 7
p(4p + 1)
43. 2c2 d, 4cd − 2d2
a2
42u2 w7
28. y 4 − y 2 , y 2
10a − 25b
42. abc − 5ac, 3ab
(y
24a6 y 3
20. 14uw3 , 21u2 w7
165h4 k 5
25. 12x − 18, 6
29. 2w, w − 7
36
(m + 2)(m + 5)(m − 1)
(y + 5)(y − 2)
72. r2 − 10r + 25, r − 5, r2 − 5r
74. p + 1, p3 + 2p2 + p, p2 + p
76. y − 3, 6y 2 − 54, 2y + 6
78. x2 − 9, x2 − x − 6
80.
c2
+ 3c − 4, c2
r(r − 5)2
p(p + 1)2
6(y + 3)(y − 3)
(x + 3)(x − 3)(x + 2)
− 4c + 3
(c − 1)(c + 3)(c − 3)
82. 5a5 − 20a4 , 2a2 − 10a + 8, 6a
30a4 (a − 1)(a − 4)
84. 3m3 , m4 − 9m2 , 4m2 + 20m + 24
12m3 (m + 2)(m − 3)(m + 3)
85. u2 + 4u − 12, 3u2 − 12, 3u + 18
3(u + 6)(u + 2)(u − 2)
86. z 2 + 8z + 16, z 2 + 4z, z 2 + 7z + 12
87. x2 − 4, 4x2 − 12x, x3 − x2 − 6x
4x(x + 2)(x − 2)(x − 3)
88. y 4 − 16y 2 , y 2 − 2y, y 2 − 6y + 8
ALG catalog ver. 2.6 – page 224 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
z(z + 3)(z + 4)2
y 2 (y + 4)(y − 4)(y − 2)
LC
Topic:
Greatest common factors.
Directions:
71—Find the greatest common factor.
1.
21, 15
3
2.
28, 42
14
3.
45, 36
5.
72, 84
12
6.
52, 78
26
7.
175, 136
9.
1776, 1976
13. 33 , 35
10. 504, 660
8
14. 56 , 53
33
18. m, m6
21. 4a2 , 8a
4a
22. 6x3 , 20x5
25. 35xy 2 , 14x2 y
34. 3k 5 , 9k 4 , 6k 2
6
37. 35x2 yz, 84xy 2 z, 49x2 z
39. 5a2 b3 , 15a3 c, 36b2 c3
41. 10y, 6y 2 + 4y
15u
46. x3 y 2 , x3 y + x2 y 2
1
49. 2x2 + x, 12x + 6
x2 y
50. r2 p − rp, 2rp − 2p
53. n2 − 16, 5n − 20
7
n
24. 3r3 , 12r2
31. 44r2 ps2 , 132r2 s
3r2
32. 28xy 2 z, 98x2 z
44r2 s
6m
14xz
36. 45u2 , 10uw, 24w2
4z
47. 3pr, p2 r2 − 3pr
w2
28. 18mn4 , 24m2
cd2
1
9k 2 m
2d2
44. ab2 + ab, a2 b
n2
ab
48. 8cd2 , 20cd + 12c
pr
4c
52. 3y 2 − 6y, 18y − 36
51. 20a + 40b, 15a + 25b
3(y − 2)
5
54. x2 + 3x, x3 − 9x
n−4
21
23. 6n, 7n2
p(r − 1)
2x + 1
1
20. w2 z 4 , u2 w4
43. n3 − 7n2 , 7n2
3
81, 110
ab
40. 20cd2 e2 , 8cd3 , 42d2 e3
42. 24w − 3x, 9wx
8.
6
19. a2 b, ab3
38. 9k 3 m5 , 18k 2 m, 27k 4 m2
1
18, 24
16. 74 , 7
82
35. 4z 3 , 16z, 8z 5
3k 2
4.
12. 1344, 357
54
27. 27cd3 , 8c2 d2
11ab
7xz
2y
45. 6s2 , 15st2 − 5t2
2x3
30. 105uv 2 , 75u2 w2
15abc3
33. 12st2 , 18t, 30s
m
26. 11abc, 55a2 b
7xy
29. 45a2 bc3 , 60abc4
15. 85 , 82
53
y5
1
11. 378, 5400
12
17. y 10 , y 5
9
55. a2 + ab, a2 − b2
56. u2 − 16u, u3 − u2
a+b
x(x + 3)
57. y 2 − 2y + 1, 3y − 3
58. x + 3, x2 + 3x + 2
1
59. m2 − 2m, m2 + m − 6
y−1
m−2
61. a2 − 5a + 6, a2 + 4a − 12
63.
x2
+ 4x − 12, x2
(d − 1)(d − 2)
70. p2 − 7p − 60, p2 − 14p + 24
c−8
+ 7y 2
74. 3z 2 − 8z − 3, 3z 2 + 7z + 2
85. a2 − ab + 3a − 3b, a2 − 2ab + b2
89. x3 + 2x2 − x − 2, x4 − 1
91. u2 − 16, u4 + 5u2 + 4
4(w + 2)
y(y − 1)
y−2
3(p + 3)
a−b
x−1
(x − 1)(x + 1)
u2 + 4
p − 12
1
3z + 1
76. 2x2 + 5x − 3, 4x2 − 12x + 5
1
− 8y
− 3xy
− 3h − 40
r+5
83. 6p3 + 18p2 , 3p2 + 6p − 9, 3p2 − 27
87. xy − y
− 15h + 50, h2
72.
81. y 2 − 4, y 2 + 6y − 16, y 2 − 3y + 2
− 3x + 3, 3x3 y
h2
a+8
77. 4w2 + 16w + 16, 4w3 + 12w2 + 8w
79.
y−5
68. 6d2 − 18d + 12, d3 − 3d2 + 2d
+ 17a + 72
+ 3y 2 , y 3
− 2y − 15
u2 + 2
75. 4n2 − 7n − 2, 2n2 − 13n + 6
− 6y 3
c+3
2(y − 5)
73. 2r2 + 9r − 5, 3r2 + 16r + 5
3y 4
− 25, y 2
66. 2y 2 − 8y − 10, 4y 2 − 12y − 20
69. c2 − 14c + 48, c2 − 18c + 80
71.
64.
x+6
y2
x(x + 1)
67. u4 + 6u2 + 8, 5u5 − 20u
− 3a − 88, a2
a+b
62. c2 + 6c + 9, c2 − 9
a−2
+ 7x + 6
65. x4 − x2 , x3 + 4x2 + 3x
a2
60. a2 + 2ab + b2 , a + b
2x − 1
78. 5x2 − 30x + 45, 5x2 − 35x − 30
80.
5z 2
+ 30z
+ 40, 9z 3
+ 54z 2
5
+ 72z
(z + 4)(z + 2)
82. w2 − w − 12, w2 + w − 6, 5w2 − 5
84. a3 − 2a2 + a, 4a2 − 4a, a3 − a2 − 2a
86. c2 − 9d2 , c2 − 3cd + c − 3d
88.
2k 2 m + 3km + m, 2k 2
90. y 4 − y 2 − 12, y 3 − 4y
a
c − 3d
+ k + 2km + m
(y − 2)(y + 2)
92. r2 + 2r + 1, r4 − 2r2 + 1
ALG catalog ver. 2.6 – page 225 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1
(r + 1)2
2k + 1
u
LD
Topic:
Values for which an expression is undefined.
Directions:
99—Give the restriction(s) on each variable. 102—Give the excluded value(s) for each variable.
103—For what value(s) is the expression undefined?
a 6= 0, b 6= 0
3.
2x2
y3
y=
6 −15
7.
5a + 3
3b − 7
11.
w2 − 9
w2 − 4w
y=
6 ±3
15.
2k 2
1 − 49k 2
none
19.
t2 − 4
5
23.
y−3
y 3 − 4y
d 6= 4, 2
27.
25 − p2
24 + 2p − p2
14 + 9z + z 2
z2 + 1
none
31.
u2 + 3u
8
22a − 11
7a2 − 5a − 2
a 6= − 27 , 1
35.
3c2 − 27
2 + 21c − 11c2
1.
k−6
2k
k 6= 0
2.
a+b
ab
5.
−3x
x−8
x 6= 8
6.
y + 21
y + 15
9.
c+8
3c3 + c2
c 6= 0, − 13
10.
24a + 60
20a2 − 16a
13.
15 − 3f
36 − f 2
f 6= ±6
14.
x2 − 1
y2 − 9
17.
2x − 14
3
none
18.
6c + c2
14
21.
21 − 3k
9k − 64k 3
22.
18x + 21
x2 − 100x4
26.
d2 − d − 20
d2 − 6d + 8
30.
34.
25.
k=
6 0, ± 38
40m + 12
+ 3m − 10
m2
a 6= 0,
4
5
1
x 6= 0, ± 10
4.
c2 − d 2
cd2
8.
−4n
2m + 11
w 6= 0, 4
12.
3z 2 − 12
10z 2 + z 3
z=
6 0, −10
k 6= ± 17
16.
−1
16n2 − 81
n 6= ± 94
20.
1 − 9y 2
2
24.
15a − 5
a5 − 25a3
28.
4t3 − 16t
15 + 8t + t2
32.
s2 − 9
s2 + 9
36.
5 + 2y − 7y 2
5 + 36y + 7y 2
y=
6 0
b 6=
7
3
none
y=
6 0, ±2
p 6= −4, 6
c 6= 0, d 6= 0
m 6= − 11
2
none
a 6= 0, ±5
t 6= −5, −3
m 6= −5, 2
29.
y 2 − 2y − 8
14
33.
x2 − 3x + 2
3x2 − 8x + 5
none
x 6= 53 , 1
none
1
c 6= 2, − 11
37.
u3
5u4 − 5
+ 12u2 + 36u
38.
u 6= −6, 0
41.
45.
2c2 − 50
3c2 − 16c3 + 20c4
c 6=
3 1
10 , 2 , 0
2y 3
14 − 7y
− 22y 2 + 56y
a4
39.
r 6= 7, 4, 0
42.
46.
2f 2 + 32f − 72
9f 3 − 12f 2 + 4f
3k 4
43.
50.
a 6= ±2
k2 − 1
− 24k 3 − 99k 2
2−w
81 − w4
w2 + 20w + 64
− 20w3 + 100w2
w4
40.
47.
x2 + 8
8x4 + 2x3 − x2
4z 4
44.
51.
b4
1 − 4a − 5a2
4a + 20a2 + 25a3
a 6= − 52 , 0
z 2 + 5z + 4
+ 56z 3 + 96z 2
48.
z 6= −12, −2, 0
w=
6 ±3
4n2 − 25
12n − 4n2 − n3
n 6= −6, 2, 0
x 6= 14 , − 12 , 0
k 6= 11, −3, 0
24
− 16
y 6= −5, − 17
w 6= 10, 0
f 6= 32 , 0
y=
6 7, 4, 0
49.
r2 − 12r + 32
− 11r3 + 28r2
r4
none
b2 − 1
− 5b2 + 4
2h − 3
5h3 + 75h2 − 80h
h=
6 1, −16, 0
b 6= ±1, ±2
52.
x4
3x + 6
− 10x2 + 9
x 6= ±1, ±3
53.
m − 10
m3 + 2m2 − 9m − 18
54.
m 6= −2, ±3
57.
a+b
a−b
61.
xy
xy + y
x2 − 4
x3 − x2 − 25x + 25
55.
x=
6 1, ±5
a 6= b
y 6= 0, x 6= −1
58.
2x − y
x+y
62.
ab + 3b
ab − 4a
3y + 12
y 3 − 5y 2 + 3y − 15
56.
d 6= −8
y 6= 5
x=
6 −y
a 6= 0, b 6= 4
d2 − d + 2
d3 + 8d2 + d + 8
59.
5c2 d
2c + 3d
c 6= − 32 d
60.
4m + 7r
7m − 4r
63.
k 2 − kn
k 2 − n2
k 6= n, k 6= −n
64.
2c − 18cd2
c2 − 4d2
m 6= 74 r
c 6= 2d, c 6= −2d
65.
w2 − 3wz + 2z 2
w2 − 2wz + z 2
w 6= z
66.
m2 + 2mr + r2
mr2 + 2mr + m
m 6= 0, r 6= −1
67.
3x3 − 12xy 2
xy − 4x + 2y − 8
x 6= −2, y =
6 4
ALG catalog ver. 2.6 – page 226 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
68.
a2 + 2ab − 8b2
− 3a + ab − 3b
a2
a 6= −b, a 6= 3
LE
Topic:
Simplifying rational expressions. See also categories DL and DM (dividing monomials), and EG (dividing
monomials and polynomials).
Directions:
6—Simplify. 9—Simplify (assume no denominator equals zero).
100—Give the restriction(s) on each variable, then simplify.
1.
6(a − 5b)
4(a − 5b)
3
2
2.
−5(r + 7)
−25(r + 7)
1
5
3.
−4c2 (c + x)
14c(c + x)
−
5.
9x
21(x + 3)
3x
7(x + 3)
6.
−5a(a + 6)
−30a
a+6
6
7.
10y 2 (a + y)
5y 2 (b + y)
2(a + y)
b+y
2c
7
4.
12p(p − 1
9p3 (p − 1)
8.
a2 (m + n)
−2a(m − n)
−
9.
w(w − 2)
(w − 2)(w + 3)
13.
(u + 9)(u − 5)
(u − 5)(u + 3)
w
w+3
u+9
u+3
10.
(c + 3)(c − 6)
c2 (c + 3)
c−6
c2
11.
(y − 7)(y + 2)
−5y(7 − y)
14.
(r + 1)(r − 1)
(r − 1)(r − 1)
r+1
r−1
15.
(n + 2p)2
(n + 2p)(n − 2p)
y+2
5y
4
3p2
a(m + n)
2(m − n)
12.
−7(a − b)
(b − a)(b + a)
16.
(x + 3)(x − 6)
(x + 3)2
7
b+a
x−6
x+3
n + 2p
n − 2p
17.
y
y2 + y
21.
15k − 45
5
25.
z2
1
y+1
z2
− 4z
3k − 9
z
z−4
29.
c − 2d
2d
33.
25cd
5cd2 + 15c2 d
18.
3
12x + 3
22.
c3 − c
−c
26.
30.
same
5
d + 3c
34.
1
4x + 1
1 − c2
−4x
+ 3x
−
x2
a2
+3
a2
4
x+3
19.
z
5z 3 − z
1
5z 2 − 1
20.
−a
ab − a
23.
nr + 4n
n
r+4
24.
7y 2 + 14
7
y2 + 2
5a2
− 3a
5a
2a − 3
27.
same
12n4
30n3 − 21n2 r
w3
+ w2
w
5w + 1
5w3
31.
3x + 5y
5y 2
35.
20ab2
−8ab − 12b
−
39.
48x3 + 16x
48x2
3x2 + 1
3x
43.
8x + 12
4x + 20
47.
6a2 − 9a
3a2 + 12a
28.
1
1−b
2a2
32.
10p
10p + 7r
36.
9xy
27x2 y + 18xy
1
3x + 2
40.
−15pr2 − 35r
−5pr
3pr + 7
p
44.
7ab − 14
21ab + 28
2a − 3
a+4
48.
5y 2 z − 10yz 2
5yz − 15yz 2
2c − d
d(c + 3)
52.
4r2 + 10r
24r2 + 16r
56.
−4kn + 2kn2 − kn3
−2kn
same
5ab
2a + 3
same
4n2
10n − 7r
37.
40w2 − 32w
4w3
38.
6x2 y − 12xy
−42xy
41.
10s + 20t
10s − 30t
s + 2t
s − 3t
42.
6k 2 − 12
6k − 12
45.
6x2 + 2x
4x3 + 2x
3x + 1
2x2 + 1
46.
8uw2 + 6uw
2u2 w − 10uw
49.
9a3 − 27a2
18a2 − 27a
50.
15x + 30
15x + 5
3(x + 2)
3x + 1
51.
8c − 4d
4cd + 12d
53.
x2 y + 2xy − 5xy 2
x2 y
54.
7ac2 − 21ac + 35c
14ac
55.
9ux2 + 12u2 x2 + 3u2 x
3ux2
10w − 8
w2
a(a − 3)
2a − 3
3n3
3n2 + 6n
+ 18n2 + 9n
n+2
n2 + 6n + 3
k2 − 2
k−2
4w + 3
u−5
ac − 3a + 5
2a
x − 5y + 2
x
57.
2−x
7
58.
4z 4 − 24z 3 + 12z 2
4z 4 + 16z 2
z 2 − 6z + 3z
z2 + 4
2x + 3
x+5
3x + 4ux + u
x
59.
4r + 2rs + 6rs2
6r − 4rs
2 + s + 3s2
3 − 2s
ALG catalog ver. 2.6 – page 227 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
ab − 2
3ab + 4
y − 2z
1 − 3z
2r + 5
4(3r − 2)
4 − 2n + n2
2
60.
2c2 d − 3cd
c2 d − 4cd + 3cd2
2c − 3
c + 3d − 4
LE
61.
4k − 12
k−3
65.
ax − ay
bx − by
69.
3w2 − w
3w + 1
73.
x2 − 9
x+3
77.
2y − 18
y 2 − 81
81.
15x2 − 6xy
25x2 − 4y 2
85.
x−2
2
x − 4x + 4
89.
n2 − 12nr + 35r2
n − 5r
3a2
18w2 + 45w
2w + 5
3
5
67.
12xy + 6y
8x2 + 4x
same
71.
5x + 4
5x2 + 4
75.
a2 − 49
a+7
79.
3u + 12
3u2 − 48
83.
64t3 − t
8t3 + t2
87.
y+9
2
y + 4y − 45
91.
a2 − 8a + 16
a−4
95.
10x2 − 7xz + z 2
5x − z
r−3
2r − 6
a
b
66.
9y − 3
15y − 5
same
70.
3a + 3b
2a − 2b
74.
y−1
y2 − 1
78.
5w2 − 20
5w + 10
3x
5x + 2y
82.
6c2 − 22c
9c3 − 121c
1
x−2
86.
n2 − 12n + 27
n−3
90.
c + 6d
c2 + 12cd + 36d2
x−3
2
y+9
n − 7r
93.
63.
62.
4
1
2
1
y+1
w−2
2
3c + 11
n−9
64.
3k − 4n
30kn − 40n2
1
10n
68.
18u2 − 48uw
5u − 15w
18u
5
same
72.
3c − 9
5c − 20
a−7
76.
z − 10
z 2 − 100
1
z + 10
1
u−4
80.
x2 − 4
11x − 22
x+2
11
8t − 1
t
84.
4a2 b − b3
2ab + b2
2a − b
1
y−5
88.
a2 + 6a + 9
a+3
x−4
92.
k+2
k 2 − 11k − 26
96.
2w − y
14w2 − 5wy − y 2
9w
3y
2x
101.
a2
a+3
1
k − 13
1
c + 6d
a+1
+ 5a + 2
1
3a + 2
94.
5b2 + 2b − 7
b−1
5b + 7
2x − z
97.
same
a2 − 4
+ 5a + 6
a−2
a+3
x2 + 2xy + y 2
x2 − y 2
x+y
x−y
98.
102.
y 2 + 7y + 12
y2 − 9
n2
y+4
y−3
n2 − 36r2
+ nr − 30r2
99.
103.
1
7w + y
w2 + 6w − 7
w2 − 1
a2
w+7
w+1
a2 − 4
− 11a + 18
a+2
a−9
100.
c2
c2 − 25
− 10c + 25
104.
y 2 + 2y − 8
y 2 − 16
108.
x2 − 7x − 18
2x + 18
c+5
c−5
y−2
y−4
n − 6r
n − 5r
105.
109.
z2 − 9
z 2 − 3z + 2
3t2
same
t2 − 16
− 11t − 4
t+4
3t + 1
106.
3d + 3
2
d − 2d + 1
110.
6r2 + 7r + 1
36r2 − 1
same
r+1
6r − 1
107.
u2 + 5u + 6
u2 − 1
111.
3x2 + xy − 2y 2
9x2 − 4y 2
same
112.
y2 + y − 6
y 2 + 5y + 6
117.
b2 + 7b + 12
b2 + 10b + 21
y−2
y+2
b+4
b+7
114.
x2 + 6x + 8
x2 + 8x + 16
118.
y 2 + 9yz + 14z 2
y 2 − 10yz − 24z 2
x+2
x+4
115.
n2 − 7n + 10
n2 − 4n + 4
119.
k 2 + 5kn + 6n2
k 2 + 17kn + 30n2
c2 + 9cd + 14d2
c2 − 4cd − 77d2
122.
a2 − 12a + 20
a2 − 16a + 28
n−5
n−2
116.
a2 + 8a + 7
a2 + 6a − 7
120.
w2 + w − 20
w2 − 12w + 32
124.
n2 + 4nx − 32x2
n2 + 3nx − 40x2
a+1
a−1
k + 3n
k + 15n
y + 7z
y − 12z
121.
4s2 − t2
− 15st + 7t2
2s + t
s − 7t
x+y
3x + 2y
113.
2s2
same
a − 10
a − 14
123.
b2 − b − 72
b2 + 9b + 8
c + 2d
c − 11d
ALG catalog ver. 2.6 – page 228 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
b−9
b+1
n − 4x
n − 5x
w+5
w−8
LE
125.
x2 + 10x + 25
x2 − 4x − 5
129.
3u2 + 4u + 1
2u2 + 5u + 3
same
126.
y 2 − 4y + 3
y 2 + 4y + 3
3u + 1
2u + 3
130.
3k 2 − 11k − 4
3k 2 + 4k + 1
same
k−4
k+1
127.
k 2 − 5k − 14
k 2 − 13k − 14
131.
5z 2 − 4z − 1
11z 2 − 12z + 1
same
128.
p2 + p − 20
p2 − p − 20
132.
8x2 + 6x + 1
2x2 + 23x + 11
5z + 1
11z − 1
133.
3b2 − 10b − 25
3b2 + 11b + 10
137.
d4 − 5d3 − 14d2
d2 − 7d
b−5
b+2
134.
2y 2 − 7y + 6
2y 2 − 13y + 15
138.
3p2 − 15p − 72
p2 + 3p
d(d + 2)
141.
2a3 − 8a
2a2 − 8a − 24
142.
4p2 + 24p + 32
2p2 + 16p + 32
146.
3z 3 − 9z 2 + 6z
3z 2 + 9z − 30
150.
2c + 3
2c − 3
136.
8r2 + 18r − 5
4r2 − 4r − 35
139.
2c2 − 16c + 24
4c2 − 8c
c−6
2c
140.
y 3 − 7y 2 + 12y
5y 2 − 15y
3x3 − 15x2 − 18x
x2 − 36
143.
4w2 + 32w + 60
2w3 − 18w
144.
6u2 − 24
9u2 + 36u + 36
10c2 − 40c + 40
2c2 − 14c + 20
147.
4x2 + 12x − 72
6x3 + 30x2 − 36x
154.
2h − 3
7h
158.
148.
y−4
5
n3 − 8n2 + 7n
n4 + 3n3 − 4n2
151.
2d3 + 4d2 − 16d
d2 + d − 12
152.
155.
a4 − 4a3 − 21a2
3a3 − 15a2 − 42a
156.
159.
14k 2 − 20k + 6
14k 2 − 8k − 6
162.
7k − 3
7k + 3
2n2 + 5n + 3
n3 + n2
2n + 3
n2
160.
(n − 6)2
5n − 30
28m2 − 32m + 4
28m2 − 18m + 2
163.
169.
x2 + 2xy + y 2
(x + y)3
ab + ac
(b + c)2
170.
(w − 3)3
w2 − 6w + 9
1
3
174.
5u + 10
−u − 2
2
5
178.
4c + 12d
−3c − 9d
1
x+y
4a2 + 14a + 10
4a3 + 10a2 + 6a
164.
2a + 5
a(2a + 3)
166.
n−6
5
4y 2 − 14y + 6
10y − 5
2(y − 3)
5
2(m − 1)
2m − 1
165.
2r3 − 16r2 − 18r
2r5 − 14r4 − 36r3
r+1
r2 (r + 2)
a(a + 3)
3(a + 2)
2s3 + 3s2 − 5s
8s + 20
h2 − 8h + 12
3h3 − 15h2 − 18h
h−2
3h(h + 1)
2d(d − 2)
d−3
5y 4 + 20y 3 + 20y 2
2y 2 + 24y + 40
3t2 − 9t + 6
12t2 − 24t + 12
t−2
4(t − 1)
y−1
3(y + 1)
5y 2 (y + 2)
2(y + 10)
2h2 + h − 6
7h2 + 14h
3y 2 + 18y − 21
9y 2 + 72y + 63
s(s − 1)
4
161.
4r − 1
2r − 7
2(u − 2)
3(u + 2)
2(w + 5)
w(w − 3)
n−7
n(n + 4)
2(x − 3)
3x(x − 1)
157.
6c2 + c − 12
6c2 − 17c + 12
5(c − 2)
c−5
z(z − 1)
z+5
153.
135.
3x(x + 1)
x+6
2(p + 2)
p+4
149.
4x + 1
x + 11
3(p − 8)
p
a(a − 2)
a−6
145.
y−2
y−5
same
a
b+c
w−3
12z 2 + 42z + 18
16z 2 + 12z + 2
3(z + 3)
4z + 1
167.
x2 − 25
(x + 5)2
168.
(a + b)2
a2 − b2
171.
(c + d)2
2c2 + 3cd + d2
172.
y 2 − 6y + 8
(y − 4)2
175.
3c + 2cd
−18 − 12d
−
c
6
176.
22x − 33
−2xy + 3y
179.
−6n − 18
5n2 + 15n
−
6
5n
180.
−24p2 r + 6pr
56p − 14
x−5
x+5
c+d
2c + d
a+b
a−b
y−2
y−4
Factors of −1
173.
−n − 4
3n + 12
−
177.
−2a − 4
5a + 10
−
−5
−
4
3
ALG catalog ver. 2.6 – page 229 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−
11
y
−
3pr
7
LE
181.
x−1
1−x
185.
a2 − 6a
12 − 2a
189.
16 − m2
m−4
193.
y2
182.
2d − 6
3−d
−2
186.
b−c
bc − b2
−
190.
1 − 5r
25r2 − 1
1
y−2
194.
h2 − 2h − 15
5h − h2
−
h−3
h
195.
a−1
2(a − 2)
198.
4 − x2
x2 − 4x − 12
−
x−2
x−6
199.
s2 + 5s + 6
9 − s2
202.
y 2 − 4y + 3
12 − y − y 2
203.
h2 + 8h + 7
14 − 5h − h2
206.
21 + 11x − 2x2
x3 − 2x2 − 35x
207.
4y − 3y 2 − y 3
10y 2 − 25y + 15
−1
−
a
2
−m − 4
2−y
− 4y + 4
−
197.
a2 + a − 2
8 − 2a2
201.
8 + 7x − x2
x2 − 10x + 16
205.
2m2 + 8m − 64
28 − 3m − m2
−
209.
−
x+1
x−2
2(m + 8)
m+7
6n2
−
−
9 − 3n
− 30n + 36
1
2(n − 2)
−
210.
1
b
−
1
5r + 1
−
y−1
y+4
183.
a−b
b−a
187.
10p − 2pr
r−5
191.
9a2 − b2
b − 3a
2x + 3
x(x + 5)
4c2 − 24c + 36
6 − 2c
−2(c − 3)
u2
−
211.
184.
3n − 9
3−n
−5p
188.
w−2
4w − 2w2
−3a − b
192.
x−y
y 2 − x2
196.
c2 − 8c + 16
4−c
4−c
200.
16n − n3
n2 + 6n + 8
−
204.
6 − a − a2
a2 + 5a + 6
−
208.
4w2 + 14w + 6
12w − 8w2 − 4w3
−1
6 − 2u
+ 2u − 15
−
−
ALG catalog ver. 2.6 – page 230 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
s+2
s−3
−
h+1
h−2
y(y + 4)
5(2y − 3)
2x2 + 16x − 40
16 − 8x
−
2
u+5
x + 10
4
−
212.
−3
−
−
1
2w
1
x+y
2w + 1
2w(w − 1)
14a − 2a2
− 36a + 56
4a2
−
a
2(a − 2)
n(n − 4)
n+2
a−2
a+2
LF
Topic:
Multiplying rational expressions. See also categories AI (fractions) and DM (monomials).
Directions:
6—Simplify. 3—Multiply. 9—Simplify (assume no denominator equals zero).
100—Give the restriction(s) on each variable, then simplify.
1.
a x
·
x b
5.
a b c
· ·
b c a
9.
3a 10b
·
5b 3a
13.
−2p2 q
2.
a a
·
b b
1
6.
bc 1 a
·
·
a ab c
2
10.
9c
4d
−
−
8d
6c
a
b
4
· 2
pq
a2
b2
2b
14. 10ab · 2
5a
8p
−
q
1
a
3
4
4b2
a
3.
x y
·
y x
7.
1 x y2
· ·
x2 y a
11.
2xy 2x
·
3y 3xy
15.
1
1
9rs3
y
ax
4x
9y
12r2 s
7r
4.
x y
·
y 2 x2
8.
a2 a x
·
·
x x2 a
12.
16.
1
20kn2
·
3k 2 n 10kn
20.
2abc
2bd
−
−
5d
7ac
5a
b
1
xy
5b
a
a2
x2
25
2
3k 2
4
21s2
17.
14x
3y 2
9y 3
7x2
6y
x
18.
2 20st2
s t
21. − 2
−
s
10st
22.
3u3
2w
−
5uw
4
5w2
− 2
6u
cd5
5c2 d
·
15cd2 3d3
19. −
15wx 12wy
·
3xy
5xz
−
12w2
xz
4b2
35
c2 d
9
23.
12a3 10a2 b4
·
8b3
27a4 b
5a
9
24.
2t2
25.
−
6a2 3b 10c
·
·
4b 5c2 8a
9a
8c
26.
5x ay 2 7xy
·
·
3y 7b 5a2
x2 y 2
3ab
27.
2a
3b2
b2 x
−2y
−6c
5x
28.
2ac
5y
29. (x − 6) ·
x+2
3x − 18
30.
x + y 2x2 y
·
6xy 2 x + y
a−3
· (a + 2)
5a + 10
8
5x2
−6x
21a2
14a3
−9x
6x3
4a
2x3
3
31.
2b − 8
1
·
b+4 b−4
35.
ax + bx
2a
·
3a
2a + 2b
39.
2a + 6 3a − 9
·
6a − 18 7a + 21
43.
12x2 y (x − 2y)2
·
3x − 6y
2xy 3
32.
1
c+1
·
c2 + c c − 1
x
3
36.
r2 + 3r
14
·
2
21r
2r + 6
1
7
40.
ab − ac bx + cx
·
bn + cn b2 − bc
44.
3w + 9 w2 + 3w
·
3w2
(w + 3)3
2
b+4
1
c2 − c
a−3
5
x+2
3
33.
8x2 y 3
−6w2
·
10xy 2 w 3wx3 y
x
3y
34.
2m − 5
40m
·
3
15m
2m − 5
1
3r
8
3m2
37.
6r + 12 5r − 15
·
3r − 9 4r + 8
41.
(a + b)3
4a
·
12ab
(a + b)2
5
2
38.
2c − 2d 6c + 6d
·
3c + 3d 8c − 8d
42.
5xy
x2 + xy
·
(x + y)2
10
x2 − 9
6
·
8
x−3
2x2 − 4xy
y2
x2 y
2x + 2y
a+b
3b
45.
1
2
3x + 9
4
46.
b+1
2b3
· 2
2
8b
b −1
b
4b − 4
47.
14x2 y 2 x2 − 9
·
3x + 9 8xy 3
1
w2 + 3w
48.
x2 − 16
4
·
2x − 8 6x + 24
52.
5a − 5b
3a2
· 2
2
9a
a − b2
7x2 − 21x
12y
49.
n2 − 9 6n2 r2
·
3nr3 n + 3
2n2 − 6n
r
50.
x2 y 2
8x − 32
·
− 16
4xy 3
x2
2x
xy + 4y
51.
x2 y
2x + 2y
·
2
−y
xy
x2
2x
x−y
ALG catalog ver. 2.6 – page 231 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
ax
bn
5
3a + 3b
1
3
LF
53.
x − 2y
2x2
· 2
3x
x − 4y 2
12n2
m2 − 9n2
·
m − 3n
15mn
54.
2a + 8 2a − 2b
·
a2 − b2 a2 − 16
3d + 9 2d2 − 50
·
2d − 10 3d2 − 27
58.
4
a2 − 4a + ab − 4b
61.
(x + y)2 ax − ay
·
x2 − y 2 ax + ay
2b + 10 3b − 15
· 2
6b
b − 25
59.
a2 − 36
a2 − 49
·
2a2 + 14a 2a3 − 12a2
1
b
56.
3x2 + x 6x − 2
·
9x2 − 1
6x
60.
xy 2 + x2 y xy 2 − 9x
· 2
y 2 + 3y
xy − x3
1
3
4mn + 12n2
5m
2x
3x + 6y
57.
55.
d+5
d−3
4p − 4q p2 − q 2
·
(p − q)2 3p + 3q
62.
1
(a + 6)(a − 7)
4a3
4
3
63.
6w − 12 (w + 2)2
·
7w + 14 w2 − 4
xy − 3x
y−x
6
7
(x − 3y)2
24x3
·
3x2 − 9xy x2 − 9y 2
64.
8x2
x + 3y
65.
a2 + 7a + 12 a − 3
·
a2 − 9
a+3
66.
x − 2 x2 − 2x − 8
·
x2 − 4
x+2
67.
y2 − 9
y+2
·
5y + 10 y 2 + 7y + 12
y−3
5(y + 4)
68.
n2 + 7n + 10 4n + 12
· 2
n+3
n −4
69.
w2 + w − 6 w2 − 2w
·
3w − 6
2w + 6
w(w − 2)
6
70.
3a − 18
3a − 3
·
a2 − 5a − 6 a2 − a
71.
x2 + 5x + 6 x2 − 7x + 12
·
x2 − 9
2x + 4
72.
y 2 − 25
5y − 5
· 2
2
y − 2y − 15 y + 4y − 5
73.
c2 − 2c − 8
4c + 4
·
c2 + 3c + 2 c2 − c − 12
4
c+3
74.
w2 − w − 20 w2 + 7w + 12
· 2
w−5
w + 8w + 16
75.
a2 − 2a − 8
2a + 8
·
a2 − 5a + 4 a2 − a − 6
2(a + 4)
(a − 1)(a − 3)
76.
m2 + 9m + 18
m−4
·
m2 + 6m + 9 m2 − 9m + 20
77.
y2 − 4
y2 − 9
·
y 2 − y − 12 y 2 − y − 6
y−2
y−4
78.
x2 − 16
x2 − 25
·
x2 − x − 30 x2 − x − 20
79.
n2 − 9n + 14 4n3 + 16n2
·
n2 + 7n + 12 3n2 − 21n
80.
a2 − 11a − 12 a2 − 11a + 24
· 2
a3 − 9a
a − 7a − 8
81.
2w2 + 5w + 2 w2 + w − 6
·
w2 − 4
2w2 + w
w+3
w
82.
y 2 − 25
2y 2 + y
·
2y 2 + 11y + 5 y 2 − 3y − 10
83.
2x2 − 3x − 9 x2 + x − 6
· 2
2x2 − 18
2x − x − 6
1
2
84.
2c2 + c − 10
c3 − 9c
·
c2 + c − 6 2c2 − c − 15
85.
a2 + 5a + 6 a2 − a − 20
·
a2 − 2a − 15 a2 + a − 2
a+4
a−1
86.
n2 + 6n + 5 n2 − 5n + 6
·
n2 + 2n − 8 n2 + 2n − 15
87.
r2 + 2r − 35 r2 + 9r + 14
·
r2 + 10r + 21 r2 − 3r − 10
88.
y 2 + 5y − 24 y 2 + 3y − 10
·
y 2 + 12y + 32 y 2 − 6y + 8
89.
y 2 + 7y + 10
y 3 − 4y
· 2
2
3y − 6y
y + 4y + 4
91.
2w2 − 6w
2w2 + 3w − 20
·
2
6w − 18w
2w2 + w − 15
93.
18r2 + 3r − 36 6r2 − r − 12
· 2
9r2 − 16
8r + 20r − 48
a+4
a+3
x−4
2
4n(n − 2)
3(n + 3)
r+7
r+3
y+5
3
90.
w+4
3(w + 3)
3(2r + 3)
4(r + 4)
x−4
x+2
4(n + 5)
n−2
9
a(a + 1)
2a3 − 2a2
a2 + 7a + 6
·
a2 − 1
+ 8a + 12
a2
5
y+3
w+3
m+6
(m + 3)(m − 5)
x−4
x−6
a − 12
a(a + 3)
y
y+2
c
n+1
n+4
(y − 3)(y + 5)
(y + 4)(y − 4)
2a2
a+2
92.
3x2 + 6x − 9
x2 − 9x
· 2
2
x + x − 2 4x − 24x − 108
94.
15w2 + 65w + 20
4w2 − 25
·
2w2 + 3w − 20 24w2 + 52w − 20
ALG catalog ver. 2.6 – page 232 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
3x
4(x + 2)
5(3w + 1)
4(3w − 1)
LF
95.
6x2 − 15x − 36
6x3 − 6x
· 2
4
3
2
3x − 9x − 12x 8x + 4x − 12
97.
y
y 2 − 6y + 5 y 2 − 4y − 5
·
·
y−5
y2 − 1
y 2 − 5y
99.
3n + 9
2n2 − 5n − 3
n2 − 2n − 8
· 2
·
2
n − n − 12 2n + 5n + 2
n2 − 9
3
2x
1
3
n+3
96.
6y 2 − 32y − 24
2y 3 − 8y
·
4
3
2
2
3y − 12y − 36y 12y − 16y − 16
98.
c2 − 144 c2 − 9
c2
· 2
· 2
c−3
c − 12c c + 15c + 36
100.
x2 − 5x + 4 x2 − 3x − 18 x2 − x − 20
·
·
x2 + 5x + 6 x2 − 6x + 5 x2 + 3x − 28
1
3y
c
(x − 6)(x + 4)
(x + 2)(x + 7)
101.
103.
r3
r3 s − rs3
1
r 2 − s2
·
· 2
2
2
− 2r s + rs r + s 2r s − 2rs2
r+s
2r(r − s)
x2 − 6xy + 9y 2 x2 − 4xy − 21y 2 x − 3y
· 2
·
x2 − 9y 2
x − 8xy + 16y 2 x − 7y
(x − 3y)2
(x − 4y)2
102.
a − b a2 + 2ab + b2 a2 b − b3
·
·
ab + b2 a2 − 2ab + b2
a+b
104.
ax − ay
x2 − xy − 12y 2
ax − 2ay
·
· 2
x2 + xy − 6y 2
a2 x − a2 y
x − 5xy + 4y 2
a+b
1
x−y
105.
107.
r2
r 2 − p2
r2 + 9r + 20
· 2
+ 4r + rp + 4p r + 2rp − 3p2
x2 + 6x + 5
x2 + x − 2
· 2
4
2
x − 5x + 4 x + 3x − 10
r+5
r + 3p
106.
ab + 3b + a2 + 3a
a2 − 25
· 2
2
ab + 5b + a + 5a a − 2a − 15
1
1
(x − 2)(x − 2)
108.
w4 − 29w2 + 100 w2 − 2w − 15
· 2
w2 − 25
w − 3w − 10
(w − 2)(w + 3)
111.
6w + 6x 3x − 3w
·
wx − x2
18
Factors of −1
109.
x2 − x
x
·
2y − 2xy y
−
x2
2y 2
110.
5 − 5c 18c2
· 2
15c
c −c
−6
−
113.
9
4−x
·
x2 − 16
21
−
117.
3
7x + 28
a+5
2a − 1
·
1 − 4a2 a2 − 25
−
114.
15y 9 − y 2
·
y+3
5y 2
−
118.
1
2a2 − 9a − 5
121.
r2 − 2r − 8
3−r
· 2
2
r − 4r − 12 r − 7r + 12
123.
9 − a2
a2 + 6a + 8
·
a2 + 5a + 6 a2 − 7a + 12
125.
6 − x − x2
(x + 4)2
· 2
2
x − 16
x − 5x + 6
127.
y 2 + 7y + 10 6 + y − 2y 2
·
2y 2 + 13y + 15
2y 3 − 8y
−
−
−
3x − 9
y
9r2 − 1 2 − r
·
r2 − 4 3r + 1
−
119.
a+4
a−4
(x + 3)(x + 4)
(x − 4)(x − 3)
1
2y
w+x
x
8 − 4x 8xy 2
·
10x2 y x2 − 4
−
3r − 1
r+2
1
r−6
−
115.
−
4x − 20 6x + 30
·
3
25 − x2
120.
16y
y 4 − 16
·
8y − 4y 2 y 2 + 4
1
4
y 2 − 6y + 8 y 2 + 9y + 18
· 2
4−y
y +y−6
124.
n2 + 3n − 10
16 − n2
·
n2 − 5n + 4 n2 + 2n − 8
128.
n2 − 16
4 − 3n − n2
· 2
+ 5n + 6 n − 5n + 4
n2
−(4y + 8)
−(y + 6)
−
−
10a + 8 − 3a2
9a3 − 81a
· 2
2
a − a − 12 3a − 7a − 6
ALG catalog ver. 2.6 – page 233 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
y
2y − 8
116.
122.
126.
y2
y+2
·
2y + 4 4y − y 2
16y
5x2 + 10x
b2 − a2 3a2 + 3b2
· 4
12
a − b4
−
112.
n+5
n−1
(n + 4)(n + 4)
(n + 2)(n + 3)
−9a
−8
LG
Topic:
Dividing rational expressions. See also categories AI (fractions) and DM (monomials).
Directions:
6—Simplify. 4—Divide. 9—Simplify (assume no denominator equals zero).
100—Give the restriction(s) on each variable, then simplify.
1.
a
b
÷
c
d
5.
a2
÷ (−5ac)
6c
9.
3x 21x
÷
4y
16y
13.
24ab
6ab2
−
÷
−
16b2
18a2 b
ad
bc
a
−
30c2
4
7
2.
x 6
÷
3
y
6.
2 a x
−10ax ÷ −
5
10.
5a 30b
÷
3b
9a
14.
2x2
4x
÷
7y
21y 2
18.
18n2 r
6r2 s
÷ −
33ns2
11nr
xy
18
50
a
a2
2b2
3xy
2
1
k2
3.
k÷
7.
1
2y
÷
2yz
z
11.
ab2
b
÷ −
−
ac
c
15.
a2 bc
abc2
÷
8m
12m
19.
60u2 x 20wx2
÷
63uw
21uw2
k3
1
4y 2
b
3a
2c
4.
1
÷ (−n)
n
8.
14ab 7bc
÷
c
a
12.
mr
nr
÷
12
9
16.
3y 2 z
2y 2
÷
−
2w
3wz
−
1
n2
2a2
c2
3m
4n
9z 2
4
9a2
2b2
17.
8xy
10wxy
÷
15wy 2
16x2
64x2
75w2 y 2
21.
−
6r3
12r
÷
3p − r
3p − r
2
r2
22.
u2
x
20. −
n2
s3
−
3a + 4 3a + 4
÷
5a2
10a
2
a
23.
x+1
x+2
÷
2x + 5 2x + 5
24.
x+1
x+2
25.
4a
16a2
÷
3a − 12 5a − 20
26.
27.
12
5d
5
12a
29.
4d + 16 5d + 20
÷
3d2
9d
3m − 12 6m − 24
÷
2m + 8
4m + 16
30.
2n − 6
3n − 9
÷
8n + 24 4n + 12
1
3
31.
x+3
x−5
x−5
34. (4p + 18) ÷
6y
12xy
÷
xy + xz
(y + z)2
38.
6p + 27
12
35.
n + 3 n2 − 9
÷
9n
6
42.
2
3n(n − 3)
45.
46.
r2 − 4
r−2
÷
r2 − 25 r − 5
39.
32.
11u
22u
÷
−1 u−1
u2
14xy 3
4x2 y 2
÷ 3
5x + 15 x − 9x
r+2
r+5
50.
m − 4 m2 − 16
÷ 2
m+3
m −9
m−3
m+4
x − 3y
(x − 3y)2
÷
x + 3y
(x + 3y)2
43.
1
w2 − 4w
÷ 2
3w + 12
w − 16
36.
4p2 − 49 4p2 + 14p
÷
2p − 7
6p2
40.
a2 − ac
a2 − c2
÷
3a + 9c 2a + 6c
2a
3(a + c)
ALG catalog ver. 2.6 – page 234 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
x2 (a − b) ax − bx
÷
y 2 (a + b)
ay + by
(a − 1)2
3a − 3
÷
(a + 1)2
7a + 7
7a − 7
3a + 3
44.
r2 − 25
÷ (2r − 10)
r2 + 5r
1
2r
48.
4xy 2
16x2 y
÷
x2 − 2x x2 y − 4y
4x + 8
3p2
51.
2x − 6y
6x − 18y
÷
12x + 4y
9x + 3y
x
y
1
3w
47.
xy 2
4xy
÷
6x − 4y
24x − 16y
1
4
x + 3y
x − 3y
2x2 (x − 3)
35y
x − 4y
2x2 y
49.
(c − d)2
4c − 4d
÷
15c
12c2 d
1
2(u + 1)
x2 − 16y 2
x2 + 4xy
÷
4xy 2
2y
(x − y)
2x − 2y
÷
cx + cy
x+y
2y − 3 2y − 3
÷
y−3
y+2
y
1
2c
c2 d − cd2
5
y+z
2x2
41.
28.
6
8
37.
10s + 10t 5s + 5t
÷
3s − 3t
9s − 9t
1
2ab2
y+2
y−3
27y
4x
1
33. (x + 3) ÷
9a − 18b 4a − 8b
÷
5x2 y
15xy 2
35ac2
14a2 b
÷
25bc3
5c
52.
ac + ad 5c + 5d
÷ 2
ac − ad
c − d2
c+d
5
LG
53.
24a2 − 6b2
2a2 + ab
÷ 2
3a − 3b
a − ab
u2 − 16
6u2 − 24u
÷ 2
2
u + 4u
u − 2u
54.
u−2
6u
4a − 2b
57.
55.
a2 − b2
(a + b)2
÷
8a
4a3
5
3
59.
a2 (a + b)
2(a + b)
3
r(r + 3)
(r − s)2
r2 + s2
÷
r 2 − s2
(r + s)2
a2 − a − 6 a2 − 9
÷
a−2
a+3
63.
y + 12
y 2 − 7y + 10
÷ 2
y−5
y − 10y + 25
65.
r2 + 5r − 6 3r + 18
÷ 2
3r − 3
r −r
a+2
a−2
9x + 36
3x + 12
÷ 2
+ 5x + 4
x −1
x2
62.
y + 12
y−2
64.
r(r − 1)
9
66.
3(x − 1)
x+4
68.
(x + 5)(x + 2)
(x − 2)(x − 5)
x−4
x+3
÷
x2 − 16 x2 + 7x + 12
w2
a2 − 4
a2 + 7a + 10
÷
2a + 6
4a + 12
2y + 4
y 2 + 3y
÷ 2
− y − 6 3y − 27
y2
n2 + 10n + 25
÷ (n2 − 25)
n2 + 5n
71.
r2 − r − 30
÷ (r2 − 2r − 24)
r+4
r+5
(r + 4)2
72. (a2 − 6a + 9) ÷
73.
y 2 − 3y − 18 y 2 − 2y − 15
÷
y2 − y − 2
3y + 3
3(y − 6)
(y − 2)(y − 5)
74.
75.
c2 − 2c − 8 c2 − c − 6
÷
2c + 6
c2 − 9
77.
d2 + 9d + 18 d2 + d − 6
÷
2d2 + 4d
d2 − 4
r2 − 1
r2 − r − 2
÷
−r−6
r2 − 4
r2
1
n(n − 5)
70. (y 2 − 3y) ÷
c−4
2
76.
1
w−3
w2 − 8w + 15
÷
+ 8w − 9
w+9
69.
79.
(x + 5)2
x2 − 25
÷
x2 − 4
(x + 2)2
60.
r 2 − s2
r 2 + s2
61.
67.
3r2 + 3s2
r 2 + s2
÷
3
r − 9r
r−3
56.
a+b
2a
3p − 6
5p + 10
÷
p2 − 4
(p − 2)2
58.
2ab2
4a
÷ 3
2
3
−a b
a − ab2
a 3 b2
y 2 − 4y + 3
y2 − y
1
(w − 1)(w − 5)
2(a − 2)
a+5
6
y
y2
a2 − 9a + 18
3a − 18
3(a − 3)
w2 − 4w + 3
w2 + w
÷
w2 + 3w − 18 w2 + 8w + 12
x2 − 25
2x + 10
÷
− 6x + 5 x2 + 2x − 3
(w − 1)(w + 2)
w(w + 1)
x+3
2
x2
78.
k2 − 9
4k 2 + 12k
÷
k 2 + 3k − 18 k 2 + 10k + 24
80.
p2 − p − 20
p2 − 16
÷ 2
2
p −9
p − p − 12
1
82.
y 2 − 4y − 21 y 2 − 6y − 7
÷ 2
y 2 + 7y + 12
y −y−6
c+4
c−2
84.
3x2 + 8x + 4 2x2 + 5x + 2
÷ 2
9x2 − 4
3x − 5x + 2
86.
x2 + 5x + xy + 5y
x2 + 6x + 5
÷
x2 + 2x + xy + 2y
x2 − 4
88.
y 4 − 13y 2 + 36 y 2 − 5y + 6
÷ 2
y2 − 4
y − 3y + 2
d+6
2d
r−1
r−3
81.
w2 + 2w − 3 w2 + 6w + 9
÷ 2
w2 + w − 2
w + 5w + 6
83.
c2 − c − 12 c2 − 6c + 8
÷
c2 + 4c + 3 c2 + 5c + 4
85.
a2 + 7a + 12
a2 + 3a + ab + 3b
÷
a2 + ab − 2b2
a2 − b2
87.
x4 − 10x2 + 9 x2 + 2x − 3
÷ 2
x2 − x − 6
x + 3x + 2
a+4
a + 2b
(x + 1)2
Factors of −1
89.
k−3 3−k
÷
9
12
−
4
3
90.
x+y
x+y
÷
w−x x−w
91.
n
1
÷
p−r
r−p
−n
92.
a−b b−a
÷ 2
c
c
ALG catalog ver. 2.6 – page 235 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−1
−c
k+4
4k
p−5
p−3
(y − 3)(y + 2)
(y + 1)(y + 4)
x−1
2x + 1
x−2
x+1
(y + 3)(y − 1)
LG
93.
y−7
÷ (7 − y)
y+7
95.
a−6
6−a
÷
3a + 6 5a + 10
97.
4 − a2
a+2
÷
a3
a
99. (20 − 4y) ÷
−
−
1
y+7
−
94. (r2 − 2r) ÷
3u + 9
u2 + 3u
÷
uw − w
1−u
98.
h2
3h
÷
h − 4 16 − h2
−12
100.
2x + 8
1
÷
2
3x − 48 36 − 9x
y(y − 1)
4(y + 1)
102.
x2 − 9
3−x
÷ 2
(x + 1)2
x −1
104.
p2 − q 2
px + qx
÷
(p − q)2
pq − p2
106.
2w2 + 9w + 4
2w2 + w
÷ 2
2
16 − w
w − w − 12
108.
6x2 + 17x + 5 3x2 − 5x − 2
÷
3x − x2
x3 − 4x
a−2
a2
101.
(y + 1)2
1 − y2
÷
8y 2
2y 3
103.
n − n2
(n − 1)2
÷ 2
10n + 8 5n + 4n
105.
a2 − 3a − 4
5 − 5a2
÷ 2
2
a + a − 20 a + 4a − 5
107.
a2 + 3a − 10
25 − a2
÷
2a2 − a − 6
2a2 − 7a − 15
−
−r2
96.
5
3
2y 2 − 50
6y + 30
2−r
r
−
n2
2(n − 1)
−
1
5
−1
ALG catalog ver. 2.6 – page 236 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−
−
3
wu
3(4 + h)
h
−6
−
(x + 3)(x − 1)
x+1
−
p
x
−
w+3
w
−
(x + 2)(2x + 5)
x−3
LH
Topic:
Adding and subtracting rational expressions (common denominators)
Directions:
6—Simplify. 9—Simplify (assume no denominator equals zero).
100—Give the restriction(s) on each variable, then simplify.
1.
2n 8n
+
3
3
10n
3
2.
1
5
+ 2
2
7x
7x
5.
3r
8r
−
p
p
5r
p
6.
6
14
−
ab ab
9.
6
4
+
a+b a+b
10.
10x
3x
+
x−y
x−y
−
10
a+b
6
7x2
−8
ab
13x
x−y
3.
11 6
−
a
a
7.
−5x 2x
+
3
3
11.
5n
3n
−
p−6 p−6
5
a
−x
2n
p−6
4.
9a 4a
−
c
c
8.
−7 −11
+ 2
n2
n
12.
3
5
−
w+7 w+7
−
13.
m + n 2n
−
x
x
m−n
x
14.
a + 2b −b
+
y
y
17.
7
8
+
10c2
10c2
3
2c2
18.
3z
5z
+
4y
4y
21.
14
4
−
12mr
12mr
22.
13
18
−
15x 15x
26.
a 5a 4a
−
+
c
c
c
25. −
8
3
4
+ −
y
y
y
5
6mr
−
9
y
15.
2a + x a − x
+
n
n
19.
8
13
+
6m 6m
7
2m
1
3x
23.
a
5a
− 2
2b2
2b
−
0
27.
4x 9x 14x
+
−
8
8
8
a+b
y
2z
y
−
3a
n
2a
b2
−
x
8
2
5
1
+
−
3x 3x 3x
2
x
30. −
−
33.
7a
5a
14a
−
+
18b 18b
18b
8a
9b
34.
9n 8n 20n
+
−
7r
7r
7r
31.
6n
3n
n
−
−
4m 4m 4m
n
2m
m
a
+
a+m a+m
38.
1
x
2x
+
3y
3y
24.
21p 16p
−
5
5
28.
9
22
6
−
+
11d 11d 11d
32.
10r − 5 r − 5
−
3r
3r
3
42.
y
a
x
y
p
7
11d
6
7
14
+
+
3w2
3w2
3w2
9
w2
11
7
5
+
−
2
2
12y
12y
12y 2
35.
6
22
8
+
−
9cd 9cd 9cd
8
3cd
36. −
−
39.
1
x
a
+
3(a + x) 3(a + x)
40.
1
3
41.
2
w+7
20.
3n
r
xy
1
−
xy − 1 xy − 1
18
n2
2y − x y − x
−
a
a
3
4y 2
37.
−
16.
−
29.
5a
c
8k 2 − 1 2k 2 + 1
+
2k
2k
5k
43.
11n 13n 17n
+
−
15x
15x
15x
n
x
−5
c
+
2(c − 5) 2(c − 5)
1
2
2x + 3y
2x − 3y
−
4xy
4xy
44.
7a + 6 3a − 6
+
10a2
10a2
48.
3r − 5p 6p − 2r
+ 2
r 2 + p2
r + p2
1
a
3
2x
45.
2 − 8n 1 − 3n
+
2n − 5 2n − 5
46.
3 − 11n
2n − 5
49.
7x − 3 7x + 1
−
8x
8x
w2 − 3w
4w2 + 6w
−
w+3
w+3
47.
a−b
3w
−1
2x
50.
2a2 + b2
a2 + 2b2
−
a+b
a+b
3a + 10b 5a − 2b
+
4
4
51.
−4u u − 6
+
3u2
3u2
r+p
r2 + p2
−u − 2
u2
52.
2w + 3
w
2a + 2b
53.
a − 2b a + 4b
+
a+b
a+b
2
54.
14x + 2 2x + 10
−
3x − 2
3x − 2
5w + 1 w − 5
−
2w
2w
4
55.
4m + 13 m − 3
+
m+2
m+2
ALG catalog ver. 2.6 – page 237 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
5
56.
9d
6c + 3d
−
c−d
c−d
6
LH
57.
−4y
3y 2 + y
+
y−1
y−1
4s2 − st
3s2
−
s−t
s−t
58.
3y
s
59.
2n2 + 3n n2 − 3n
−
n+6
n+6
3r2 − 1 6r + 1
+
r+2
r+2
60.
3r
n
61.
8a − 9 3a a − 3
−
−
10
10
10
63.
r
7r − 3 2r + 5
+
−
6x
6x
6x
65.
4y + 3 5(y − 1) 2(5y − 2)
−
+
3y − 4
3y − 4
3y − 4
67.
4(3x + 2) 2(5x + 2) 2x + 4
−
−
x+y
x+y
x+y
69.
3x2 − 2x + 9 7x2 − 10x − 11
−
6x2
6x2
71.
3r − 4 −4r + 1 7 − 5r
+
−
3r + 2
3r + 2
3r + 2
73.
4(y 2 − 2y − 5) y(4y − 11)
−
y+2
y+2
75.
(b − 1)(b + 1) (b − 2)2
−
b
b
77.
p2 r + 3pr
p2 r + 2p
−
pr + 2p
pr + 2p
2a − 3
5
3r − 4
3x
9y + 4
3y − 4
−2x2 + 4x + 10
3x2
4r − 10
3r + 2
3y − 20
y+2
4b − 5
b
78.
ab − 4b2
3ab + 4b2
+
2a + 4ab
2a + 4ab
mn
n2
+
(m + n)2
(m + n)2
82.
x2
xy
−
(x − y)2
(x − y)2
n
m+n
85.
y2
1
−
y+1 y+1
−5y − 11 8y + 2 y + 1
+
+
8
8
8
64.
4p −12p + 17 p − 4
−
+
15
15
15
66.
3(3a − 2)
a−4
5(1 − 4a)
+
−
2a − 1
2a − 1
2a − 1
68.
2(2m + 1)
3(m + 3)
2m + 7
+
−
mn + n
mn + n
mn + n
70.
−7a2 + a − 5 8a2 − 2a + 4
−
3a
3a
−5a2 + a − 3
a
72.
9y − 14 12y − 17 7 − 3y
−
+
4y − 5
4y − 5
4y − 5
−6y + 10
4y − 5
74.
3(2n + 5) −5(n2 + n − 3)
−
2n − 1
2n − 1
76.
(c + 3)2
(c − 2)(c + 2)
−
5c
5c
79.
n2 + 2n
n2 + 8
−
3n − 12 3n − 12
83.
a2
1
−
(a − 1)2
(a − 1)2
2
3
y−2
2
7 − 3p
5
x
x−y
6c + 13
5c
80.
y2 + y
y2 − 3
−
5y + 15 5y + 15
84.
−25
x2
+
(x + 5)2
(x + 5)2
86.
y−1
a2
4
−
2+a 2+a
x−5
x+5
a+1
a−1
a−2
87.
y 2 − 4y
y − 10
+
y+2
y+2
90.
x − 6 x2 − 6
+
x+4
x+4
x−3
91.
c
5
+
c2 − 25 c2 − 25
2r
3
+
4r2 − 9 4r2 − 9
88.
93.
2w2 + 2w
w2 + 2w + 9
−
w−3
w−3
95.
c + 14
c−6
+
c2 − 16 c2 − 16
97.
x2 + 3x 2(2x + 1)
−
x−2
x−2
99.
2(n2 − 3n) n2 − 9
−
n−3
n−3
101.
2a2 − 5a − 2 a2 − a − 5
−
a2 − 9
a2 − 9
2a2 − 1
3a2 − 5
− 2
+ 5a + 6 a + 5a + 6
a2
1
2r − 3
u2 + 7u 3u + 12
−
u−2
u−2
−12 − 2a 3 − a2
−
a−5
a−5
92.
u+6
y−5
103.
5m + 4
mn + n
5n2 + 11n
2n − 1
1
c−5
89.
15
2b
2b + 1
3r + 2
r+2
81.
0
62.
a+3
94.
2n2 − 5n n2 − 5n + 25
−
n+5
n+5
96.
2x + 8
x−2
+ 2
x2 − 4
x −4
x+1
98.
r2 + 5r
−2(r − 1)
+
r+1
r+1
n−3
100.
−2(y 2 − 4) 3y 2 + 6y
+
y+4
y+4
a−1
a+3
102.
8w − 12 w2 − 5w + 8
+
w2 − 1
w2 − 1
a−2
a+3
104.
w+3
2
c−4
u2
ALG catalog ver. 2.6 – page 238 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
u−5
3
x−2
r+2
y+2
w+4
w+1
u2 − 4u − 10
4u − 6
+ 2
− 5u + 4
u − 5u + 4
u+4
u−1
1
5
LH
Factors of −1
106.
s
r
−
r−s r−s
108.
1
x
−
4(x − 1) 4(x − 1)
110.
1
1
+
x−4 4−x
0
112.
x
x
+
x−5 5−x
0
1
114.
u2
3u
+
u−3 3−u
u
−1
116.
xy
y2
+
y−x x−y
−y
−(m + 3)
118.
d2
c2
−
c−d c−d
1
a+b
120.
105.
3
2x
−
2x − 3 2x − 3
107.
q
p
−
x(p − q) x(p − q)
109.
2
5
−
a−b b−a
111.
n
k
−
n−k
k−n
113.
c
3
+
c−3 3−c
115.
b
a
+
a−b b−a
117.
9
m2
−
m−3 m−3
119.
a2
−1
−
1
x
7
a−b
n+k
n−k
b
a
− 2
2
−b
a − b2
−
n2
ALG catalog ver. 2.6 – page 239 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−1
−
1
4
−(c + d)
6
n
− 2
− 36 n − 36
−
1
n+6
LI
Topic:
Adding and subtracting rational expressions (different denominators)
Directions:
6—Simplify. 9—Simplify (assume no denominator equals zero).
100—Give the restriction(s) on each variable, then simplify.
1.
1 1
−
a
b
5.
a
b
+
8 12
9.
5
2
+
xy
yz
13.
5
8
− 2
ax3
a x
17.
3w
w
w
+ −
4
5
8
b−a
ab
3a + 2b
24
5x + 2z
xyz
5a − 8x2
a2 x3
33w
40
2.
a
c
+
b
d
6.
x
3x
−
10
4
10.
b
a
−
2x 3x
14.
4
3
+ 4
x2 y
xy
18.
x 2x
x
+
+
5
3
10
2y − 3x
xy
4.
w
5
+
5
w
11a
18
8.
r
s
−
15 10
2r − 3s
30
12.
2
7
+
a2
ab
7a + 2b
a2 b
16.
n
3
−
8m2
2nm
20.
a c
b
− −
c
b a
3.
2
3
−
x y
−13x
20
7.
4a a
+
9
6
3a − 2b
6x
11.
9
4
−
np n2
15.
2
c
+
3cd 6d2
19.
2 2 2
− +
a
b
c
ad + cb
bd
3y 3 + 4x
x2 y 4
29x
30
9n − 4p
n2 p
4d + c2
6cd2
2bc − 2ac + 2ab
abc
21.
4
5
6
+
+
b2
ab bc
22.
2
4
3
− 2 2 + 3
3
r s r s
rs
26.
n+3 n−2
−
5
3
30.
−2n + 19
15
33.
x+y
2x − y
+
6
14
34.
2a − 5b a + 4b
−
7a
3a
38.
2
3
+
a−5 a−7
42.
k
k
−
k−1 k+3
b
a
+
a−b a+b
31.
2
a2 bc
−
35.
3
2
−
2
ab c abc2
28.
m − 2n 3m + n
+
4m
5m
8
5
−
x+2 x+5
46.
u
3u
+
u+4 u−5
32.
50.
c
d
−
c − 3d 2c + d
2c2 + 3d2
2c2 − 5cd − 3d2
a + 2b 2a + b
−
2
5
a + 8b
10
2a + 2b
45
36.
4m − 1 m + 4
+
4
18
38m − 1
36
39.
a−b b−c
−
ab
bc
40.
2ac − bc − ab
abc
43.
12
6
+
p+1 p−4
47.
2b
8
−
b − 3 b − 12
44.
m
2r
−
2m − r
m+r
m2 − 3mr + 2r2
2m2 + mr − r2
ALG catalog ver. 2.6 – page 240 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
3
3
−
y−3 y+6
27
y 2 + 3y − 18
48.
2b2 − 32b + 24
b2 − 15b + 36
51.
m − 3n n − 2p
+
mn
pm
n2 − 5np + mp
mnp
18p − 12
p2 − 3p − 4
4u2 − 11u
u2 − u − 20
a2 + b2
a2 − b2
1
1
1
+ 2 2 + 3
3
6xy
5x y
2x y
5x2 + 6xy + 15y 2
30x3 y 3
3w + 1 w − 8
+
3
4
a+b a+b
−
9
15
c
b
a
−
+
ab ac bc
c2 − b2 + a2
abc
15w − 20
12
−3x + 9
x2 + 7x + 10
4k
k 2 + 2k − 3
49.
n+7 n−5
−
12
8
24.
2bc − 3ac − 2ab
a2 b2 c2
17m − 6n
20m
5a − 29
a2 − 12a + 35
45.
x−y
x − 5y
+
2
3
2x
y
3
+
−
3y
2x xy
−n + 29
24
−a − 43b
21a
41.
27.
5x − 13y
6
13x + 4y
42
37.
2
1
4
+
− 2
2
ab
7ab 5b
n2 − 12m
8nm2
a2 b − c2 a − b2 c
abc
4x2 + 3y 2 − 18
6xy
−28a + 10b + 35
35ab2
4s2 − 3rs + 2r2
r 3 s3
29.
23.
an2 − bmn − cm2
m2 n2
4ac + 5bc + 6ab
ab2 c
25.
a
b
c
−
−
m2
mn n2
w2 + 25
5w
6
2w
+
w−9 w+3
2w2 − 12w + 18
w2 − 6w − 27
52.
7x
3y
+
x+y
x−y
7x2 + 10xy − 3y 2
x2 − y 2
LI
53.
x+1 x−1
−
x−1 x+1
4x
x2 − 1
54.
a + 2b a + 3
+
a − 3b a + 2
55.
2a2 − ab + 5a − 5b
a2 − 3ab + 2a − 6b
57.
1
5r
+
r + 4 2(r + 4)
58.
10x
3
−
4(x − 1) x − 1
61.
5
x + 20
−
6 3x + 12
3x − 20
6x + 24
62.
a
2a
+
10y − 5
5
59.
4
c
+
a2 c a2 + a
66.
1
3
+
2u2 − u 2u
63.
69.
r
5r
−
2r − 4 12r − 24
70.
w−3
2
−
30w + 5 18w + 3
r
12r − 24
73.
74.
a
4
+
6a − 4 3a2 − 2a
2
3d
+
c2 + c c2 − cd
78.
2
5
−
6x − 6y
3x + 3y
2c + 3cd + d
c(c + 1)(c − d)
81.
82.
1
3(x + y)
+ 2
3x + y
9x − y 2
−5a
(a + 5)(a − 5)
85.
6
u − 7w
+
u2 − 4w2
5u − 10w
86.
91.
75.
10
(x + 5)(x + 1)
1
y
+
+ 2yz − 3z 2
y + 3z
y 2 − yz + 1
(y + 3z)(y − z)
k 2 + 2k − 1
k 2 + 4k + 4
3u
u
+
7w
7w + 21
64.
2
5
−
3xy
xy − 3x
x
y+4
+
2x + 2y
3x + 3y
68.
r−1
1
−
rs − s s2
72.
2n
m
+
3m − 6n 4m − 8n
0
3m + 8n
12m − 24n
b
3
−
ab − 5b2
2a − 10b
76.
2
3y − 2
+
2y + 6 4y 2 + 12y
79.
7y − 2
4y 2 + 12y
4
n
−
r2 + 2r
nr − 3r
80.
2
1
+
10s + 5t 5s + 15t
2n − rn − 12
r(r + 2)(n − 3)
83.
4s + 7t
5(2s + t)(s + 3t)
3
4 − 3m
+
16 − m2
m−4
84.
5
6a
−
a2 − b2
a−b
a − 5b
(a + b)(a − b)
6m + 8
(m − 4)(m + 4)
87.
a+5
3
+
4a − 2 4a2 − 1
88.
2x
x2 + 9
− 2
2x + 6 x − 9
−3x
x−3
8a + 13
2(2a + 1)(2a − 1)
90.
3a
4
−
a2 − a − 6 a − 3
92.
2
2w + 3
+
w − 4 w2 − 5w + 4
4w + 1
(w − 4)(w − 1)
94.
2
3x + 9
−
x2 + 3x − 10 x + 5
x + 13
(x + 5)(x − 2)
96.
5r
3
−
r2 + 8rs + 12s2
r + 6s
−a − 8
(a − 3)(a + 2)
3y
4
+
y 2 + 5y − 14 y − 2
95.
4
z−5
+
z + 2 z 2 − 2z − 8
97.
a−8
7
+
a2 − 6a + 9 4a − 12
11a − 53
4(a − 3)2
98.
5m
1
+
2m + 6 m2 + 4m + 3
99.
4
3
−
x2 − 6x + 5 5x − 25
23 − 3x
5(x − 1)(x − 5)
100.
8
y+1
−
3y − 18 y 2 − 5y − 6
5z − 21
(z − 4)(z + 2)
a+4
1
−
4a − 4 8a
2a2 + 7a + 1
8a2 − 8a
93.
7(y + 4)
(y + 7)(y − 2)
k
1
−
k + 2 (k + 2)2
1
10b − 2a
2n2 + 16n
3(n + 4)(n − 4)
1
x−5
−
x + 1 x2 + 6x + 5
y2
2n2
4n
−
− 16 3n + 12
n2
60.
3x + 2y + 8
6x + 6y
2
3x − y
11u − 23w
5(u + 2w)(u − 2w)
89.
71.
x + 9y
6(x − y)(x + y)
a
a2
− 2
a + 5 a − 25
8a
2
+
(a − b)2
a−b
−13y − 6
3xy 2 − 9xy
a2 + 8
6a2 − 4a
1
xy + 4y
77.
67.
3w − 19
90 + 15
2y + 1
6
−
xy + 4y
3x + 12
5a2 + 11a − 15
2a2 + 15a + 25
4uw + 9u
7w2 + 21w
6u − 1
4u2 − 2u
ac2 + 4a + 4
a3 c + a2 c
2a − 1
a−2
+
a+5
2a + 5
10a − 2b
a2 − 2ab + b2
−a + 4ay
10y − 5
65.
56.
2x2 + 2y 2
x2 − y 2
5x − 6
2x − 2
5r + 2
2r + 8
x−y
x+y
+
x+y
x−y
ALG catalog ver. 2.6 – page 241 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
2(r − 3s)
(r + 6s)(r + 2s)
11m + 1
2(m + 3)(m + 1)
5
3y − 18
LI
101.
4
1
−
w2 + 7w + 10 w2 + w − 20
102.
3
2
+
r2 − 11r + 18 r2 + 4r − 12
5r
(r − 2)(r − 9)(r + 6)
103.
6
1
−
c2 + c − 6 c2 + 11c + 24
104.
2
5
+
n2 + 7n + 12 n2 − 4n − 21
7n + 6
(n + 3)(n + 4)(n − 7)
105.
5
2
−
y 2 − 9 y 2 + 8y + 15
25 − 3y
(y + 3)(y − 3)(y + 5)
106.
7
2
+
z 2 − 6z + 7 z 2 − 1
107.
3
4
+
p2 − 4p − 12 p2 − 4
7p − 30
(p − 6)(p + 2)(p − 2)
108.
6
1
−
x2 − 16 x2 + 8x + 16
109.
c−3
c−7
+
c + 1 c2 − 1
110.
a2 + 8 a + 1
−
a2 − 4 a − 2
111.
n+2
4
+
n + 3 n2 + 10n + 21
112.
d+3
d2 − 5
− 2
d + 2 d + 3d + 2
113.
a+3
a+3
+
a2 − a − 6 a2 − 5a + 6
114.
y−2
y+1
−
y 2 + 2y + 1 y 2 − y − 2
115.
x+4
x−4
−
x2 + 3x − 10 x2 − 6x + 8
116.
c+5
c−6
+
c2 − 5c − 36 c2 − 11c + 18
117.
4
1
3
−
−
x2 + x − 12 x2 + 7x + 12 x2 − 9
118.
6
2
4
−
−
n2 − 4 n2 + 7n + 10 n2 + 3n − 10
3(w − 6)
(w + 2)(w + 5)(w − 4)
−5(c − 4)
(c + 3)(c − 2)(c + 8)
c−4
c−1
n+6
n+7
2a(a + 3)
(a − 2)(a − 3)(a + 2)
−1
(x + 5)(x − 2)
3
(n − 3)(x + 3)(x + 4)
119.
r2
3(3z − 7)
(z − 7)(z + 1)(z − 1)
5x + 28
(x − 4)(x + 4)2
−3
a+2
4
d+1
−3(2y − 1)
(y + 1)2 (y − 2)
2c2 + c − 34
(c − 9)(c + 4)(c − 2)
26
(n − 2)(n + 2)(n − 5)
r
1
3
+ 2
+ 2
+ 8r + 15 r − r − 12 r + r − 20
120.
r2 + 14
(r + 3)(r + 5)(r − 4)
y2
5
3
1
− 2
− 2
+ 2y − 3 y − 6y + 5 y − 2y − 15
y − 33
(y − 1)(y + 3)(y − 5)
Factors of −1
122.
b
a
+
a−b b−a
17
w−4
124.
n
2n
−
a−5 5−a
7
5
+
2 − x x2 − 4
−7x − 9
(x + 2)(x − 2)
126.
3m
8
−
m2 − n 2
n−m
127.
5
a
−
b − a a2 − b2
−6a − 5b
(a + b)(a − b)
128.
k−3
2
+
k 2 − 25 5 − k
129.
1
3
3a
−
+
a − 6 a + 6 36 − a2
130.
2
6
7
+
+
y 2 − 25 5 − y
y+5
131.
3
4x
5
+
−
1 − x x2 + 11x − 12 x + 12
132.
2
4
6
−
+
a + 3 5 − a a2 − 2a − 15
121.
5
8
+
y−3 3−y
123.
11
6
−
w−4 4−w
125.
3
3−y
−5a + 24
(a + 6)(a − 6)
−4x − 31
(x + 12)(x − 1)
ALG catalog ver. 2.6 – page 242 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−1
3n
a−5
11m + 8n
(m + n)(m − n)
−k − 13
(k + 5)(k − 5)
y − 63
(y + 5)(y − 5)
6a + 8
(a − 5)(a + 3)
LJ
Topic:
Simplifying complex fractions.
Directions:
6—Simplify.
1.
7.
1
2
3
4
2
3
15
25
4
3
9
20
13.
25.
31.
3
7
+
1
7
5
21
−
11
21
7
m
9
3a
4
15a
12
4
x
1
x
3
4
+r
3
4
−r
u
w
u
w
1−
2
7x
3
14x
57.
61.
65.
69.
73.
3r
2
7
n
9
2
25
15
3
2
10
9
3
4
−
7
2
+2
3
2
−n
u
2
+
2
u
1
2u
+
−
5a
6
−
a
3
+
y
4
y
3
3b
2
b
6
n2 −25
n
n−5
4
5
−
5
7
2
21
−
10
21
x
3
u
v2
2u
v
42.
2w + u
w−u
46.
4r − 7
3r + 4
54.
14 − 4n
3n − 2n2
58.
+4
4x + 3y
6x − 4y
5a − 9b
2a + b
n+5
n
1
a
−a
3
a
a
c
a
c
b+
7
x
−6
2
x
+3
2a
5b
+
7a
15b
x
5
7
15
1
10
+
r+
66.
70.
74.
3
5
+
1
5
x+
62.
7+
1
2
5−
1
2
1
4
+
1
8
1
4
−
1
8
a
b
r−
s
t
s
t
1
x
2
n
1
n
−
−
−
1
3x
1
r
3
r
2x
5
+
3x
2
−
3c2
10d3
3y
10
7y
10
3−r
r
r2 − 9
5
3
3
4.
22.
34.
43.
7 − 6x
2 + 3x
47.
9
2
2r − n
r − 3n
4x + 3y
15x − 7y
1
− 2
r + 3r
1
6
4+
1
5
1
6
+
1
3
1
6
+
1
2
6w
3
7w
23.
14w2
29.
x
2
t
4
+s
t
4
−s
1
y
x
y
59.
63.
67.
71.
75.
a2
6
2
c
+
+3
c+
1
2c
c
2
x
y
y
x
1
y
−
−
1
a
−
a
2b
+
1
x
1
5
+
2
5
6
10
+
3
10
5
6
−
5
7
5
6
+
5
7
10b4
21a2
15b6
14a2
3u + 4
8u − 3
6a + 2
a + 84
30 − 35x
7x + 105
30.
4
9b2
36.
52.
56.
x+y
68.
b
2a
−1
5−
1
n
5+
1
n
−a
1
b
−1
3
y
+
72.
76.
1
2
+
1
4
1
2
+
1
8
m2
8
5m
8
6
5
m
5
4n
9r3
8n4
15r2
5
6rn3
1
6
a
5
2y
5
2y 2
4
y2
−
4w
3
8
9
+
16w
9
−4
y−
7
9
3−
2
y
2w
3
5w
3
1
a
3
7
5n − 1
5n + 1
a
b
a
b
−
7
1
21 + 21
3
−
4
+w
60.
64.
1
7
w
5
w
5
15
4
1 32
24.
9
x
−
5
3c2
2b − 2a
a2 + b2
6 14
18.
1
13
−6x
y
2x2
3y
2
3
1
2
15
−20
12.
48.
x2 −y 2
x+y
ALG catalog ver. 2.6 – page 243 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
3 21
−3
8
6.
4
5
1
x
1
b
y−x
2 54
44.
+ 14a
−x
−10
15
t + 4s
t − 4s
2a
3
6
7
x
5
2
−
3
40.
3
2u2
−
4
9
2−x
2
u2
+
a2 +
55.
35.
4x4 y 3
1
2
4
u
17.
3
4
7x
6y 5
3
2u
11.
25
18
14x5
3y 2
x+
5.
1
4
6−
1+
51.
1
2
6
1−
bc − a
bc + a
2x + 1
10x + 2
−7
10
16.
39.
6a + 7b
7a + 9b
25
7
1+
28.
6
d
5
−6
10.
1 − a2
3
rt + s
rt − s
3
x
5
3
b
a
9c2
5d4
33.
b−
50.
4−
2
5
1
27.
1
2v
3 + 4r
3 − 4r
u2
21.
39
x
38.
4 − 3x
3 − 2x
6
15.
3
2
15
−
28
4
5
9.
5
8
1
7
13
−3
7
3.
4
3x − 1
1
7
−2
x
3
x
2
32.
3
14
−
20.
26.
3
5
2r −
53.
−2
7
9m
2+
49.
1
6
2−
3−
45.
3
4
14.
6
5
8
37.
41.
8.
3
4
3+
19.
2.
9—Simplify (assume no denominator equals zero).
6y + 5
5y − 8
3w + 2
4w − 9
9y 2 − 7y
27y − 18
+2
15
x
−
+
b
a
+
1
b
c
3d
−
d
3c
1
3c
+
1
3d
a−b
a+b
b2 − a 2
2wx + 6x
5wx − 45
a2 + b2
a+b
c−d
−
1
(a + b)2
LJ
77.
3
2
−
m+
85.
x
3
w+5
w
5
w
1−
1
7
1
49
m2 −
89.
a
a+3
a
1 − a+3
2b
b−a
3−
97.
4a
b+a
1−
105.
94.
b+a
b−a
98.
2
x−5
−
4
5
y
+
1−
4
y2
109.
8
n2 −4
−
9+
117. 2 +
121. 1 −
125.
129.
1
1+
2
1
x+ x
1
1
1− a−2
6
a−2
+
5
a+2
7
a2 −4
+
2
a−2
1
n+5
+
−2
c2
36
c
6
− d2
5
x
+
1−
4
x2
6
x2
w
w−z − 1
w
w−z
+1
1
r+1
+1
y−3
y+2
114.
3x2 + 4x + 3
x2 + 2x + 1
1
3−a
1
n−3
2n2 −3n−5
n2 +2n−15
110.
1
x
1
x2
1
x2 −1
1
x−1
118. 1 −
122. 2 +
11a + 2
2a + 11
2
2n − 5
126.
130.
4
x
99.
−
3
c2 −9
+
2
c+3
4
c+3
+
2
c−3
−
−1
y2
x
y
x
3u2 w − 1
w
1
w
2−
1
x−y
+
1
x+y
1
x−y
−
1
x+y
a
a+1
+
1
a−1
96.
3x + 3
x+4
100.
−
1
a+b
1
a
+
3
a2
1+
1
a
−
1+
c
c−1
−
2
a2 −1
1
a−2
8a − 3
5a − 2
1
1
2− a
2r − 5
r−2
1
1
2− 3−r
y 2 +8y+15
y 2 +y−6
y+1
y−2
y 2 +2y−15
y 2 −2y−3
p+4
p−6
+
u+
u−2
u
1−
3
n
1−
9
n2
n
n+3
r2 −
1
s2
r+
1
s
x
5
1
2
−
2x2
5
2u
u2 − 1
rs − 1
s
1
2x + 5
5
2
−
5
x+y
+3
3
x+y
+7
3
c+1
+1
1
c+3
+1
x
x−y
−
3x + 3y + 5
7x + 7y + 3
c+3
c+1
x
x+y
2
xy
x2 −y 2
1
2+
2
u+1
a+1
108.
c+1
2c + 1
1
1−
1
2b
6
a2
3
c2 −1
2+
x
y
1
a2 −b2
1
a−b
92.
2a − 1
2a − 9
1
a+1
123.
s2 −49
s2 +6s−16
88.
6
x+4
x
x+1
5x + 12
2x + 5
131.
x−y
3−
4+
s
s+7
84.
2−
119. 2 −
s−1
s−2
x
x+4
5
a−2
111.
127.
80.
3
a−2
115.
2c − 3
6c − 6
2
1 − 2a
1
w2
3u2 +
z
z2 + z + 1
1
z+ z1
1
1
2+ 2+x
2s+4
s+8
107.
x
x+1
1
1+
3
2
x
−
2x − 1
+4
+1
16
x2
2+
r+1
r−1
+1
−1
9u4 −
91.
95.
−2
−
4
x
1+
z
w
1
3+a
−
−a
87.
x+3
x−2
3a
a2 −9
1
3−a
1
4a
x−
c + 6d
6
−d
3
r−1
83.
104.
106.
r−2
3r − 4
20
r2 −4
1
e
1 + 2e
e
n−2
−5
6
y2
2
r+2
3−
113.
−4
1
2a
1+
79.
102.
x2 −25
1−
1
e2
u−4
u−3
1
n−2
2
x+5
y−1
2y 2
4y 2 + 6y
1+
a
3
2
n+2
−
90.
2
u−2
u−1−
n
n−2
y+2
y−1
4
u−2
u−2−
101.
103.
86.
2
y
y+1−
82.
7
7m − 1
4
y
y+4+
93.
78.
−6x
25
w2
1−
81.
3−y
y
2y −
2x2 − 9x
p+1
p+2
2p2 +3
p2 −4p−12
p+2
2p2 + 3
6
c
+
+
1
c2
9+
112.
116.
3
c
1
p+1
−1
1
p2 −1
+1
1−p
p
1−
2
1
1− n
1−
1
1
1− 1−a
1
124. 1 −
132.
3c + 1
2
120. 1 −
128.
1
c2
x2 −4x+3
x2 +3x−18
x2 −6x+5
x2 +13x+42
a+3
a+4
+
3n − 1
n+1
1−a
x+7
x−5
a+9
a−7
2a2 −5
a2 −3a−28
2a2 + 9a + 15
2a2 − 5
x−2+
9x+11
x+3
x+5+
x−15
x+4
133.
x+4
x+3
w+2+
2
w+5
w+6+
6
w+1
134.
w+1
w+5
u−4
135.
u−1−
ALG catalog ver. 2.6 – page 244 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
15
u+1
u+1
u+4
136.
d+3
d+5+
4
d+1
d+1
d+3
LK
Topic:
Rational equations (first-degree solutions).
Directions:
15—Solve. 16—Solve and check.
101—Give the restriction(s) on each variable, then solve.
1.
63
=9
y
5.
−
1
4
=
6
3r
−8
9.
5
=1
n−6
11
13. 3 =
7
12
4a + 5
17.
2x − 5
5
=
9
3
21.
2
7
=
y+5
y
25.
−2 =
6.
2
1
=
5y
10
10.
14
= −2
x−3
14. 1 =
− 14
22
a
2.
−48
x
−11
3.
12 =
4
7.
6
3
=
8w
5
11. −1 =
−4
5
8k − 1
3
4
−4
5
16
8
r+5
28
= −4
5y + 3
2
2y + 1
=
3
7
11
6
19.
9
1
=
4a + 2
2
4
−7
22.
6
2
=
n
10 + n
−15
23.
w
2
=
3
w−2
−4
5
3
=
2p
p+1
5
26.
4
1
=−
9+c
3c
27.
7
3
=
5n − 2
2n
29.
m−7
3
=
3−m
5
11
2
30.
3
p−1
=
2
p−6
31.
r+2
2
=
r−2
7
33.
15
8
=
z+3
z−4
34.
w+2
w+1
=
12
13
−14
35.
m−5
m−7
=
9
5
37.
2c − 3
3c − 2
=
4
5
38.
5
1
=
8h + 7
4h − 3
11
6
39.
9
6
=
2r − 3
3r − 7
41.
v+1
v + 10
=
v−1
v+3
42.
z−5
z−1
=
z−1
z−5
43.
k−8
k+8
=
k−3
k+3
45.
5h − 4
5h − 2
=
h+2
h+4
46.
2x − 1
3x − 2
=
2x + 3
3x + 5
− 12
47.
2p + 3
2p + 2
=
7p − 1
7p − 3
49.
3d
3
=
2d2 + 2
2d − 5
50.
4
2v
= 2
2v + 1
v +3
6
51.
2x
1
=
6x2 − 5
3x + 10
12
− 72
13
5
0
− 25
9
− 13
16
3
8.
−
16.
−2
18.
10
−15
= −3
n
5
4
2
=
33
11a
12. 4 =
−13
15.
4.
8
y+3
− 32
−1
18
=2
2w + 3
20. −
3
5
1
=
9
7 − 3p
− 44
15
24.
k−4
1
=
k
3
28.
5
7
=
3y
2y − 5
32.
2
1−y
=
3
4+y
19
2
36.
4
7
=
x−3
x+2
3
40.
m+5
2m + 1
=
2
3
44.
c−2
c−3
=
c−5
c+5
48.
2z − 2
2z − 1
=
z−3
z−2
52.
4
10h
= 2
2h − 1
5h − 7
56.
n n
− −9=0
3
6
60.
2x x
− = −4
15
6
120
64.
5a
3a 2
=
+
7
5
7
5
2
68.
c
c
3c
+
− 10 =
3
10 5
−60
72.
1 3y + 5
4y
−
=
6
20
15
− 15
76.
u + 11
u+3
−2=
4
3
6
− 18
5
0
7
3
6
− 25
11
−1
29
3
13
25
11
−1
14
5
− 14
53.
a a
− = 10
4
9
57.
7w
3w
+
− 26 = 0
10
5
61.
1
5
1
y− = y
2
9
9
65.
z
z
3z
+ =
−3
2 3
4
72
20
10
7
−36
54.
x
x
=1−
8
12
58.
5
5
z + z = −50
6
9
62.
u
u 4
= +
3
5
3
66.
w
7w
w
+
− −9=0
5
20
4
55.
24
5
−36
10
1
1
y + y = −6
2
4
59. 13 =
2r
4r
+
3
7
−8
21
2
63.
2x x 5
− − =0
3
4
6
67.
a
5a a
= 18 −
+
4
6
3
71.
3n + 2 n
2
− =
4
6
3
75.
w−2
w−6
=2−
2
10
2
24
54
30
69.
2 r
2r + 11
= −
9
3 3
73. 4 +
−13
−1
h+2 h+4
−
=0
2
6
70.
b−6
2b 3
=
−
8
9
4
74. 10 =
0
a−4 a+2
+
5
2
14
ALG catalog ver. 2.6 – page 245 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
2
7
6
−3
LK
77. 1 −
4 − 3m
2m − 1
=
6
5
78.
79.
2x − 3 1
x+2
− =
6
3
4
82.
16
2y − 5
y−7 7
=
+
6
4
2
83.
31
85.
n−2 n−3
n+7
+
=
5
3
15
87.
u+5
u − 1 2(u − 4)
=
+
6
4
3
89.
7
5
=3−
2b
b
93.
3
7
− −4=0
k
k
97.
2
4
1
+
=
y
3y
3
101.
1
1
3
=
+
4x 2x2
8x
19
6
1
10
− 45
1 − 2b b − 2
−
−3=0
4
8
80.
−4
10
− 16
3
81.
2c + 1
5 − 2c
=2−
10
6
5
4m + 6
m − 10 13
+
=
8
4
3
84.
0
4
5
86.
h+1
5h − 2 3h + 1
−
=
2
3
6
88.
2(w − 3)
4w − 7 9 − 2w
=
−
3
9
6
3
3
−
+2=0
r
5r
94.
10
6
+ = 12
x
x
4
3
95.
17
2
=5+
c
c
98.
11
3
5
= −
2n
n 2
−1
99.
1 1
7
+ =
v
3
12
102.
2
7
5
=
+
x2
9x 6x2
103.
1
5
3
+
=0
−
x 10x2
6x
91. 4 =
3
2
5
3
+
8 h
8
9
3
2
5 a−1
+
=2
a
2a
109.
9
m+4
5
−
=
2m
4m
36
106. 5 −
3
9
110.
6
w − 28
=
w
7w
7
17
1
3−y
1
− =
2y
4
3y
6
5
2
92.
3
2
+ =1
x 5
96.
3
9
+ 14 = −
m
m
100. 0 =
4
104. −
1
5
105.
z+5 7−z
7
−
− =0
4
6
4
4
90.
− 65
5h − 9 4h − 8
−
=4
2
3
5
9
1 6
− −
2b 4
b
− 76
−6
8
2
1
= 2
−
10x 2x2
5x
− 21
2
107.
z−5
10
−2=−
3z
z
111.
13
5
x−2
−
+
=0
54 6x
9x
108. 0 =
5
5
9−k
− +3
4k
k
112.
2 r−1
2
+
=
r
3r
5
116.
5 1
5 3
+ = −
y
3
6 y
120.
2x + 5 1
3
=
−
8x2
4x
2
1
25
3
113.
1
1
4
2
= + −
3u
3 u 2
117.
x−3
2
11
+
=
x2
5x
15x
−2
9
2
1
1
7
1
−
= −
6 2s
2 3s
114.
5
5 7 4
= + +
2
a 4 a
118.
10
x−2
4x − 1
x+1 x−4
x
+
= 0 119.
−
−
=
3x
x2
6x2
12x
3x
6x2
115.
12
2
3
1
4
3 c−1 c−2
13
+
+
=
c
2c
4c
12
6
122.
8
x+6 x−4
8
+
+
=
x
3x
6x
9
123.
r−3
5
1
6−r
−
+ =
3r
12 r
6r
12
124.
2n − 3 3n + 1 5n − 9
+
+
=0
n
2n
4n
125.
3
x + 15
=
−2
x+3
x+3
126.
127.
3
3y + 16
+4=
y+1
y+1
5
2d + 3
+
+3=0
d−2
2−d
4
129.
v
4
+
−2=0
v−4 4−v
130.
a
a
−5=
a−3
a−3
1
x+3
3
+ =
x+5 2
x+5
Ø
131.
2t
4
=
+2
t−4
t+5
−14
128.
1
2a + 5
4
=3−
a−3
3−a
10
w
5
= −3
+
w−5 5−w
132.
3
h
=
+2
h−3
h−3
136.
2
5
7
−
=
8−d
3d 8 − d
Ø
Ø
5
134.
5
n+6
5
=−
−
n+2 n+2
6
135.
138.
4z
5
−
= −4
z+2 z−2
7
3
4
=
+
a−2
2a a − 2
−2
4
137.
24
9
Ø
133.
3
10
4
3
121.
6
16
139.
3p
8
=
−3
p+2
p−4
− 26
3
ALG catalog ver. 2.6 – page 246 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
5
4
31
140.
w
2
−
−1=0
w−4 w+5
−14
LK
141. 1 +
2y
3y
=
y+2
y−2
142. 3 =
143.
145. 2 −
5
4m
=
m+3
2m + 1
146.
1
8
2u
3u
=2+
u−2
2u + 3
147.
4
1−a
=
−1
1+a
1−a
150.
2w
w−1
−1=
w−1
w+1
0
151.
154.
5x − 1 x − 3
−
=4
x+2
x−2
158.
9
2
161.
1
1
2
+
=
y+1 y+3
y+4
163.
1
1
4
+
=
x−1 x−2
2x − 1
165.
n
18
+1=
n2 − 9
n+3
167.
a
3a2 − 10
−1=
2a2 − 5a
2a − 5
169.
4
16
6
= 2
+
b
b−6
b − 6b
152.
8
3x − 2
−3=0
+
x−3
x+3
19
y − 1 2y − 1
+
y+1
y−1
3
156.
2z − 1 z − 2
−
=1
z+2
z+4
3
1
2
− =
p−2 p
p+3
159.
160.
2
3
2
+
=
2d − 3 3d − 2
d
− 94
12
13
2
3
1
−
+
=0
k−1 k+4 k+5
5
4
164.
8
1
3
−
=
2a + 3 a − 4
a+2
166.
6
4y 2
−
=4
y 2 − 4y + 4 y − 2
168.
c
7c2 + 8
= 2
−2
3c − 4
3c − 4c
−1
170.
4
7
5
=
+
u2 − 3u u − 3
u
−19
172.
1
2
19
+
= 2
3x x − 2
3x − 6x
174.
1
3 6x − 35
= −
x−3
2
2x − 6
8
176.
2y
22
=
+2
2y + 3
6y + 9
− 20
3
178.
9
5
4
− 2
=
r+1 r −1
1−r
10
9
180.
a+1
3
4
=
−
a2 − 4
a+2 2−a
1
2
182.
2
5
11
+ 2
−
=0
y + 2 y − 4 2y − 4
4
184.
2
5
1
=
−
m+1
3 − 3m2
3 − 3m
10
−6
186.
a − 6 5a2 − 12
6a
=
+ 2
a+3
a−3
a −9
3
188.
r+1
r
r + 20
−
− 2
=0
r+2 r−2
r −4
−3
6
5
3
= +
+ 8k
k
k+8
4
1
3
+
=
2c + 1 c − 3
c
162.
2
4
− 17
4
1
3a
5
+ −
=0
7a + 14 7 a + 2
175.
5
8u − 16
−1=
2u − 1
5 − 10u
q2
3
11
− 52
173.
177.
2y
4
=1−
2y − 1
3y + 1
12
5
5
1
6
+
=
a+3 a−3
a
k2
148.
3
2n − 5
=2−
n−1
n+1
155. 3 =
− 12
171.
x
1
+
−1=0
x − 1 2x + 7
5
2
r+2 r−3
+
−2=0
r−2 r+2
t
3t
+
=4
t − 10 t + 3
−2
− 45
18
157.
144.
3
2
−120
− 13
153.
d
2d
+1=
d+6
d+1
−45
− 25
149.
s
2s
+
s−3 s+5
− 38
7
6
3
4
+
+
=0
−9 3−q
q+3
179.
4
2x − 3
5
− 2
=
x−2
x −4
x+2
181.
4
4
9
=
+ 3
n−3
n + 3 n − 9n
183.
a+2
1
9
+
= 2
a − 5 a2 + 5a
a − 25
185.
z+3
3
z2
=
− 2
z−1 z −1
z+1
187.
y−4
y+4
48
=
−
y+4
y − 4 y 2 − 16
15
7
3
8
15
4
ALG catalog ver. 2.6 – page 247 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
− 17
3
− 17
4
2
5
3
2
− 11
2
8
LK
189.
191.
c+2
2c2 + 4
c+5
− 2
+
=0
c − 1 c − 4c + 3 c − 3
v 2 + 12
2v + 3 v + 6
=
−
− v − 20
v+4
v−5
v2
190.
5
4
46
+
= 2
w+1 w+3
w + 4w + 3
−3
192.
10
5
25
+
= 2
c−3 c+1
c − 2c − 3
3
194.
2
3b
3
−
=
8 − b b2 − 4b − 32
b+4
5
13
3
2
193.
y+3 y−2
2y 2 + 9
+
= 2
y−6 y+2
y − 4y − 12
195.
2h + 7
4
3
−
=
h2 − 10h + 16 h − 2
h−8
9
196.
2
12x − 1
4
−
=
x2 − x − 6 x + 2
x−3
197.
x+2
x−1
3x2 + 6
+
= 2
x + 3 2x − 3
2x + 3x − 9
5
198.
p2 − 3p + 6
p
3p
=
−
2p2 + 7p − 4
2p − 1 p + 4
199.
y − 2 2y + 3
5y 2 − 3y + 5
+
= 2
y + 2 3y + 5
3y + 11y + 10
200.
a − 1 3a − 2
5a2 − 3a − 11
=
+
2a2 + 5a + 3
a + 1 2a + 3
201.
4
2
5
= 2
+
r2 + 5r − 6
r + 7r + 6 r2 − 1
−8
202.
7
6
2
−
= 2
y 2 − 7y + 12 y 2 − 16
y + y − 12
203.
2
1
3
= 2
+
d2 − d − 6
d + 5d + 6 d2 − 9
3
2
204.
3
4
8
+
= 2
x2 + 2x − 15 x2 − 9
x + 8x + 15
205.
2
4
5
= 2
−
a2 + 2a − 24
a + 6a a2 − 4a
206.
5
3
4
= 2
−
c2 + 7c + 12
c + 4c + 3 c2 + 5c + 4
207.
12
7
5
+
= 2
4r2 − 12r + 9 4r2 − 9
4r + 12r + 9
208.
4
2
7
+
= 2
2v 2 − 17v + 30 v 2 − 2v − 24
2v + 3v − 20
209.
1
1
5
1
7
(3x + ) = (11x − 5) − (x − )
2
3
4
3
5
210.
h
4
h
2
1
(9 − ) = (5 + ) − (7h − 10)
6
h
3
h
9
211.
2
1
3w
5
(9w2 + 5) + (2w − 7) =
(w + )
6
3
2
6
212.
2
n
3
n
1
(5 + ) − (4 + ) =
(10n − 13)
n
4
n
3
2n
1
− 28
3
−5
46
− 51
14
ALG catalog ver. 2.6 – page 248 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
2
1
2
3
8
− 65
54
53
1
2
4
9
11
−48
LL
Topic:
Advanced rational equations (quadratic solutions).
Directions:
15—Solve. 16—Solve and check.
101—Give the restriction(s) on each variable, then solve.
1.
4
x
=
x
9
5.
a−2
1
=
2
a−3
9.
t2
±6
1, 4
5
−3
= 2
+ 4t
t − 16
2.
a
5
=
5
a
6.
y−2
1
=
5y − 16
y−2
10.
5
2
g2
±5
4, 5
2
7
= 2
−9
g − 3g
3.
w
5
=
20
w
7.
1
n+1
=
4n − 3
n + 13
11.
−7
3
= 2
8h − h2
h + 8h
14.
−20
17.
8
6
= 2
+ 7k
k − 49
k2
4
13
=
n2 + 3n
5n + n2
15.
8
y
=
y
18
8.
x
1
=
3x + 4
x
12.
− 17
16.
3
2
18.
2w + 10
15 − 3w
=
w2 − 25
5w − w2
4, −1
4
3
=
− 25
5n − n2
12
7
= 2
m2 + 7m
m − 3m
17
− 19
9
4
8
=
p2 + 6p
3p + 18
n2
±12
−15
7
2
5
=
a2 + a
2a − a2
5
3
= 2
8 − 4r
r − 2r
− 12
5
19.
9
1
=
6y + 12
3y − y 2
1,
21.
±2
4.
−21
− 21
5
13.
±10
22.
5p + 20
6p − 24
= 2
p2 − 16
p − 4p
26.
−3
k−2
=
2−k
3
24
20.
5
7
=
3a − a2
4a − 12
− 28
5
4
3
23.
3g + 6
5g + 25
= 2
g2 − 4
g + 5g
27.
v − 10
−2
−
=0
16
v+8
5
24.
4r + 6
r+5
=
4r2 − 9
25 − r2
28.
4
c+1
−
=0
c+5
3
13
4
15
25.
m−1
5
=
5
m−1
6, −4
5, −1
−6, 8
29.
2
3y − 1
−
=0
y + 2 3y + 4
30.
31.
1, 5
2, − 53
33.
x−3
x+1
−
=0
x − 2 2x − 1
3
p−1
=
p2 − 4p − 5
p−5
34.
37. r − 5 =
14
r
n+4
9
= 2
n−2
n + 2n − 8
41. 4n − 3 =
38. z + 8 =
7, −2
n + 13
n+1
32.
35.
20
z
4x + 4
x
=
x2 + 3x + 2
x+2
36.
c−8
c−3
43. 4s −
2
6
2−y
−4, 1
2s − 4
−2=0
s+2
40. w − 7 =
44. y + 5 +
0, −1
45. 5 =
x+5
−x
x−2
46. 14 =
−5, 3
0,
49.
y+3
3y 2 + 2
+1=
5
2
50.
1, − 16
k − 14
+ 4k
k−1
1
3
−
=1
x2
4x
57.
c2 − 4
c−2
=2−
c+3
c+3
−4, 1
4
11
w+3
8, −4
6
=0
y−2
−4, 1
d
2
=
2
5−d
47. 1 −
1, 6
48.
6
3
+1=
a+2
a
52.
h+8
2h + 1
+4=
6
2h
−6, 1
17
4
a2
a
a2 + 1
+ =
8
6
16
51.
1
3 , −3
53.
2y
10
= 2
y−2
y − 6y + 8
−1, 5
39. y + 5 =
2, −10
1
2x
+
=0
x − 4 5x − 2
2, − 12
4
42. 5c − 4 =
±2
a
1
+
=0
2a + 1 4a + 3
−1, − 14
−1, −7
±2
−7, 1
x + 5 x − 10
−
=3
5
3x
10,
54.
1
1
2
= + 2
x
3 3x
58.
6
3r
+4=
r−2
r−2
1, 2
Ø
1, −12
5
3
55.
5
2
+ =3
x2
x
59.
y+2
9y
−
=2
y
y+4
−1,
ALG catalog ver. 2.6 – page 249 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
4
5
−1,
5
3
56.
1
4
− −5=0
x2
x
60.
x
2x + 1
=
+2
x
x+2
2, −1
−1,
1
5
LL
61.
3
2k
=
−3
k+5
k+1
4
3y
+2=
y−2
y−4
62.
−9, −2
65.
63.
7
h+3
4
+3=
h−1
h+1
66.
64.
Ø
67.
1
2
=
−2
r−2
r+1
68.
1
2,1
69.
3
1
−1= 2
a
a −a
2
n−4
1
− =6
n2 + 2n n
70.
−1
71.
7
10
s+2
−
=
s − 1 s2 − s
s
2
1
1
− =
w2 − 2w
3
w
74.
75.
3, −4
2, 4
4
4
+2= 2
y−1
y −y
72.
16
2
f
− =
f 2 − 4f
f
f −4
p
p2 + 9
3
+
= 2
p−3 p+3
p −9
Ø
80.
y2 + 1 y + 3
y
+
=
y2 − 1 y − 1
y+1
−4
82.
3x
5
2x − 24
=
+
x2 − 8x + 12
x−6 x−2
79.
2x2 − 3 2x − 3
10x
= 2
+
x+2
x −4
x−2
81.
1
10
y
+
=0
−
y 2 − 8y + 12 y − 6 y − 2
83.
3
2w
15
= 2
−
w+4
w + 3w − 4 w − 1
− 32
84. 0 =
f2
f −3
3f − 5
2f + 2
+
=
+ 4f + 3
f +3
f +1
−6
86.
y2
5
y
y+1
=
−
− 3y − 4
y+1 y−4
85.
87.
89.
Ø
2
3n
n + 44
+
=
2n2 + 9n + 10 2n + 5
n+2
91. 0 =
−4, 2
4
10
2b + 6
+
+
b − 9 b2 − 4b − 45
b+5
6, −2
93.
2y
4
3y
+
= 2
y 2 + 3y − 10 y 2 + 6y + 5
y −y−2
95.
x
5
2x
+
= 2
x2 − x − 6 x2 + 5x + 6
x −9
97.
3n
2n
1
+
=
n2 + 2n − 8 2n2 + 19n + 44
2n2 + 7n − 22
99.
d2
76.
78.
2
z
13
+
=
z + 2 z2 − 4
z−2
−1, 8
−8
−5
3d
7
2d
− 2
= 2
+ 2d − 3 d − 6d + 5
d − 2d − 15
Ø
11, 1
a2
x2 + 7x
36
−4=
x−2
2x − 4
c − 25
c
5
+ =
2c2 − c
c
2c − 1
5, 6
77.
1
2,3
14
3
−5
−6
±3
n
10
=
+5
n+2
n−6
0,
−2
73.
2
6
+1=
a−3
a−8
2, 13
13, 6
0, −2
8
2d
+2=
d+3
d−2
30
18
=
+1
k−3
k−4
−1
4
7
g
−
−
g 2 + 5g + 6 g + 3 g + 2
a2 − 5
a+1
a
+
=
+ 3a − 10 a + 5
a−2
−5, −1
7, −1
88.
x
3x2
x
− 2
=
x + 3 x − x − 12
x−4
90.
7
1
f
+
+
=0
f − 4 f 2 − 9f + 20 f − 5
92.
2 − 4n
2
2n
−
=
n2 + 7n + 12
n+4
n+3
94.
s
1
4
−
= 2
3s2 + 17s + 10
3s − 4s − 4 s2 + 3s − 10
96.
3m
4m
6
−
= 2
m2 + 5m + 6 m2 + 6m + 8
m + 7m + 12
3
98.
3
5p
2
= 2
−
2p2 + 11p − 6
2p + 7p − 4 p2 + 10p + 24
−5,
100.
3a2
ALG catalog ver. 2.6 – page 250 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
0, − 73
1, 3
−2, 1
4a
5
3a
+ 2
= 2
+ 20a + 12 a + 5a − 6
3a − a − 2
6, −3
2, 5
4
5
LM
Topic:
Solving for other variables (rational equations).
Directions:
19—Solve for the indicated variable.
1.
r
r1
= ; for r1
t
t1
rt1
t
2.
x
x0
= 0 ; for y 0
y
y
3.
D
R
= ; for r
d
r
Rd
D
4.
F1
m1
=
; for F2
F2
m2
5.
t
t
1
+ = ; for t
b c
b
6.
n n
1
+ = ; for n
a
b
ab
1
a+b
7.
k
k
+ = 1; for k
m n
8.
w
x
+ w = ; for w
y
y
x
y+1
9.
c−y
2
= ; for y
c+1
3
10.
u
a−u
=
; for x
x+u
a−x
11.
4
5
=
; for a
a−c
a−d
5c − 4d
12.
a
4
=
; for b
a−b
b+3
13.
1
1
1
+
= ; for x
ax bx
ab
a+b
14.
1
1
1
+ = ; for y
xy
y
x
15.
a
b
+ = 1; for y
x y
16.
b
c
− = a; for x
x y
17.
1
1
1
+
= ; for R1
R1
R2
R
18.
1
1
1
=
+ ; for D
f
D
d
fd
d−f
19.
1 1
1
+ = ; for f
p q
f
20.
1
1
1
= ; for t2
−
t1
t2
3
3t1
3 − t1
21.
p q
+ = 1; for p
r
r
22.
p+x
+ 3 = x; for p
2
23.
x y
+ = 1; for y
a
b
24.
m−n
= t; for n
rn
25.
d D
E
+
=
; for t
t
t
t+1
26.
a
b
a
+ =
; for r
r
r
r−1
27.
1
1
1
1
+
+
= ; for R
R1
R2
R3
R
28.
1
1
1
1
+
+
=
; for R2
R1
R2
R3
RT
S
L
29. P =
31. V =
1
s
1
s
c
b+c
mn
m+n
c−2
3
bx
x−a
R2 R
R2 − R
pq
p+q
r−q
ab − bx
a
d+D
E−d−D
+w V
; for L
100
+
−
1
t
1
t
; for s
Vt−t
V +1
R1 R2 R3
R1 R2 + R1 R3 + R2 R3
VS
100P − W V
30. K =
32.
1
=
p
ALG catalog ver. 2.6 – page 251 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
L s+
M
1
r
1
r
+
−
1
2
1
2
1
2
x0 y
x
F1 m2
m1
a
a+4
x+1
by
ay + c
4−x
m
rt + 1
; for s
; for r
u2
a
a+b
b
2Km − L
2L
2 + 2p
1−p
R1 R3 RT
R1 R3 + R1 RT + R3 RT
LN
Topic:
Word problems involving rational equations. See also categories KD (ratio) and KE (proportion).
Directions:
0—(No explicit directions.) 16—Solve and check.
40—Write an equation and solve.
39—Translate and solve.
Work problems
1.
Michael can mow the lawn in half an hour, and Alex
can mow it in three-quarters of an hour. Suppose
Michael spends 20 minutes on the job, and then Alex
finishes it. How long does Alex have to work?
2.
Enrique can paint the fence around his house in
12 hours. His sister can do it in 10 hours. If Enrique
paints for 5 hours and then turns over the job to his
sister, how long will it take her to finish? 5 hr 50 min
15 min
3.
A computer operator can enter 75 data forms each
hour. After the operator has been working for
20 minutes, a second operator starts working also.
Two hours later, they have entered 295 forms. How
long would it have taken the first operator to do the
entire job? 3 hrs 56 min
4.
A clerk can process 200 payroll vouchers in an hour.
Another can do 160 in an hour. After the first clerk
has been working for 30 minutes, the second clerk
begins. How long after the second clerk begins will
they have processed a total of 400 vouchers? 50 min
5.
Arthur and Bradley can complete a job in 12 minutes
if they work together. By himself, Bradley would
take 36 minutes to do the same job. How long would
Arther take working alone? 18 min
6.
Working together, Edward and Frances can wallpaper
an apartment in 12 hours. It would take Frances
30 hours to do the same job by herself. How long
would it take Edward to do it alone? 20 hrs
7.
A construction crew can frame a house in five 8-hour
workdays. They can do the same job in three
workdays with the help of a another crew. If the
other crew worked alone, how many hours would it
take to frame the house? 60
8.
Simon can landscape a yard in two 8-hour workdays.
He can do the same job in 10 hours if he works with
Raoul. If Raoul works alone, how many hours will it
take him to landscape the yard? 2 2
Together, Dan and Laura can write their wedding
invitations in 9 hours. It would take Laura 15 hours
to do the job by herself. How long would it take
Dan to do the job? 22 1 hrs
10. Working together, Ken and Joelene can mow a lawn
in 7 hours. It would take Ken 10 hours to do it
alone. How long would it take Joelene to do it alone?
9.
2
11. Alice can wash and wax her car in 3 21 hours. If
Bernice helped her, Alice could do the job in 2 hours.
How long would it take Bernice working alone?
4 32 hrs
3
23 13 hrs
12. Rosa can shovel the snow off her driveway in
1 12 hours. If her father helped, the job would take
only half an hour. How long would it take her father
working alone? 3 hr
4
13. At a water treatment plant, one of the overflow tanks
can be emptied in 10 hours by pipe A or in 15 hours
by pipe B. If both pipes were used, how long would
it take them to empty the tank? 6 hrs
14. The main water pipe can fill a spa in 4 hours. It
takes a garden hose 12 hours to fill the same spa. If
both the water pipe and hose were used, how long
would it take to fill the spa? 3 hrs
15. Jake can put up 100 ft of fence in 12 hours. His
father can do the same job in 9 hours. If they work
together, how long should it take Jake and his father
to put up the fence? 5 1 hrs
16. Don can hoe the garden in 6 hours. His brother,
West, can hoe it in 4 hours. If they work together,
how long should it take Don and West to hoe the
garden? 2 2 hr
17. Nancy spent 3 hours painting her dormitory room.
But her roommate, Marcia, didn’t like the color. So
Marcia did the job over again in 5 hours. Working
together, how long would it have taken them to paint
the room? 1 7 hrs
18. Mrs. Hanson’s 1st period class set up chairs for
an assembly in 25 minutes. The next day her
2nd period class did the same job in 15 minutes. How
long would it have taken the two classes, working
together, to set up the chairs? 9 3 min
19. According to work records, plumber A does a certain
job in 8 hours, while plumber B does the same job
in 6 hours. If they worked together, how long should
the job take? (Round your answer to the nearest
minute.) 3 hr 26 min
20. A contractor notices that crew A can stripe a certain
length of road in 1 12 hours, while crew B takes
1 34 hours to do the job. If the crews worked together,
how long should the job take? (Round your answer
to the nearest minute.) 48 min
7
8
5
8
ALG catalog ver. 2.6 – page 252 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
LN
21. To do a job alone, it would take Mindy 3 days,
Wendy 4 days, and Cindy 6 days. How long would it
take to do the job if they all worked together?
22. To do a job alone, it would take Dan 6 hours, Fran
2 hours, and Stan 3 hours. How long would it take
to do the job if they all worked together? 1 hr
1 13 days
23. Floyd can collate a set of papers in 45 minutes.
Pheobe can do it in 50 minutes. But Anita is the
fastest of all; she can do the collation in 40 minutes.
How long would it take all three, working together,
to do the job? (Approximate your answer to the
nearest minute.) 15 min
24. Janet can enter some data into the computer in
70 minutes. Ramon can do the same work in
56 minutes. But Ira tops them both; he can do
the data entry in 49 minutes. How long would it
take all three, working together, to do the job?
(Approximate your answer to the nearest minute.)
19 min
25. Working together, Anna, Brian, and Chelsea can
complete a certain job in 3 hours. Anna can do the
job in 8 hours and Brian can do it in 10 hours. How
long (to the nearest quarter hour) would it take
Chelsea to complete the job? 9 1 hrs
4
26. If they work together, Mark, Naomi, and Yvonne can
sort some library books in 1 21 hours. Working alone,
it would take Yvonne 5 hours and Mark 6 hours to
do the same job. How long (to the nearest quarter
hour) would it take Naomi, if she worked alone?
3 14 hrs
27. Minh can do a job in 2 hours, and Ruth can do the
same job in 4 hours. How long would it take Isabel
to do the job if, working together, all three can do it
in just one hour? 4 hrs
28. To do a job alone, it would take Karl three 8-hour
workdays and Inez five 8-hour workdays. How long
would it take Melanie to do the job if all three of
them can do it in one workday? (Give answer in
hours and minutes.) 19 hrs 12 min
29. Working together, it takes Stuart and Tracy
48 minutes to stuff some envelopes. Doing the job
alone, Stuart would take twice as much time as
Tracy to stuff the envelopes. How long (in hours and
minutes) would it take him? 2 hrs 24 min
30. Working together, Jeanette and her brother can
deliver newspapers in 56 minutes. But, working
alone, it takes Jeanette three times longer than her
brother to deliver the papers. How long does it take
her (to the nearest minute)? 75 min
31. Grace and Yee can take inventory of their store in
1 34 hours, if they work together. It takes Yee half as
much time as Grace, if they do the work individually.
How long does it take Yee (to the nearest minute)?
32. Working together, Fred and Ned can assemble 500
gadgets in 3 12 hours. Working indvidually, it takes
Fred two-thirds as much time as Ned to complete
the job. How long does it take Ned (in hours and
minutes)? 8 hr 45 min
79 min
33. One pipe can fill a tank in 5 hours. Another can
empty the tank in 8 hours. If both pipes are left
open, how long will it take to fill the tank? 13 1 hrs
34. One pipe can fill a tank in 2 21 hours. Another can
empty the tank in 3 hours. If both pipes are left
open, how long will it take to fill the tank? 15 hrs
35. A pipe can fill a swimming pool in 20 hours, and
an outlet pipe can empty it in 32 hours. How
long would it take to fill the pool if the pipes are
operating at the same time? 53 1 hrs
36. A pipe can fill a swimming pool in 24 hours, and an
outlet pipe can empty it in 16 hours. How long
would it take to empty a pool if the pipes are
operating at the same time? 48 hrs
37. Mr. Jenkins can dig a ditch in one hour less time
than his son. Working together, they can do the job
in 32 hour. How long does it take Mr. Jenkins to do
the job alone? 1 hr
38. Using a large and a small drain pipe, a fish pond can
be emptied in 56 minutes. Working alone, the small
pipe takes 15 minutes longer than the large pipe.
How long does it take the large pipe? 1 3 hr
39. Together, Louise and Stacy can do a job in 1 hour
and 12 minutes. Individually, Louise takes an hour
longer than Stacy. How long does it take Stacy to
complete the job by herself? 2 hrs
40. Working together, two pipes can fill the tank in
2 hours and 6 minutes. Working alone, the larger
pipe fills the tank in 4 hours less time than the
smaller one. How long does the larger pipe take?
3
3
4
3 hrs
ALG catalog ver. 2.6 – page 253 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
LN
Ratio and proportion
41. Separate 99 into two parts that are in a ratio of 4 : 7.
36, 63
42. Separate 154 into two parts that are in a ratio
of 9 : 5.
99, 55
43. Separate 132 into two parts that are in the ratio 3 : 8.
36, 96
44. Separate 105 into two parts that are in the
ratio 11 : 4.
77, 28
45. Jeanelle and her brother need to divide $12 in a 5 : 3
ratio. How should the money be divided?
46. Mike and his sister need to divide $24 in a 7 : 8 ratio.
How should the money be divided? $11.20 and $12.80
$7.50 and $4.50
47. Mr. and Mrs Maloy are going to divide a $180 cash
rebate in a 7 : 5 ratio. How should the money be
divided? $105 and $75
48. Fred and Nina are going to divide a $576 tax refund
in a 4 : 5 ratio. How should the money be divided?
49. Jamie and Kendra earn $42 for shoveling snow. If
Jamie works 3 hours and Kendra works 4 hours, how
should they split the money so each receives the
correct share? $18 and $24
50. Craig and David earn $65 for doing yardwork. If
Craig works for 4 21 hours and David works for
3 hours, how should the money be divided so that
each receives the correct share? $39 and $26
51. Mr. Richards spends 5 hours on a consulting job and
Ms. Samuels spends 7 hours on the same job. They
are paid a total of $600. How should the money be
divided so that each receives the correct share?
52. Mr. and Mrs. Sibayan spend 12 hours and 8 hours,
respectively, on an upholstering job. They are paid a
total of $740. How should the money be divided so
that each receives the correct share? $444 and $296
$256 and $320
$250 and $350
53. A nickel and iron alloy weighs 144 kilograms. The
ratio of iron to nickel is 3 : 1. How many kilograms
of iron are in the alloy? 108
54. An aluminum and magnesium alloy weighs
26 kilograms. The ratio of aluminum to magnesium
is 8 : 5. How many kilograms of magnesium are in
the alloy? 10
55. A mixture of flour and sugar weighs 15 pounds. The
ratio of flour to sugar is 5 : 1. How many pounds of
sugar are in the mixture? 2.5
56. A mixture of oil and vinegar fills a 6 oz jar. The
ratio of oil to vinegar is 7 : 5. How many ounces of
oil are in the jar? 3.5
Number problems
57. The sum of a number and its reciprocal is
the number. 2 or 5
5
29
10 .
Find
2
58. The sum of a number and its reciprocal is
the number. 5 or 1
26
5 .
Find
5
19
3 .
59. The sum of a number and four times its reciprocal
is 4. What is the number? 2
60. The sum of a number and twice its reciprocal is
What is the number? 6 or 1
61. The difference between a number and its reciprocal
is 15
4 . What is the number? 4 or − 1
62. The difference between a number and its reciprocal
is 56 . What is the number? 3 or − 2
63. A number is 12 more than three times its reciprocal.
Find the number. 2 or − 3
64. A number is 3 12 more than twice its reciprocal. Find
the number. 4 or − 1 .
4
2
3
2
3
2
Time, distance, rate
65. A freight train travels along a regular route that
is 140 mi. One day the train leaves on its route a
half-hour late. In order to reach its destination on
time, the train goes 5 mph faster than usual. What
is the train’s usual speed? 35 mph
66. A delivery truck follows a regular route that
is 270 km. One day the driver begins the route a
half-hour late. In order to finish on time, she drives
the truck 6 km/hr faster than usual. What is the
truck’s usual speed? 54 km/hr
ALG catalog ver. 2.6 – page 254 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
LN
67. An airplane travels between two cities that are
350 miles apart. One day the airplane leaves
15 minutes behind schedule. In order to arrive at its
destination on time, the airplane flies 25 mph faster
than usual. What is the airplane’s usual speed?
175 mph
69. Alfred drove his car 100 miles in the same time
that Brigit drove her car 125 miles. If Alfred drove
10 mph slower than Brigit, find the speed of each.
A 40 mph, B 50 mph
68. A ferry boat goes between two ports that are 24 km
apart. One day the ferry begins the trip 6 minutes
behind schedule. In order to reach the other port
at the correct time, the ferry travels 8 km/hr faster
than usual. What is the ferry’s usual speed?
40 km/hr
70. Silvia rode her bicycle 40 km in the same time Tracy
rode 28 km. Tracy’s rate was 3 km/h slower than
Silvia’s. How fast did each woman ride?
S 10 km/hr, T 7 km/h
71. Judy and Karen work at the same office. Judy drives
30 miles to work and Karen drives 20 miles. Both
take the same time to get there since Judy drives
15 mph faster than Karen. Find the speed of each
person. J 45 mph, Susan 30 mph
72. It took Mr. H as much time to drive 180 km as it
took Mr. G to drive 135 km. Mr. H’s speed was
15 km/h faster than Mr. G. How fast did each person
drive? G 60 km/hr, H 45 km/hr
73. A sightseeing bus travels 40 miles to a scenic
overlook. The rate returning is twice the rate going,
and the travel time for the round-trip is 2 hours.
Find the rate for the return part of the trip. 60 mph
74. A garbage truck drives 56 km to its destination,
the city dump. If the rate returning is twice the
rate going, and the travel time for the round-trip is
2 hours, find the rate returning. 84 kph
75. Sarah walked 10 km into the country. She returned
walking 3 km/h slower. The total time for the round
trip was 7 hours. How fast did she walk going out to
the country? 5 km/hr
76. Luke cycled 36 miles to the beach. On the way back,
he cycled 6 mph slower than on the way out. If the
round-trip took 5 hours, what was his rate returning
from the beach? 12 mph
Wind and current problems
77. A canoe goes 6 miles upstream in the same time that
it takes to travel 10 miles downstream. The current
is flowing at 1 mph. Find the rate of the canoe in
still water. 4 mph
78. A plane can fly 600 kilometers with the wind in the
same time that it can fly 520 kilometers against the
wind. The wind is blowing at 30 kilometers per
hour. Find the rate of the plane in still air.
420 km/h
79. In still water, Lanette sails her boat at an average
rate of 30 km/hr. One day, she spent as much time
sailing 104 miles down a river as she did sailing
91 miles back up the river. What was the rate of the
river current? 2 km/h
80. A blimp can go 32 miles against the wind in the
same time that it takes to go 88 miles with the wind.
The speed of the blimp in still air is 30 mph. What
is the speed of the wind? 14 mph.
81. A woman can row her boat 3 21 km with the current
in the same amount amount of time it takes to row
1 12 km against the current. If she rows her boat
2 km/hr in still water, what is the speed of the
current? 0.8 km/hr
82. A man can swim three-fourths of a mile down river
in the same amount of time it takes him to go half
a mile up river. If he swims at an average rate
of 2 mph, what must be the rate of the river current?
83. It takes an airplane half the time to fly 852 kilometers
with a tail wind as it does to fly 1560 kilometers
with a head wind. Find the wind speed, if the
airplane’s speed is 408 km/hr. 18 km/hr
84. A jet flies 852 miles with a tailwind in half the time
it takes to fly 1560 miles against the same wind.
Find the jet’s speed, if the wind speed is 18 mph.
85. An airplane flies 900 km with a tailwind and then
returns the same distance against the wind. The
round-trip lasts 7 hours. If the airplane’s speed in
still air is 280 km/hr, what is the speed of the wind?
86. A jet’s speed in still air is 240 mph. One day it
flew 700 miles with a tailwind, and then returned
the same distance against the wind. The total flying
time was 6 hours. Find the speed of the wind.
80 km/hr
0.4 mph
408 mph
40 mph
ALG catalog ver. 2.6 – page 255 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
LN
87. A helicopter flew 15 miles against a 25 mph
headwind. Then it flew back with the wind at its
tail. The round-trip lasted 27 minutes. Find the
helicopter’s speed in still air. 75 mph
88. An airplane flew 200 km with a tailwind of 60 km/hr.
Then it returned against the wind. The total flying
time was 45 minutes. Find the speed of the airplane
in still air. 540 km/hr
89. Mr. Wiley averages 12 mph when his boat is in still
water. One day he goes 12 miles upstream and the
same distance back. The round-trip takes two hours
and 15 minutes. What is the speed of the current on
that day? 4 mph
90. Maria swam one kilometer up a river and the same
distance back. The workout lasted 40 minutes. If
Maria swims 4 km/hr in still water, what was the
speed of river current? 2 km/hr
91. Eugene can row 10 km/hr in still water. One day it
took him 4 hours longer to go a certain distance
upstream than the same distance downstream. If the
speed of the current was 5 km/hr, how far upstream
did he go? 30 km
92. Josh rows his kayak in still water at a rate
of 6 km/hr. One day when the river current
was 3 km/hr, he rowed a certain distance downstream
and then back upstream the same distance. The
entire trip took 40 minutes. How far downstream did
he go? 1.5 km
ALG catalog ver. 2.6 – page 256 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
MA
Topic:
Rational roots. See also category MB (roots of variable expressions).
Directions:
6—Simplify.
1.
√
9
√
± 64
9.
√
− 121
13.
17.
√
√
2.
3
5.
6.
±8
−11
−49
225
not real
29.
√
√
324
−100
−120
√
37. ± 5776
±76
√
41. − −900
45.
1
16
not real
r
1
49
± 17
53. −
r
9
16
− 34
r
25
9
61.
r
25
121
65. −
69.
73.
r
r
√
√
77. ± 0.81
81.
√
1.44
34.
38.
7.
√
− 49
±10
11.
√
169
not real
15.
√
−121
6
√
√
−81
25
√
19. ± 196
400
20
23.
− 54
37
21
√
√
±0.9
110
7056
r
1
9
1
6
54.
r
4
25
2
5
r
r
49
36
196
121
70. −
r
169
1849
0.49
−0.05
86.
√
not real
16.
√
−225
±14
√
20. − 256
−16
√
24. ± 900
±30
50
−72
− 13
43
−0.2
r
1
81
r
r
1
4
81
64
r
r
324
289
71.
r
529
1089
not real
9
8
± 10
13
√
0.0121
1
100
25
81
1
− 10
5
9
64
25
64. −
r
121
400
r
± 85
− 11
20
81
169
r
not real
1
5
r
76.
0.11
88
60. ±
±0.3
−0.01
1
25
r
r
72. ±
87.
ALG catalog ver. 2.6 – page 257 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
r
23
33
0.8
1000
√
44. − −625
68.
18
17
−48
√
7744
56.
± 47
0.06
56
40.
52. −
√
83. − 0.0001
not real
√
1000000
1
9
±1.1
12
36.
48.
√
75. ± 0.09
√
0.64
±500
−56
100
169
9
√
3136
± 12
16
49
67.
79.
0.7
r
28.
±4
√
32. − 2304
35
√
43. − −289
63. ±
± 14
11
0.0036
√
144
√
35. ± 250000
59.
7
10
r
√
not real
− 76
49
100
12.
13
√
1225
55. ±
66. ±
78.
31.
51.
1
36
√
81
√
27. − 5184
47. ±
r
8.
−7
√
2500
− 13
50.
√
± 16
√
39. − 3136
84
√
82. ± 1.21
1.2
±42
12100
√
74. − 0.04
0.1
√
85. − 0.0025
±24
4.
5
625
√
42. − −441
62.
5
11
225
144
0.01
22.
√
58. −
5
3
1369
441
14.
46. −
1
4
49. ±
57.
36
√
25
√
30. ± 1764
36
√
33. − 14400
r
√
125—Which are real numbers?
3.
−2
√
26. ± 576
18
1296
√
− 4
√
10. ± 100
18.
15
√
21. − 10000
25.
7—Simplify, if possible.
9
13
289
576
√
0.25
± 17
24
0.5
√
80. − 0.36
−0.6
√
0.0004
0.02
84.
√
88. ± 0.0144
±0.12
MA
√
89. ±7 25 ±35
√
93. 10 36 60
√
97. −6 144 −72
p
101. (27)2 27
105.
p
(−80)2
p
(7)2
113. −
p
(−44)2
q
(− 35 )2
121. −
125.
−7
p
(1.7)2
p
(−6)4
−1.7
36
p
(−3)8
133. ±
p
25(64)
√
141. −
145.
p
√
149. ±
49 · 121
(25)(49)(16)
82 · 32
p
−140
p
(50)2
−50
114. −
p
(−5)2
−5
q
( 72 )2
±40
7
2
122. −
p
(−42.6)2
126. −
p
(3)6
p
(2)14
√
142.
150.
144 · 9
−42.6
p
±27
70
6
108.
111. −
p
−16
112. −
p
(36)2
115. −
p
116. −
p
(−90)2
−90
119. −
q
− 14
120. −
q
2
(− 10
7 )
− 10
7
0.08
124.
p
128.
p
123.
(16)2
(−72)2
( 14 )2
p
(0.08)2
p
(−9)4
p
(2)10
±12
139.
p
(16)(81)
120
143. ±
p
(6)2 (7)2
±42
√
147. − 52 · 112
52 · 32 · 62
90
151.
p
ALG catalog ver. 2.6 – page 258 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−81
(−8)2
(4)6
136.
36
140. −
±144
−55
(4)2 (3)2 (7)2
84
−36
0.39
64
p
(10)8
√
64 · 36
−20
(4)(81)(64)
8
(−0.39)2
132. −
p
36 · 100 · 4
√
−72
32
√
135. − 4 · 100
36
p
54
131.
128
√
92. 4 49 28
√
96. −7 64 −56
√
100. ±4 256 ±64
p
104. (45)2 45
99
(−54)2
127. −
−27
p
(36)(4)
√
146. ±
24
(4)2 (2)2 (5)2
p
138. ±
77
15
110. −
134.
±40
150
107.
130.
−81
22
32
118.
3
5
129. −
137.
−44
√
91. 11 81
√
95. ±9 9
√
99. 5 196
p
103. (6)2
−12
(−32)2
106.
80
109. −
117.
√
90. −3 16
√
94. 2 121
√
98. 10 225
p
102. (15)2
−10000
48
p
9(49)
144.
√
49 · 9 · 25
148.
p
(9)2 (4)2
−21
105
36
√
152. − 32 · 72 · 52
−105
MB
Topic:
Simplifying square roots.
Directions:
1.
5.
9.
13.
17.
21.
25.
29.
33.
37.
41.
45.
6—Simplify. 7—Simplify, if possible.
125—Which are real numbers?
√
√
√
√
± 12 ±2 3
2.
18 3 2
√
√
√
√
32 4 2
6. − 45 −3 5
√
√
√
√
− 60 −2 15
10. ± 72 ±6 2
√
√
√
√
96 4 6
14.
99 3 11
√
√
√
√
− 120 −2 30
18.
132 2 33
√
√
−117 not real
22. − −68 not real
√
√
√
√
147 7 3
26. ± 160 ±4 10
√
√
√
√
250 5 10
30. − 288 −12 2
√
√
√
√
± 800 ±20 2
34.
825 5 33
√
√
√
√
4851 21 11
38. − 2548 −14 13
√
√
√
√
± 6174 ±21 14
42.
6075 45 3
√
√
√
√
− 4335 −17 15
46.
6760 26 10
√
√
49. 6 8 12 2
√
√
53. ±5 75 ±25 3
√
√
57. 5 108 30 3
√
√
61. 11 135 33 15
√
√
65. 3 5103 81 7
5 √
√
69. ± ( 99 ) ±5 11
3
√
50 √2
73.
3
15
r
1 √5
1
77.
4 2
r
3
√
81. ± 3
± 35 10
5
√
√
85. − 0.72 −0.6 2
√
√
14.4 1.2 10
89.
p
√
93.
(10)3 10 10
97. ±
p
(3)7
√
101. −5 −72
105. −
p
√
±27 3
not real
(5)4 (3)3
√
−75 3
√
√
50. ±2 24 ±4 6
√
√
54. 10 40 20 10
√
√
58. −2 98 −14 2
√
√
62. −4 117 −12 13
√
√
66. ±4 6804 ±72 21
70.
74.
78.
82.
86.
90.
94.
4√
√
50 4 2
5
√
√
44
±
± 211
4
r
√
3
− 2
− 211
4
r
1 5√
8
3
3 3
√
√
0.32 0.4 2
√
√
± 0.24 ±0.2 6
p
√
(5)5 25 5
98. −
p
(2)11
√
102. 6 −150
106.
p
(6)3 (3)6
√
−32 2
not real
√
162 6
10—Simplify (assume variable expressions are positive).
3.
7.
11.
15.
19.
23.
27.
31.
35.
39.
43.
47.
√
√
− 20 −2 5
√
√
48 4 3
√
√
84 2 21
√
√
± 112 ±4 7
√
√
135 3 15
√
− −136 not real
√
√
− 198 −3 22
√
√
± 300 ±10 3
√
√
936 6 26
√
√
2511 9 31
√
√
− 3456 −24 6
√
√
± 6534 ±33 6
√
√
51. −5 27 −15 3
√
√
55. 3 88 6 22
√
√
59. ±6 128 ±48 2
√
√
63. ±2 150 ±10 6
√
√
67. 2 1728 48 3
1 √
√
71. − ( 80 ) − 5
4
√
76 √19
75.
5
10
r
√
8
79. ± 3
± 335
9
r
2 2√
83.
2
6
3 3
√
√
87. ± 0.27 ±0.3 3
√
√
91.
2.42 1.1 2
p
√
95. − (7)3 −7 7
99.
p
(5)7
103. −
p
107. ±
p
ALG catalog ver. 2.6 – page 259 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
125 5
(−5)5
not real
(10)5 (2)4
√
±400 10
4.
8.
12.
16.
20.
24.
28.
32.
36.
40.
44.
48.
√
√
28 2 7
√
√
± 52 ±2 13
√
√
90 3 10
√
√
− 125 −5 5
√
√
± 140 ±2 35
√
−198 not real
√
√
242 11 2
√
√
405 9 5
√
√
− 1350 −15 6
√
√
± 3744 ±12 26
√
√
3825 15 17
√
√
3549 13 21
√
√
52. 3 32 12 2
√
√
56. −3 54 −9 6
√
√
60. 9 112 36 7
√
√
64. 3 162 27 2
√
√
68. −5 3312 −60 23
72.
76.
80.
84.
88.
92.
96.
3√
√
68 3 17
2
√
117
√
−
− 13
3
r
4 √22
2
3
9
r
2
√
− 2
− 47 7
7
√
√
0.54 0.3 6
√
√
− 1.6 −0.4 10
p
√
± (6)5 ±36 6
100.
p
104.
p
108.
p
(2)15
√
128 2
(−10)3
(8)4 (3)3
not real
√
192 3
MB
109.
p
113.
p
117.
121.
√
48 3
(6)3 (2)5
√
240 30
(2)6 (6)3 (5)3
√
2 3
144
√
137. ±
a2
161.
m6
p
165.
p
√
126.
130.
p8
138.
±p4
p4 q 2 r 6
x3
p
y7
177.
√
193. ±
197.
−p2 qr3
√
−ab2 ac
√
5p2 p
182.
115.
p
(2)3 (10)3 (9)2
√
±4q r
√
3xz 2yz
√
√
198.
159.
171.
206.
175. −
5x − 5y
116. −
p
(3)5 (2)8 (7)3
−5x3
36y 8
p
y5
120.
p√
124.
√
9c2
136.
7ab5
p
64y 6
±3d2
8y 3
√
a5
√
±m4 m
168.
p
√
x2 yz y
√
−10y 3 y
−7k 4
√
160. ± 121k 6 m8
164.
100y 7
−9x
√
144. ± m12 ±m6
√
148. a30 a15
√
152. 64d10 8d5
p
156. − 4m2 p6 −2mp3
√
−y 2 y
p
3c
√
140. − 49k 8
6y 4
x4 y 3 z 2
√
5 2
2500
±11k 3 m4
√
a2 a
p15
√
p7 p
√
172. ± hk 6 m9
√
±k 3 m4 hm
√
176. ± 144m
√
±12 m
√
k 2 10
√
179. ± 3x3
18y 2
√
3y 2
183.
√
45a4
√
3a2 5
√
184. − 24c6
√
−2c3 6
20m
√
2 5m
187.
√
27r5
√
3r2 3r
√
188. ± 32k 7
√
±4k 3 2k
191.
p
10k 4
p
9x4 y 3
√
−3xy 3 6
√
3x2 y y
12ab5 c4
√
2b2 c2 3ab
p
40x10 y 3 z 5
√
4a2
√
49a2 b10
p
p
(30)3 (2)7
√
128. − 81x2
√
132. ± 9d4
7m
√
167. ± m9
√
p3 w2 rpw
p
54x2 y 6
√
p
163. −
√
2x x
p
√
wx4 y 2
214.
p
(10a2 d)2
p
√
±2k 2 m4 6k
√
−kn2 5k
125m8 r12 p2
180. −
192.
207. ±
p
81w20 x11 y 27
a2 (a2 + 1)2
a3 + a
211.
p
(7x3 )4
215.
p
(b − c)2 (b + c)2
b2 − c2
ALG catalog ver. 2.6 – page 260 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
49x6
√
44cd7
√
−p3 2p
√
2d3 11cd
√
−2ab 5b
√
200. ± 3h8 m2
√
±h4 m 3
204.
√
75u9 w6 z 14
√
5u4 w3 z 7 3u
√
208. − 144a26 cd17
√
−12a13 d8 cd
√
±9w10 x5 y 13 xy
10a2 d
p
2p7
√
196. − 20a2 b3
√
5m4 r6 p 5
√
14a9 b15 c7 c
p
√
2m2 p 10mp
40m5 p3
√
199. − 5k 3 n4
203.
196a18 b30 c15
210.
√
±x 3x
√
195. ± 24k 5 m8
√
±2x5 yz 2 10yz
6z
25(x − y)2
194.
4x3
±2y
√
a12 a6
√
147. ± x20 ±x10
√
151. − 144m12 −12m6
√
155. c4 d2 c2 d
√
−a5 a
rp7 w5
4y 2
143.
√
±p p
p
202. ±
√
11kn12 r4 nr
(6z)2
139.
w4
p
p3
p
√
135. − 25x6
±k 3
w2 x8 y 4
190. −
121kn25 r5
p
w8
√
2 2
64
√
49m2
√
131. 16a4
p
√
5a2 b 2
16rq 2
127.
√
360 5
√
±480 15
112. ±
√
−1008 21
p√
4c
√
186.
√
−4ab4 c6 3a
213.
√
√
−2p 3p
12p3
18x2 yz 3
p
170.
178.
√
±2x 2
p
209.
158.
174.
√
m 6
50a4 b2
p
√
(3)3 (15)3
123. ±
x10 x5
√
146. − r18 −r9
√
150. ± 81c10 ±9c5
√
154. 9a2 b2 3ab
142.
√
201. − 48a3 b8 c12
205.
p
119.
x2
√
166. − a11
√
181. ± 8x2
189.
x4
√
y3 y
6m2
p
√
√
135 5
111.
−x
16c2
162. ±
25p5
185. −
√
√
2 5
400
√
x x
p
√
p√
√
134. ± k 6
m3
√
169. − a3 b4 c
173.
p
(4)3 (8)2 (3)3
√
122. − x2
a
√
141. − b10 −b5
√
145. b16 b8
√
149. 121a14 11a7
p
153. ± x2 y 2 ±xy
157. −
114. ±
118.
√
125. ± 25d2 ±5d
p
129. − y 4 −y2
133.
p
(5)5 (10)3
√
±192 3
p√
√
√
−1250 2
110. −
212.
p
216.
p
(2y)6
8y 3
(w + 4)2 (w − 4)2
w2 − 16
MB
217.
√
16y + 48
221.
p
225.
p
(x − 5)2
218.
x−5
222.
p
226.
p
50(5p − 2)2
√
(25p − 10) 2
229.
√
x2 − 12xy + 36y 2
234.
237.
230.
p
a+2
p
x − 6y
241.
245.
y 2n
√
x6a+2
p
50m − 175
(a + b)2
√
5 2m − 7
a+b
x3 (x + 3)2
(x2
a2 + 4a + 4
233.
√
√
4 y+3
√
√
219.
√
n3 + 2n2
223.
p
(y + 1)3
227.
p
n5 (m + 1)3
231.
p
y 2 − 6y + 9
235.
√
9d2 − 24d + 16
√
+ 3x) x
(mn2
x2 − 2x + 1
x−1
4a2 + 20a + 25
238.
yn
x3a+1
x2n y 8n+1
√
xn y 4n y
242.
246.
x4a
√
a5 − a4
√
(y + 1) y + 1
224.
p
228.
p
√
nm + 1
y−3
y 2m−10
a2n+1 b4n
y m−5
√
an b2n a
(r − p)3
√
(r − p) r − p
48(c + d)3
232.
√
c2 + 10c + 25
236.
√
100w2 − 20wz + z 2
10w − z
2a3n
243.
√
x4a+1
247.
√
r3x
ALG catalog ver. 2.6 – page 261 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
a2 a − 1
√
(4c + 4d) 3c + 3d
+ n2 )
√
239. 4a6n
x2a
p
√
220.
3d − 4
2a + 5
√
√
n n+2
√
x2a x
√
rx rx
√
240. 25r2x
5rx
244.
√
k 6n+1
√
k 3n k
248.
√
a5n+2
√
a2n+1 an
c+5
MC
Topic:
Multiplying square roots.
Directions:
6—Simplify. 3—Multiply. 7—Simplify, if possible.
10—Simplify (assume variable expressions are positive).
1.
5.
9.
√
11 ·
√
11
11
√
(− 36 )2
36
√
( 5 )2
18.
22.
√
√
25. (− 12 )(− 15 )
√
√
29. − 50 · 18
7·
√
√
( 81 )2
15
81
√
14. (− 3 )8
−100
√
√
6.
12 ·
√
√
6 5
√
√
3·
√
3·
√
14 3
√ √
− 6· 6
7.
√
( 25 )2
√
15. ( 5 )6
81
13
√
39
25
21
√
3 7
√
√
26. − 50 · 15
14
√
√
27. ( 39 )( 52 )
31.
60
√ √
√
34. (− 5 )( 7 )(− 10 )
√
√
( 22 )( 22 )
8.
√
−( 9)2
√
√
45 · 125
22
−9
26
√
16. −(− 2 )10
125
√ √ √
5· 2
10
√
√
23. ( 6 )(− 30 )
√
−5 30
4.
√
12. ( 26 )2
19.
−32
√
√ √
11 · 7
77
√
√
√
24. − 11 · 22 −11 2
20.
√
−6 5
√
26 3
28.
√
√
35 · 14
√
7 10
√
√
32. (− 128 )(− 8 )
75
√
√
√
35. ( 6 )(− 20 )( 3 )
√
5 14
√
√
√
37. (− 10 )( 3 )(− 30 )
−6
11. (− 14 )2
−7
√
√
30. ( 75 )( 48 )
−30
7
√
3.
√
10. −(− 7 )2
5
√ √
7· 3
21
√
√
√
21. ( 14 )( 7 ) 7 2
33.
√
√
(− 15 )(− 15 )
√
√
13. −( 10 )4
17.
2.
36.
√
√ √
10 · 5 · 6
32
√
10 3
√
−6 10
√ √
√
38. − 6 · 18 · 3
−18
39.
√ √ √
7 · 5 · 35
√
√ √
40. ( 24 )(− 2 )( 12 )
35
−24
30
√
√
√
−49 7
41. (− 7 )5
√
45. ( 2 )11
42. ( 3 )7
√
46. (− 2 )15
√
32 2
√ √
49. −2 6 · 84
√
−12 14
√
√
53. (3 24 )( 34 52 )
√
9 78
√
√
27 3
43. −(− 21 )3
√
47. ( 3 )9
√
−128 2
√
44. ( 34 )3
√
21 21
√
34 34
√
48. −(− 5 )7
√
81 3
√
√
50. ( 45 )( 13 10 )
√
5 2
√
√
51. (− 32 2 )(− 60 )
√
√
54. −8 108 · 2 6
√
−288 2
√
√
55. (7 54 )(− 20 )
√
3 30
√
125 5
52.
√
√
22 · 3 33
56.
1
3
60.
2
7
√
√
√
33 6
√
72 · 10 99
√
60 22
−42 30
√
√
57. 2 40 · 140
√
√
58. (− 35 133 )(−5 7 )
√
√
59. ( 66 )(−2 132 )
√
√
61. (− 12 11 )(−4 6 )
√
√
62. 8 2 · 3 3
√
√
63. 6 3 · 23 5
√
65. (3 6 )2
√
66. (−5 10 )2
√
40 14
√
−132 2
√
21 19
√
24 6
√
2 66
54
√
√
√
69. (−7 2 )(3 10 )( 5 )
√
√
67. −(8 2 )2
250
√
√
70. ( 6 )(2 3 )(6 2 )
√
22 15
77.
√
x·
√
x
a3 d ·
√
93. ( y )8
√
ad7
y4
4c7
a2 d4
86.
√
√
t·
√
t5
27x5 ·
√
3x7
9x6
p
√
90. ( xy )( x3 y 9 )
√
94. ( m )2
√
√
√
76. ( 38 )(3 18 )(2 2 )
√
36 38
m
√
√
5w · 5w
x2 y 5
87.
p
10p3 ·
√
ALG catalog ver. 2.6 – page 262 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
q2
84.
z5
10p
√
√
91. ( c7 n )( c5 n )
√
95. ( q )4
√ √
80. ( a )( a )
5w
√
√
83. ( z 3 )( z 7 )
t3
48
√
√
72. (−2 6 )( 34 24 )(− 16 )
√
79.
√
12 6
√
144 5
3y
63
128
√
√
√
75. (−3 3 )(−4 5 )(2 12 )
√
−15 210
82.
r2
√
√
85. ( 2c9 )( 8c5 )
√
√
√
√
74. ( 32 6 )(−5 14 )( 10 )
√
√
78. ( 3y )( 3y )
x
√
√
81. ( r3 )( r )
89.
√
68. (4 3 )2
−128
63
√
√
√
73. (6 11 )( 33 )( 31 5 )
√
√
−20 26
√
√
71. (3 28 )( 14 36 )( 7 )
−210
168 ·
√
√
64. (5 2 )(−4 13 )
√
4 15
√
72
√
10p2
c6 n
√ √
k · k7
a
k4
√
√
88. ( 6z 7 )( 6z 3 )
92.
√
√
rs9 · rs5
√
96. ( h )6
h3
rs7
6z 5
MC
√
97. ( n5 )4
101. (
105.
p
√
2p4 )12
5k ·
√
√
98. ( a2 )14
n10
√
k 15
3k
√
√
12a7 · 21a
p
p
113. 7p3 · 7p4
109.
117.
121.
√
7p3 p
p
33y 3
√
√
√
11y 2 3y
6bw4 ·
√
12b4 w2 2w
106.
√
6a4 7
p
11y 2 ·
102. (
64p24
110.
114.
118.
48b7 w
122.
p
5y 3 )6
√
√
√
√
√
s3 ·
√
18d ·
√
24t5 ·
√
13mp ·
120x5
√
10k k
5k
p
107.
√
6d5 3
√
18t2
52mp2
√
√
125. 4 2x9 · 3 50x
√
s2 11
6d9
√
12t3 3t
52mp3
√
√
126. 5 15a3 · 2 15a3
√
28ay 2 · 2 63a5
p
130. 3
p
44p3 r4 ·
√
√
133. x 3x · 6x2
√
3x2 2x
134. y
p
2y 3 · y
√
10y 4 y
√
√
137. 2a 10a · 3b 2ab2
a·
18pr
√
a2 + a
50y 2
√
√
138. cd c3 d · 5d 24d3
√
12a2 b2 5
√
p
142.
√
3·
√
√
n4 10
√
√
10x3 · 15x3
√
√
115. 19s · 19s2
√
5x3 6
√
19s s
119.
√
√
8x · 12x6
123.
√
√
40a2 c · 10a3 c2
√
4x3 6x
√
20a2 c ac
100. (
p
104. (
p
108.
√
3 n−5
y 20
10q 7 )4
100q 14
√
√
2u · 3u5
√
u3 6
√
√
8m · 20m
√
√
116. 2u2 · 8u3
√
4m 10
112.
√
4u2 u
120.
√
√
14k 7 · 35k 2
124.
p
√
7k 4 10k
24r2 y 3 ·
√
6ry
√
12ry 2 r
108y 4
√
√
128. 7 5w · 3 5w5
105w3
√
√
32bx5 · 4 2b2 x
√
√
132. 5 5h2 · 72hk 4
135.
√
√
18a3 · a 8a
√
√
136. m m5 · m 12m
√
32bx3 b
p
12a3
p
3xy 3 · 2 12xy 2
√
30hk 2 10h
√
2m5 3
√
√
140. 4r 20n3 r · r 5nr
√
72xy 2 y
3n − 15
y 4 )10
131.
139. 6y
√
10c2 d4 6c
√
a a+1
√
√
10n7 · n
111.
√
18p2 r2 22r
84a3 y
141.
√
32w10
p
√
127. 2 27y · 6 3y 7
150a3
129.
x28
√
103. ( 2w2 )10
125y 9
11s
20k 2 ·
√
99. ( x7 )8
a14
143.
√
√
2n2 − n · n
40r3 n2
144.
√
√
5k + 10 · 5
148.
√
√
w+4· w−4
√
5 k+2
√
n 2n − 1
145.
149.
√
√
x+y·
√
6x + 3y ·
p
3 4x2 − y 2
x+y
√
x+y
6x − 3y
146.
150.
√
√
k−3·
√
k−3
√
a − ab · a + ab
√
a 1 − b2
k−3
147.
√
√
a−b· a+b
√
√
√
151. 12u − 4 · 12u − 4
4(3u − 1)
ALG catalog ver. 2.6 – page 263 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
a2 − b2
152.
w2 − 16
√
√
5p + 10r · 5p + 10r
5(p + 2r)
MD
Topic:
Dividing square roots.
Directions:
6—Simplify. 4—Divide. 7—Simplify, if possible.
9—Simplify (assume no denominator equals zero).
10—Simplify (assume variable expressions are positive).
√
√
4.
1
√
21
√
21
21
8.
10
√
10
√
12.
√
72
√
18
16.
√
90
√
5
20.
√
7 13
√
5 2
24.
15
√
3
7
14
28.
1
√
12
√
6
3
15
√
2 5
√
3 5
2
32.
10
√
3 2
√
5 2
3
35.
√
96
√
54
4
3
36.
√
48
√
75
4
5
2
2
39.
6
√
48
40.
25
√
75
√
5 3
3
√
2 42
7
43.
√
5 3
√
6 10
44.
√
11 2
√
5 22
47.
√
132
√
48
48.
√
84
√
60
105
5
51.
√
5 260
√
65
52.
√
2 600
√
80
√
4 3
5
55.
√
96
√
240
56.
√
90
√
6 126
59.
r
1
117
60.
r
1
88
√
22
44
63.
r
7
2
64.
r
6
11
√
66
11
67.
r
11
132
68.
r
21
168
5
5
2.
1
√
17
√
11
6.
6
√
6
10.
√
125
√
5
14.
√
48
√
6
18.
√
5
√
17
22.
5
√
15
26.
1
√
3 10
30.
2
√
3 6
34.
30
√
20
38.
4
√
32
42.
√
6 14
√
7 3
46.
√
75
√
10
50.
√
7 12
√
140
54.
√
3 112
√
5 21
58.
r
1
11
√
11
11
62.
r
13
10
√
130
10
66.
r
19
133
1.
1
√
5
5.
11
√
11
9.
√
27
√
3
13.
√
60
√
3
17.
√
7
√
11
2
√
10
√
21.
1
√
5 2
√
25.
29.
5
√
2 15
33.
24
√
8
37.
8
√
50
41.
√
8 15
√
5 2
45.
√
70
√
105
49.
√
9 75
√
108
53.
√
5 21
√
7 70
57.
r
1
5
√
5
5
61.
r
5
3
√
15
3
65.
r
17
85
3
√
2 5
√
77
11
10
5
2
10
√
15
6
√
6 2
√
4 2
5
√
4 30
5
√
6
3
15
2
√
30
14
√
5
5
17
17
√
6
5
√
2 2
1
√
7
7.
23
√
23
11.
√
98
√
2
15.
√
54
√
2
19.
√
2 11
√
3 5
23.
24
√
6
1
√
28
√
27.
31.
√
85
17
√
15
3
√
10
30
√
6
9
√
3 5
√
√
30
2
√
√
7
7
√
7
7
3.
ALG catalog ver. 2.6 – page 264 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
23
7
√
3 3
√
2 55
15
√
4 6
√
3
2
√
30
12
√
11
2
10
√
10
5
√
13
39
√
14
2
√
3
6
10
2
√
3 2
√
7 26
10
√
5 3
√
11
5
√
35
5
√
30
√
35
42
√
2
4
MD
√
4 2
5
69.
r
32
25
73.
r
1
·
6
77.
r
4 1
·
5 2
81.
a
√
a
√
85.
7y
√
7
89.
y
√
10y
93.
a3
√
a
97.
a2
√
a5
√
3
2
70.
r
27
36
74.
r
2
·
3
78.
r
9 5
·
10 9
a
82.
x
√
x
√
√
y 7
86.
r
r
8
49
75.
r
3
·
4
79.
r
4 14
·
7 3
83.
1
√
m
√
x
15p
√
p
√
15 p
87.
2c
√
2
√
c 2
3
√
3k
√
90.
91.
2
√
2w
94.
x2
√
x
95.
u4
√
u9
98.
c3
√
c11
99.
p5
p
p5
√
1
3
2
6
√
10
5
√
10y
10
√
a2 a
√
a
a
√
r
44
102. p
11y 3
3x
105. √
15x3
√
15x
5x
12u2
106. √
8u
18u3
109. √
24u7
√
45n2
113. √
5n7
√
40k
117. √
20k 11
√
3 6u
2u
20a9
110. √
45a5
√
8c6
114. √
32c9
p
6y 9
118. p
2y 4
121.
√
5 2d3
√
d d5
1
125. √
xy
√
5 2
d2
√
cd2
129. √
c3 d 4
15a4 b
133. √
3ab
√
c
c
3r4 p5
137. p
12r2 p3
√
a2 x5
141. √
3a7 x3
√
c
a3
√
4 11y
y2
√
3u 2u
√
4a6 3a
3
1
√
4c c
√
y 2 3y
√
6
x
√
√
r3 p3 3p
2
√
x 3a
3a3
√
x2 y
6m2 n
134. √
6m3 n
12a3 b5
138. √
26a7 b2
√
142. √
2hk 7
12h4 k 5
√
6mn
√
6b4 26a
13a
√
k 6h
6h2
√
6
5
72.
r
24
100
76.
r
1
·
5
80.
r
4 3
·
3 7
1
√
y
√
84.
88.
11y
√
y
√
11 y
c
√
7c
√
92.
√
u
u
96.
d7
√
d5
√
y4 y
√
p2 p
x3
100. √
x3
√
x x
r
√
10
4
5
6
√
2 6
3
m
m
√
2w
w
r
√
21
15
7
15
√
2 7
7
y
y
7c
7
√
12 5b
5b2
14
104. √
7w5
√
2 7w
w3
9r3
107. √
18r5
√
3 2r
2
33y 4
108. √
11y
√
3y 3 11y
2k 7
111. √
40k
√
32u8
115. √
2u5
p
18p5
119. p
36p2
√
k 6 10k
10
103. √
24
20b3
√
2u u
√
p 2p
2
√
2k 3k 8
123. √
12k 3
c
127. √
cd
ab
ab
x3 y
130. p
x2 y
√
5a3 3ab
√
2
2
√
6x2 x
122. √
6x7
1
126. √
ab
xy
xy
5
2
√
x x
5x
15x3
√
2
k5
√
15
8
3k
k
5
101. √
45x5
√
3 n
n2
√
2 2
7
71.
√
k3 k
√
cd
d
ab
131. √
a5 b
√
b ab
a2
2c5 d3
135. √
8cd5
18mp5
139. p
3mp3
√
32uw4
143. √
2u5 w
ALG catalog ver. 2.6 – page 265 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
c4 2cd
2
√
6p3 3mp
√
4w w
u2
112. √
√
2 3m
3m3
6m3
27m11
√
2x6
116. √
50x3
√
5n3
120. √
35n7
√
x x
5
√
7
7n2
√
a a3
√
4 6a
√
a2 6
24
m
128. √
mp
√
mp
p
124.
u2 w
132. √
uw3
x2 y
136. p
7x6 y
30x3 y 2
140. p
6x3 y
√
144.
54c3 d3
√
3cd6
√
u uw
w
√
7y
7x
√
5xy 6xy
√
3c 3d
d2
MD
√
a
a
145.
r
1
a
149.
r
45
8y
153.
r
x4
6
157.
r
5c2
15c7
161.
r
3m
4n
165.
r
c2
d3
169.
r
ab4
8
√
b2 2a
4
r
st6
√
173.
r
177.
r
4xy 2
3z 5
181.
s
54x6 z
w3 y 4
r
25
x
√
3 10y
2y
150.
r
75c
28
√
x2 6
6
154.
r
21
a2
√
3c
3c3
158.
r
22r5
14r3
√
3mn
2n
162.
r
16k
7n
166.
s
h6
p7
170.
s
48x2
y
√
4x 3x
y
174.
r
k9
m2 n
√
k 4 kn
mn
178.
r
72b
a7 c2
√
6 2ab
a4 c2
182.
r
ab8
5c2 d
1
a−1
√
c d
d2
rs
st3
1
185. √
x+y
189.
s
197.
√
2y 3xz
3z 3
√
3x3 6wz
w2 y 2
√
x+y
x+y
20p3 + 8p2
5p + 2
√
c+d
193. √
c2 − d 2
r
√
5 x
x
146.
2p
√
c−d
c−d
4ab2 − 8b3
a2 + ab − 6b2
√
2b a + 3b
a + 3b
186. √
190.
r
r
1
4d
152.
r
49u
44
156.
r
8
w5
160.
r
30y
18y 2
164.
r
a
11b
√
11ab
11b
m
m2 n2
168.
r
1
x8 y
√
y
x4 y
90
cd3
√
3 10cd
cd2
172.
r
z5
32w
√
z 2 2wz
8
175.
r
xy 7
z5
√
y 3 xyz
z3
176.
s
n2 r 3
p
√
nr rp
p
179.
r
c2 d
6a3
√
c 6ad
6a2
180.
r
u3 w8
88r
√
uw4 22ru
44r
√
b4 5ad
5cd
183.
s
m2 n2
40r4 p3
184.
r
7s5 t
2u2 w
√
s2 14stw
2uw
√
a−1
a−1
a−b
187. √
a−b
p
3
151.
r
50
9m
155.
r
p3
5
√
r 77
7
159.
r
12a
20a9
√
4 7kn
7n
163.
r
5x
y
167.
r
1
m3 n4
171.
r
√
5 21c
14
√
21
a
√
h3 p
p4
198.
√
√
y+5
y 2 + 4y − 5
m2 − 1
m2 − 5m − 6
m2 − 7m + 6
m−6
3p
3
√
2 5m
3m
√
p 5p
5
√
15
5a4
√
5xy
y
√
√
mn 10p
20r2 p2
√
√
y−1
y−1
195.
a−b
a2 b − ab2
r
c2 − d 2
199. p
(c − d)3
ALG catalog ver. 2.6 – page 266 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1
ab
x−2
2
x − 4x + 4
d
2d
√
7 11u
22
√
2 2w
w3
√
15y
3y
n+2
188. √
n+2
a−b
√
n2 − 25
√
n+5
196.
r
200.
x2 + 4xy + 4y 2
p
(x + 2y)5
√
x−2
x−2
√
c2 − d2
c−d
√
n+2
192.
√
191. √
√
w+3
r
√
148.
r
w2 − w − 12
w−4
194. p
√
147.
√
n−5
r+3
2
r + 6r + 9
√
x + 2y
x + 2y
√
r+3
r+3
ME
Topic:
Adding and subtracting square roots.
Directions:
6—Simplify. 1—Add. 2—Subtract. 7—Simplify, if possible.
10—Simplify (assume variable expressions are positive).
98—Perform the indicated operation(s).
1.
5.
9.
√
81 +
√
√
− 5−
√
9
√
2.
12
√
−2 5
5
√
11 + 4 11
√
3−
√
3
49 −
√
√
− 15 −
36
√
1
√
−2 15
15
√
√
10. 10 6 + 2 6
√
5 11
√
√
13. − 35 − 4 35
17.
6.
√
√
−5 35
14.
18.
0
√
√
21. − 17 + (− 17 )
√
√
√
2−4 2
19 −
√
0
√
√
22. − 5 + (− 5 )
√
√
16 − 25
7.
√
√
14 + 14
√
−2 5
4.
√
√
− 121 + 64
√
2 14
8.
√
√
7+ 7
√
4 7
√
√
12. 8 23 − 23
−1
√
√
11. 7 7 − 3 7
√
12 6
√
−3 2
19
3.
−3
√
2 7
√
7 23
√
√
15. 7 21 − 11 21
√
−4 21
√
√
16. −18 5 + 10 5
√
√
19. − 7 − (− 7 )
0
√
√
20. − 13 − (− 13 )
√
2 10
24.
√
√
6 − (− 6 )
28.
√
√
7− 6
32.
√
√
√
6−3 6−7 6
23.
√
√
10 − (− 10 )
√
−8 5
0
√
2 6
√
−2 17
25.
√
10 +
√
5
26.
same
√
√
√
29. − 5 + 3 5 − 5 5
√
5 26
24 +
√
6
√
√
√
45. − 50 − 98
√
−12 2
46.
53.
√
96 +
√
√
− 10
√
9 6
150
√
√
57. 6 135 − 5 60
√
√
61. 12 2 + 8 3
65.
69.
√
12 +
√
√
11 3
√
2−
√
32 +
√
48
√
72
√
32 + 5 2
75 −
√
48
√
240 −
√
58. 4 27 +
√
√
66.
√
3 2
√
√
√
73. − 90 + 160 − 10
70.
74.
√
50 +
√
√
15 2
√
√
48
180 −
√
√
150 −
√
47.
55.
√
40.
√
5 5
√
−7 13
√
14 2
√
−5 2
√
√
48. − 125 + 80
√
− 5
√
√
52. 5 48 − 2 108
√
8 3
56.
√
√
99 − 44
√
√
√
60. −4 8 + 162
0
11
√
2
√
√
35 − 5 7
√
√
√
80 + 180 + 45
68.
√
√
√
24 + 54 + 150
√
√
√
71. − 50 − 18 + 98
72.
√
√
√
48 − 3 − 75
√
√
√
75. − 125 − 5 + 180
76.
√
√
√
32 + 18 − 98
same
√
13 5
√
− 2
0
√
3 17
64.
67.
5
√
√
17 + 68
√
√
44. −2 2 − 18
√
4 6
√
√
√
63. − 66 + 33
same
54
√
√
128 + 72
√
−9 6
√
3 13
3
59. − 500 + 5 20
√
18
24 −
√
√
20 + 45
√
√
16 3
√
√
√
√
51. − 117 − 2 52
0
√
−2 15
45 −
√
√
12 − 3
same
√
√
√
36. − 13 + 5 13 − 13
7
√
√
43. − 54 + 7 6
3
540
98 +
√
2 5
39.
√
9 2
√
√
62. −2 10 − 10 2
same
75 +
√
√
√
√
−6 5
√
√
50. −4 63 + 6 28
54.
√
8 15
√
√
√
√
31. 2 15 − 15 − 5 15
√
√
√
35. − 7 − 10 7 + 12 7
√
6 2
42.
same
−4 15
√
√
√
34. 17 2 − 4 2 − 7 2
5
√
√
49. −2 40 + 90
√
√
27. − 2 − 14
same
√
√
38. − 5 − 125
√
3 6
√
√
41. 3 5 − 20
√
6
√
− 39
√
√
√
33. 10 26 − 6 26 + 26
√
5+
√
√
√
30. −6 39 + 39 + 4 39
√
−3 5
37.
√
0
same
√
10 6
√
−2 3
0
0
√
√
√
77. 2 15 + 60 − 3 15
√
√
√
78. − 28 − 8 7 − 112
√
√
√
79. 2 8 − 72 + 3 2
√
√
√
81. 5 50 + 2 72 − 3 18
√
√
√
82. 7 45 − 4 80 + 2 20
√
√
√
83. − 108 + 5 48 − 3 75
√
√
√
84. 3 96 + 6 54 + 2 150
√
√
√
85. 2 5 + 7 5 − 3 11
√
√
√
86. − 2 + 3 5 − 6 2
√
√
√
87. 6 3 − 10 2 + 7 3
√
√
√
88. −8 6 − 10 + 6 10
√
15
√
28 2
√
√
9 5 − 3 11
89.
√
48 +
√
50 +
√
√
6 3+5 2
√
12
√
−14 7
√
9 5
√
√
−7 2 + 3 5
90.
√
80 +
√
20 +
√
√
4 10 + 6 5
√
160
√
− 3
√
√
13 3 − 10 2
91.
√
√
√
24 − 54 + 27
√
√
− 6+3 3
ALG catalog ver. 2.6 – page 267 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
2
√
√
√
80. −6 48 − 2 3 + 243
√
−17 3
√
40 6
√
√
5 10 − 8 6
92.
√
√
√
72 + 90 − 18
√
√
3 2 + 3 10
ME
93.
√
25 −
√
√
5+2 5
45 +
√
125
√
√
√
97. −3 32 + 4 50 + 6 8
√
√
−8 11 + 12 7
113.
√
72
95.
√
√
54 − 2 24 + 4 96
√
√
√
125. − 16a4 + 25a4
129.
√
36a2 +
√
6a + 9a a
√
−6 5
√
√
√
108. −7 80 + 245 − 180
√
√
5 10 − 19 5
√
√
18 2 + 3 22
√
37 3
110.
114.
126.
√
81a3
p
144p2 +
p
64p2
20p
√
√
130. 2 9c5 − 100c4
6c
2√
√
x3 y 3 + xy 16xy
134. 5
√
√
137. 6 18u4 w + u 27u2 w
√
√
138. −ab 75a + 2 12a3 b2
p
√
−ab 3a
120.
3√
4√
1√
48 −
27 −
192
4
3
4
√
−3 3
√
√
141. 5 3a − 10 3a + 2 3a
√
√
√
142. 3 n + 11 n − 12 n
√
−3 3a
√
2 n
√
√
√
145. −4 36y + 100y − 8 4y
√
√
√
147. 5 b − 2 49b − 9b
131.
p
151.
√
√
√
48r − 3 75r + 8 12r
√
√
√
153. x 49y + 4x 9y + 7x y
√
−30 y
135.
√
5 3r
√
26x y
√
6a x
√
√
√
157. 2c2 24d4 + c2 d 6d2 − cd 150c2 d2
0
7n
3d2
√
√
25a3 b2 + ab 121a
√
√
136. −c 36cd2 + 4d c3
√
16ab a
√
−2cd c
√
√
140. 2st 8st3 + 50s3 t5
p
4
139. −3x
p 45xy −
y 5x2 y 3
√
2
√
√
81n2 − 4n2
√
2r
√
√
132. d2 169d2 − 5 4d6
√
49y 3 − y 64y
√
−y y
5x − xy
2√
√
9st2 2st
5y
√
√
√
√
√
√
143. − xy − 8 xy + 15 xy 144. 14 c + c − 18 c
146.
√
32 2m
√
√
√
155. −7a 16x + 7a x + 3a 81x
128.
−3c3
√
−3 c
√
6 xy
√
−12 b
√
√
√
149. 3 18m − 50m + 7 32m
√
√
124. 7 2r − 6 2r
√
−8 a
√
√
9c6 − 36c6
−9xy
√
√
4 5
0
127.
√
√
3 2 + 36 3
√
√
2 5−5 2
2√
1√
2√
32 −
200 +
18
2
5
3
√
9xy xy
√
√
18u2 2w + 3u2 3w
√
1√
3√
80 +
50 − 2 32
2
5
√
√
123. 2 a − 10 a
√
√
− 33 + 12 19
√
√
√
6 7 − 2 6 + 15 3
118.
c − 10c2
p
√
133. 4y 2 49yz − 3y 64y 3 z
√
4y 2 yz
√
√
√
√
24 + 3 28 − 96 + 5 27
√
1√
5√
116. − 45 +
500 +
20
5
2
√
−5 cd
√
√
√
132 − 3 33 + 6 76
√
−27 5
√
√
√
√
112. −3 12 − 18 + 72 + 8 75
√
10 15
√
√
122. −6 cd + cd
a2
−18
√
√
√
107. 10 12 + 300 + 147
√
−2 3
√
5 x
√
√
√
100. − 96 − 3 36 + 2 24
√
√
√
106. 2 176 − 5 11 − 99
√
√
√
240 − 2 54 + 540 + 3 24
√
x+4 x
√
5 15 + 20
104.
√
4√
5√
√
√
98 −
72 8 6 − 2
115. 4 24 +
7
6
√
√
√
125
20
80 √
117.
+
−
5
3
6
4
√
√
√
96
24
54 √6
119. −
+
+
6
3
2
6
121.
√
√
√
375 + 225 + 25
96.
√
6 3−3
√
√
√
99. 6 40 − 4 90 − 3 20
0
√
1√
2√
108 − 147 +
27
3
3
√
√
√
36 + 108 − 81
√
√
√
103. 6 18 − 198 + 3 88
√
√
−11 13 + 4 17
√
7 6
√
49 −
√
√
√
102. −7 52 + 117 + 2 68
√
√
√
√
109. 5 40 + 125 − 6 180 − 250
111.
√
16
√
√
√
101. −2 99 + 3 112 − 44
√
98 −
√
2−7
√
√
√
98. 2 64 + 10 12 − 5 48
√
20 2
105.
√
94.
√
√
√
25a − a + 3 121a
√
37 a
√
√
√
148. − 81w + 7 16w + 3 w
√
22 w
√
√
√
150. −2 20k + 5 5k − 80k
√
−3 5k
√
√
√
152. 9 24y − 4 54y + 150y
√
11 6y
√
√
√
154. 8c w − c 4w − 3c 36w
√
−12c w
√
√
√
156. p r − p 100r − 5p 25r
√
−34p r
p
p
p
158. x 20y 3 − 2y 4x3 y + 80x2 y 3
ALG catalog ver. 2.6 – page 268 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
√
6xy 5y − 4xy xy
ME
√
√
√
159. n 27m3 n − m 3mn3 − 75m3 n3
√
−3mn 3mn
√
√
√
√
161. m 5n − m 20n + 3r 2m − r 8m
p
p
√
√
163. x 16y − xy 2 − x2 y + y 25x
165.
p
0.64p4 r + p
p
0.04p2 r
√
√
167. b 0.09ab2 − 0.25ab4
√
p2 r
√
−0.2b2 a
√
√
r 2m − m 5n
√
√
3x y + 4y x
√
√
√
160. −5ax 40ax2 + 2x a3 x2 + a 90ax4
√
√
2ax2 a − 7ax2 10a
√
√
√
√
162. −3b xy + a 4xy + b 9xy − 6a xy
√
−4a xy
√
√
√
√
164. 4k 10k + k 32k + k 40k − 2k 8k
√
6k 10k
166.
p
√
1.21x3 y 3 + xy 0.81xy
√
√
168. cd2 0.49cd − d 0.01c3 d3
ALG catalog ver. 2.6 – page 269 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
2xy xy
√
0.6cd2 cd
MF
Topic:
Combined methods (simplifying square roots).
Directions:
6—Simplify. 7—Simplify, if possible.
98—Perform the indicated operation(s).
1.
√
√ √
2( 8 − 2 )
2
2.
√ √
√
6( 6 + 24 )
10—Simplify (assume variable expressions are positive).
3.
18
√ √
√
− 10( 40 + 10 )
4.
√ √
√
− 3( 27 − 3 )
−6
8.
√ √
√
3( 75 − 6 12 )
−21
12.
√
√
√
14(2 7 + 14 )
−30
5.
√
√
√
− 11(2 11 + 44 )
6.
√ √
√
− 5( 45 − 5 5 )
10
7.
√ √
√
7(4 7 + 28 )
42
−44
9.
√ √
√
− 5( 10 + 3 5 )
10.
√
−5 2 − 15
13.
√
√
√
6(2 15 − 42 )
√
5 − 1 5 − √5
√
5
5
√
3 − 1 3√2 − √6
√
6
6
√
√
6 + 10 √ √
√
3+ 5
2
√
√
3 6 − 10
√
30
49.
√
2+2 3
3
−140
√
√
90 − 15 3 − 45 2
√
12 10
30.
34.
38.
42.
46.
54.
57.
√
√
3 5 + 4 35
√
2 5
√
√
7 10
10
√
3+4 7
2
√
−45 − 30 2
√
√ √
√
28. 4 5(3 15 − 60 + 70 )
√
√
20 3 + 20 14
31.
35.
39.
43.
47.
√
30 + 4 66
12
55.
√
√
55 7 − 84 5
35
58.
2
−
3
r
3
2
32.
36.
40.
44.
√
√
5 3+2 5
5
51.
√
√
11 20 − 6 112
√
2 35
r
√
1 + 2 √2 + 2
√
2
2
√
2 + 1 2√3 + √6
√
6
6
√
√
14 + 2 √
√
7+1
2
√
√
3 5+2 3
√
15
√
√
2 10 − 15
√
4 5
√
6
−
6
√
48.
√
√
6−3 3
52.
56.
√
√
3 5 − 10 3
3
59.
ALG catalog ver. 2.6 – page 270 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
r
5
3
√
√
20 7 + 10 2
√
2 5
√
√
2 35 + 10
√
√
2 75 − 5 80
√
2 15
3
+
5
√
21 + 6 42
√
2 7
√
√
3+6 6
2
√
√
8 3 − 12 6
√
4 2
r
√
1 − 3 √3 − 3
√
3
3
√
5 + 3 5√15 + 3√5
√
15
15
√
√
3 − 21
√
√
1− 7
3
√
√
2 5+3 2
√
10
√
√
5 2+3 5
5
√
√
2 2− 3
4
√
5 + 4 11
√
2 6
√
√
√
48 7 + 98 3
63
2
5
√
1 + 7 √7 + 7
√
7
7
√
2 − 5 2√10 − 5√2
√
10
10
√
√
30 − 15 √ √
√
6− 3
5
√
√
14 − 2 3
√
21
√ √
√
√
26. 5 3( 27 − 3 12 − 24 )
√
√
7 6−6 7
21
√
√
4 48 + 7 28
√
3 21
r
√
√
90 5 − 18 11
√ √
√
√
24. 2 10( 10 − 2 40 − 160 )
50.
5
+
2
√
40 − 10 7
210
√
√
3 13 − 7
√
7 2
r
√ √
√
20. 6 3(5 15 − 33 )
−212
√
√
3 26 − 14
14
53.
√
√
−30 6 + 30 5
√ √
√
√
22. −6 2(2 2 + 4 50 − 32 )
√
√
9 5−5 3
15
√
√
6 2+2 6
√
3 6
√
√
√
16. − 30(6 5 − 5 6 )
50
√ √
√
√
25. −3 15( 5 − 2 15 + 30 )
√
√ √
√
27. 2 6(5 6 + 60 − 150 )
√ √
√
21( 7 + 3 35 )
√ √
√
19. 2 5(2 20 − 35 )
√
√
18 3 + 90 2
√ √
√
√
23. 3 7( 28 + 5 7 + 63 )
45.
15.
√
14 2 + 14
√
√
7 3 + 21 15
√ √
√
18. 3 6( 18 + 5 12 )
√ √
√
√
21. 2 5( 45 − 2 20 + 6 5 )
41.
√
−2 15 + 10
√
√
√
14. − 10(4 2 − 15 )
√
−110 + 165 2
37.
√ √
√
11. − 2( 30 − 5 2 )
√
√
−8 5 + 5 6
√ √
√
17. −5 11( 44 − 3 22 )
33.
√
√
6(3 2 − 6 )
√
6 3−6
√
√
6 10 − 6 7
29.
√
√
8 15
15
√
√
3 40 + 13 27
√
3 30
√
√
60 3 + 117 10
90
60.
r
7
−
4
r
4
7
√
3 28
28
MF
√
√
1
11 10
+ 10
10
10
r
r
√
1
7
7
65. 2
−
−
21
7
9
r
r
√
8
3
69. 10
− 75 + 6
25
4
61.
r
√
√
4 2−2 3
73.
r
5
+2
2
r
r
9
−5
10
1
10
√
1 √
5 6
− 6 −
6
6
r
r
4
5 37√5
66. 3
+5
10
5
4
r
r
8 √
1
70. 6
+ 54 − 24
3
6
62.
r
√
3 10
5
√
√
√
√
83. 5 ab(2 a + 3 ab + 6 b )
r
−
a2
+
81d
86.
√
5a d
18d
a2
36d
√
18a 2b
97.
1
−
x2
3
−
16
p
90. −
p
x2 y + 2 xy 2
√
3 xy
49
+
2c3
r
2
9
√
2
6
25
+
3
r
1
x
r
50a2
b
√
1− x
x
94. 6wz
r
r
3
2
r
√
9
18
33 2
−3
−
+
20
8
25
r
√
2√
3
√
189 + 14
+ 2 84 8 21
3
7
r
r
√
1
5
√
−2 20 − 10
+3
−5 5
5
9
r
√
2
3√
√
12
+ 3 24 −
150 7 6
3
5
r
1
−
2
r
√
√
√ √
84. 2 x( 9x − 5 x3 + 25x )
16x − 10x2
87.
50
c3
√
3 2c
2c2
wz 3 √
− 72w3 z 5
18
98.
1
+
y3
r
√
5
10
√
√
5 4cd − 3 32d
√
2 cd
88.
√
√
18m
8mn
√
−
6n
n
92.
√
5c − 6 2c
c
r
91.
95.
x2 y
s
p
36x3
+ 5 x7 y
y
√
11x3 xy
1
y4
√
y+1
y2
r
√
9y 2 − 15y
√
−5wz 2 2wz
r
1 6√7
7
7
r
r
3
1 5√3
68.
+
6
4
3
r
r
r
5
4
1
72. −
+
+
4
5
5
√
7−
64.
√
√ √
√
82. −3 y( 8y − 50y + 5 y )
√
x+2 y
3
r
1
3
r
√
√
6
17 3
−
2
12
80.
√
2 3
3
r
√
8 2mn
3n
√
93. 4a 98b − 2b
r
71.
r
78.
√
√
15ab + 10a b + 30b a
√
r
9
−4
2
76.
−72c
√
√
a b + 2a
a
89.
67.
r
74.
√ √
√
√
81. 4 3c( 3c − 12c − 75c )
85.
√
3−
√
3 6
r
√
2√
3
√
75. −6 240 +
375 + 10
−20 15
5
5
r
r
√
9
8
√
+ 2 32 + 15
9 + 18 2
77. 12
16
9
r
r
1√
9
2
√
79.
72 + 4
− 10
7 2
2
2
25
√
√
2 ab2 + 8b
√
2 ab
63.
99.
d
cd
√
−√
2
cd
c3 d
√
√
c − cd
c
ALG catalog ver. 2.6 – page 271 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
√
4 45k − 10 kd3
√
10kd
√
√
6 2d − d2 10
d
p
48p3
1
p
−p
p5
3p2
√
11 3
3p
√
96. 3hk 20h2 k + 5k 2
√
7h2 k 5k
√
√
b
ab
100. √ + √
ab
a3
r
h4
5k
√
√
a+ b
a
MG
Topic:
Multiplying binomials with square roots.
Directions:
3—Multiply.
1.
(1 +
√
2 )(2 +
√
2)
6—Simplify.
2.
√
4+3 2
5.
√
√
(4 3 − 5)(3 3 + 2)
6.
√
7 )(3 7 − 5)
√
√
14. (5 5 + 2)(8 + 5 )
2+
√
√
√
√
7 3−5 5
x )(2 −
√
x)
2a −
√
√
√
√
27. (x 2 − y 5 )(2x 2 − y 5 )
29. (1 −
1−
√
√
3 )(1 −
5−
√
3+
√
√
5)
√
√
33. ( 10 − 4)( 2 + 1)
√
√
4x 2 − 3xy 10 + 5y 2
√
√
30. ( 2 + 2)( 3 − 2)
34. (2 +
7.
√
√
( 5 + 7)(3 5 + 1)
√
√
11. (4 3 + 1)(5 − 2 3 )
15. (10 −
√
√
6 )(3 6 − 2)
√
5 )(1 +
√
10 )
√
√
(3 + 2 2 )(1 − 2 )
√
2
√
√
12. (3 + 4 7 )(5 7 + 3)
√
√
16. ( 10 − 6)(2 10 + 5)
√
−10 − 7 10
√
√ √
20. ( 7 − 2 2 )( 14 + 1)
√
√
5 2−3 7
√
√
23. (2 a − 1)( a + 2)
24. (c −
√
√
c )( c − 3)
√
√
−4c + c c + 3 c
√
√
√
√
28. (p q − 2 r )(p q + r )
√
8.
√
149 + 27 7
√
√
√
19. ( 10 + 4)( 5 + 2 )
31. (2 −
√
√
( 3 − 2)( 3 − 6)
−1 −
√
√
26. (3 + k m )(3 + 2k m )
√
ab − b
√
√
√
−4 − 2 2 + 2 3 + 6
15
4.
√
15 − 8 3
√
2a + 3 a − 2
√
y+5 y+6
√
√ √
√
25. (2 a + b )( a − b )
√
2)
√
√
9 2+6 5
√
√
22. ( y + 3)( y + 2)
x−x
2 )(5 +
√
−38 + 32 6
√
√
√
18. ( 3 − 5 )(2 − 15 )
√
√
5 2−3 3
√
√
−19 + 18 3
√
41 + 42 5
√
√
√
17. (2 6 + 1)( 3 − 2 )
(4 −
√
22 + 22 5
√
−13 − 2 2
√
−9 + 13 7
21. (1 +
√
√
(2 − 3 3 )(4 − 3 )
√
√
10. (1 − 2 2 )(4 2 + 3)
√
−22 + 11 5
√
3.
√
18 − 2
√
17 − 14 3
√
√
(2 5 − 3)(4 − 5 )
13. (6 +
√
√
( 6 + 1)( 6 − 4)
√
2−3 6
√
26 − 7 3
9.
10—Simplify (assume variable expressions are positive).
7 )(5 +
√
2)
√
√
√
10 + 2 2 − 5 7 − 14
√
√
35. ( 2 − 4)( 6 − 3)
√
9 + 9k m + 2k 2 m
√
p2 q − 3p qr − 2r
√
√
32. ( 5 + 3)( 2 + 1)
√
√
√
3 + 3 2 + 5 + 10
36. (1 +
√
√
6 )(1 − 3 )
√
√
√
1−3 2− 3+ 6
√
√
√
−4 − 4 2 + 2 5 + 10
√
√
√
2 + 5 2 + 5 + 2 10
√
√
√
12 − 3 2 + 2 3 − 4 6
√
√
√
√
37. ( 2 + 14 )( 7 + 2 )
√
√ √
√
38. ( 6 − 3 )( 2 + 6 )
√
√ √
√
39. ( 15 + 5 )( 5 − 3 )
√
√
√
√
40. ( 5 − 10 )( 2 − 5 )
√
√
√
2 + 7 2 + 2 7 + 14
√
√
√
6−3 2+2 3− 6
√
√
√
5 + 5 3 − 3 5 − 15
√
√
√
−5 + 5 2 − 2 5 + 10
Squares
41. (3 +
√
5 )(3 +
√
5)
√
14 + 6 5
√
3+2 2
√ √
√
√
45. ( 6 − 2 )( 6 − 2 )
√
8−4 3
49. (1 +
√
√
√
42. ( 2 + 1)( 2 + 1)
√
4+2 3
√
√
53. ( 10 − 2 )2
√
12 − 4 5
√
3 )(4 −
√
3)
√
19 − 8 3
√
50. ( 5 + 2)2
√
5+2 6
√
9+4 5
√
√
54. ( 6 − 3 )2
√
9−6 2
√
51. ( 7 − 3)2
√
√
44. ( 7 − 2)( 7 − 2)
√
11 − 4 7
√ √
√
√
√
√ √
√
46. ( 10 − 7 )( 10 − 7 ) 47. ( 3 + 2 )( 3 + 2 )
√
17 − 2 70
3 )2
43. (4 −
√
√
√
√
48. (2 2 + 3 )(2 2 + 3 )
√
11 + 4 6
√
16 − 6 7
√
√
55. ( 5 + 10 )2
52. (4 −
√
10 )2
√
26 − 8 10
√
√
56. ( 2 + 6 )2
√
8+4 3
√
15 + 10 2
√
√
57. ( 3 − 2 5 )2
√
√
58. (2 2 − 3 7 )2
√
√
59. (2 5 + 6 )2
√
√
60. (4 3 + 3 4 )2
√
23 − 4 15
√
71 − 12 14
√
26 + 4 30
√
84 + 48 3
√
√
61. (2 22 − 5 2 )2
√
√
62. ( 10 − 5 2 )2
√
√
63. (2 7 + 14 )2
√
√
64. ( 15 + 10 3 )2
√
138 − 40 11
√
60 − 20 5
√
42 + 28 2
√
315 + 60 5
√
√
67. (3 r − 2 s )2
√
√
68. (a b + b a )2
√
√
65. ( x − y )2
√
x − 2 xy + y
√
√
66. ( a + b )2
√
a + 2 ab + b
√
9r − 12 rs + 4s
ALG catalog ver. 2.6 – page 272 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
a2 b + 2ab ab + ab2
MG
Conjugates
69. (1 +
√
2 )(1 −
√
2)
√
√
73. ( 6 − 3)( 6 + 3)
70. (4 +
−1
√
5 )(4 −
√
5)
√
√
74. ( 3 − 7)( 3 + 7)
−3
√
√ √
√
77. ( 5 + 3 )( 5 − 3 )
11
−46
√
√ √
√
78. ( 2 + 6 )( 2 − 6 )
−4
2
81. (−7 −
√
10 )(7 −
√
−39
−23
√
√
√
85. (2 5 + 3)(2 5 − 3)
√
2 )(5 −
√
2)
31
√
√
89. (10 − 2 7 )(10 + 2 7 )
√
3)
√
√
75. ( 2 + 6)( 2 − 6)
72. (5 −
1
√
√ √
√
79. ( 7 − 5 )( 7 + 5 )
83. (−1 +
√
6 )(1 + 6 )
84. (−3 +
√
√
√
91. (1 + 6 2 )(1 − 6 2 )
√
√
√
√
96. ( 6 − 15 )(− 6 − 15 )
9
√
√ √
√
97. (− 6 − 2 5 )( 6 − 2 5 )
14
√
√
√
√
98. ( 15 − 3 10 )(− 15 − 3 10 )
103. (−4 5 −
5 )(4 5 −
9
√ √
√
√
100. (− 5 + 3 2 )( 5 + 3 2 )
13
−60
√
√
√
√
102. (−6 2 + 6 )(6 2 + 6 )
−66
√
√
√
−75
104. (2 10 −
√
√
√
√
105. (2 3 − 3 2 )(2 3 + 3 2 )
−6
√
√
√
√
106. (5 2 + 2 5 )(5 2 − 2 5 )
30
√
√
√
√
108. (2 6 + 5 2 )(2 6 − 5 2 )
−26
109. (a +
√
3 )(a −
√
22
110. (x −
3)
a2 − 3
113. (2 −
√
√
5 )(x +
√
5)
xy )(2 +
√
xy )
114. (3 +
4 − xy
a−b
9k − m2
√
√
√
√
123. ( w + xy )( w − xy )
√ √
√
√
125. ( c − a b )( c + a b )
√
√
127. (5 + 3 m )(5 − 3 m )
cd )
√ √
√
√
118. ( a − b )( a + b )
p−r
√
√
121. (3 k − m)(3 k + m)
cd )(3 −
√
9 − cd
√
√ √
√
117. ( p + r )( p − r )
w − xy
c − a2 b
25 − 9m
√
y )(2 +
√
y)
2)
−38
112. (4 +
4−y
x2 − 5
√
111. (2 −
2 )(−2 10 −
√
75
5)
√
√
√
√
107. (2 10 − 3 2 )(2 10 + 3 2 )
√
√ √
√
116. ( n − 7 )( n + 7 )
a−6
n−7
√
√
119. ( c − d)( c + d)
120. (x +
c − d2
√
√
122. (r + 2 p )(r − 2 p )
√
x2 − y
r2 − 4p
√
√
√ √
124. ( ab − c )( ab + c )
ab − c
√
√
√
√
126. (x y − w )(x y + w )
√
√
128. (2 p − 1)(2 p + 1)
√
√
w )(4 − w )
16 − w
√ √
√
√
115. ( a + 6 )( a − 6 )
ALG catalog ver. 2.6 – page 273 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
5)
136
−7
√
5 )(3 +
√
√
92. (12 + 2 2 )(12 − 2 2 )
√
√
√
√
95. ( 10 − 3 )(− 10 + 3 )
√
√
52
−23
√
6
√
√
88. (3 6 − 2)(3 6 + 2)
87. (5 3 − 5)(5 3 + 5)
√
√
√
√
94. ( 30 + 7 )(− 30 + 7 )
√
18
−4
√
17
√
√
√
√
101. (3 7 + 3 )(−3 7 + 3 )
√
7)
−1
√
√
√
√
√
93. ( 2 + 21 )(− 2 + 21 )
√
√
√
√
99. ( 15 + 2 6 )(− 15 + 2 6 )
7 )(5 +
√
√ √
√
80. ( 2 − 3 )( 2 + 3 )
−71
44
√
√
√
76. ( 7 + 1)( 7 − 1)
−34
50
√
√
90. (8 − 2 5 )(8 + 2 5 )
72
3 )(2 +
5
√
86. (4 2 + 1)(4 2 − 1)
17
√
2
82. (−5 −
10 )
71. (2 −
4p − 1
x2 y − w
y )(x −
√
y)
MH
Topic:
Simplifying complex fractions.
Directions:
6—Simplify. 9—Simplify (assume no denominator equals zero).
10—Simplify (assume variable expressions are positive).
3.
−7
√
3+ 2
√
−12 + 6 5
7.
10
√
3 − 14
−20
√
3+3
√
10 3 − 30
3
11.
√
14
5−1
14.
1
√
3− 7
√
3+ 7
2
15.
√
6
11 + 2
18.
√
19.
√
1.
2
√
1− 3
−1 −
3
2.
√
9
7−4
−4 −
5.
−4
√
6+2
√
4−2 6
6.
√
6
5+2
9.
36
√
10 − 8
10.
13.
−3
√
4+ 5
17.
5
√
√
2− 3
√
√
−2 10 − 16
3
√
−12 + 3 5
11
√
√
−5 2 − 5 3
21.
35
√
√
10 − 5
25.
√
3
√
5− 3
29.
√
2
√
2+1
33.
√
3
√
6−6
37.
√
2 2
√
5 + 10
41.
√
5
√
3−1
45.
√
21
√
√
3+ 7
√
√
−3 7 + 7 3
4
49.
√
− 15
√
√
6− 3
√
√
− 10 − 5
53.
√
√
7 10 + 7 5
√
5 3+3
22
√
7
8
√
6− 5
√
√
8 6+8 5
−20
√
2+ 6
√
√
5 2−5 6
22.
√
26.
√
− 6
√
6+4
√
3−2 6
5
23.
√
2
30.
√
√
− 2−2 3
10
√
√
10 2 − 4 5
15
√
√
5
√
√
30 − 2
15 +
2
√
5
10
√
3 − 10
√
√
−10 − 3 10
√
√
−6 − 2 14
8.
18
√
3− 3
√
9 + 3 13
12.
11
√
7+ 5
√
7− 5
4
√
6 11 − 12
7
16.
√
10
13 − 4
3
√
8+ 7
√
√
6 2−3 7
20.
√
−9
√
10 + 11
28
√
3+ 7
√
√
−7 3 + 7 7
√
15 − 4 15
32.
√
36.
√
14
√
7−4
40.
√
38.
√
2 6
√
3+2
√
√
−6 2 + 4 6
39.
√
5 3
√
6 − 15
42.
√
− 2
√
5+4
√
√
10 − 4 2
11
46.
√
50.
26
√
√
13 + 10
√
43.
7−
7
√
2
√
√
6−3 2
2
√
√
10 3 + 5 5
7
√
√
7 7 + 14
47
√
− 10
√
√
2+ 5
51.
√
33
√
√
11 + 6
√
√
2 5−5 2
3
44.
48.
√
√
− 10 + 2 3
55.
√
2 2+1
7
√
√
−7 2 − 4 14
9
√
3 5
10 + 2
6+
√
√
−3 − 2 3
3
√
2
√
√
5 2−2 5
2
√
√
6 3− 6
34
√
6
√
2− 3
√
15
√
√
2− 3
√
√
− 30 − 3 5
√
√
−2 3 − 3 2
10
√
35 − 15
52.
√
56.
√
− 6
√
√
11 + 2
√
2
√
√
5+ 6
2
3
3−2
√
√
√
11 3 − 3 22
5
√
4−
2
√
√
√
5 5+5 3
√
47.
√
√
13 2 − 2 65
3
54.
√
√
− 15
√
4 + 15
35.
√
5
√
−10 13 − 40
3
10
√
5− 3
31.
34.
√
√
7 2+2 7
5
√
28.
√
− 6
√
1+ 3
14
√
7− 2
24.
27.
√
√
5 10 + 2 5
23
−1 +
√
√
9 10 − 9 11
√
11 − 11
10
10
√
5− 2
√
√
7 5+7
2
11
√
11 + 1
√
√
√
5 6 + 10
28
4.
√
√
2−
√
4
5+1
√
2
−3 +
√
√
14 + 6
4
√
√
− 66 + 2 3
9
57.
4
√
3 2−2
√
6 2+4
7
58.
8
√
5−2 5
√
40 + 16 5
5
59.
−13
√
4 3+3
√
−4 3 + 3
3
60.
7
√
5 2+8
√
−5 2 + 8
2
61.
11
√
3+2 5
√
−3 + 2 5
62.
−2
√
3 2+4
√
−3 2 + 4
63.
2
√
5−2 6
√
10 + 4 6
64.
4
√
7−3 5
√
7+3 5
ALG catalog ver. 2.6 – page 274 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
MH
65.
69.
73.
√
14
√
√
2 7− 2
√
2− 3
√
1+ 3
√
6+4
√
2−2
√
√
7 2+ 7
13
66.
√
−5 + 3 3
2
70.
74.
81.
√
6 − 11
√
7 − 2 11
√
3 10 + 2
√
2 5+1
4+
√
11
78.
√
√
√
30 2 + 4 5 − 3 10 − 2
19
85.
√
−11 + 3 15
7
√
5+2
√
5−3
√
1 − 10
√
6− 2
√
−11 − 5 5
4
71.
75.
√
√
3 7+9
√
3 7+5
√
6−5 3
√
4+3 6
√
9+6 7
19
79.
83.
√
√
√
−24 + 20 3 + 18 6 − 45 2
38
86.
√
√
2 2− 6
√
√
3− 2
89.
√
1− a
1−a
90.
√
3
√
√
x− 3
87.
93.
√
x
√
x+2
√
x−2 x
x−4
94.
√
97.
√
√
c+ d
√
√
c− d
98.
√
√
a− b
√
√
a+ b
√
c + 2 cd + d
c−d
√
m
m−1
√
m+ m
m−1
√
a − 2 ab + b
a−b
√
17 + 9 2
47
68.
72.
76.
√
4−3 5
√
5+2 5
√
15 − 12
√
4 3−7
√
50 − 23 5
5
80.
84.
√
√
6+ 3
√
√
2 6− 5
91.
w
√
w− w
95.
√
99.
√
m+1
√
m−1
√
ALG catalog ver. 2.6 – page 275 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
a
√
a+ b
√
a − ab
a−b
√
m+2 m+1
m−1
√
− 3
√
15 + 2 6
√
7+1
√
7+4
√
2− 5
√
5 + 15
√
√
5−2 2
3
−1 +
3
√
7
√
2+1
√
3 2−6
√
11 − 2 6
√
10 − 5 3
√
−4 − 3 2
6
√
√
√
22 − 4 6 + 11 3 − 6 2
5
88.
√
w+ w
w−1
√
√
√
√
10 − 5 5 − 2 15 + 5 3
10
√
√
√
12 + 6 2 + 30 + 15
19
√
3x + 3
x−3
√
2+ 2
√
7− 2
√
3+8
√
6+1
√
√
− 7−7 3
20
√
√
√
84 + 48 3 − 12 5 − 7 15
√
√
√
2 6+4−6 3−6 2
1
√
1+ a
√
21
√
3−3 7
√
√
√
3 2− 3+8 6−8
5
√
√
√
6 − 6 10 + 2 − 2 5
34
82.
√
√
5− 3
√
√
5+2 3
67.
√
√
3 15 + 5 3
12
√
√
√
−2 2 − 3 − 6 − 4
77.
√
5
√
√
3 3 + 15
√
√
14 + 2
√
√
7+ 2
√
√
√
7 2 + 14 − 2 7 + 2
5
92.
√
1
√
a− b
√
96.
p
√
1+2 p
100.
√
1− w
√
1+ w
√
√
a+ b
a−b
√
p − 2p
1 − 4p
√
1−2 w+w
1−w
MI
Topic:
Solving equations with square roots.
Directions:
15—Solve.
1.
5.
9.
3=
√
√
x
√
2.
9
y = 16
−5 =
a
√
17. 2 = 5 p
25.
√
49. 8 =
√
57.
√
65.
√
√
58. 5 −
Ø
√
−5
66. 2 =
3f − 5
1+r =5
70.
42
a + 5 + 11 = 16
77. 8 −
h+7
7.
9=
11.
√
x+4=0
23.
36
√
20
8
√
√
2c
81
y
3
=
2
5
47. 9 =
25
2
51. 7 −
12
√
√
59.
Ø
√
2p + 1 = 0
13
63. 5 =
−20
67.
24 + a
4 − 5s = 8
71. 5 =
−12
√
n−6
√
78. −2 = 6 − h + 1
75. 3 +
55
√
k
Ø
121
49
16
√
x = 10
c
=1
3
√
44. 3 − k = 8
48.
√
5y = 10
52. 11 = 3 +
1
2
56. 15 = 3 +
2
60. 0 = 6 +
Ø
64
16
25
9
Ø
20
√
√
√
10m
6r
√
3−p
68. 4 =
8
√
72.
4 + 7y = 9
3 − 4y − 4
− 32
76. 5 =
√
Ø
46
−99
101
32
5
24
3g
√
64.
x+3=7
17
3x + 1
√
625
40. 2 −
√
20 + k = 14
79. −1 =
63
8+y
√
1 − c = 10
√
p
√
27
2k = 6
√
√
√
1
b 1
28.
=
16
20
5
√
y
32. −4 =
Ø
5
√
36. 1 = 5 h − 3
1
4
√
55. 5 2c − 3 = 7
1
12
25 =
24. 2 +
121
36
25
3x
8.
36
√
20. 4 k = 7
1
9
√
y
5
1
39.
−
=
36
2
3
2
√
43.
w − 6 = −10 Ø
4
9
√
n=6
√
16. 0 = 2 m − 22
81
√
2
=− a Ø
3
√
35. 8 = 10 a + 3
25
4.
12. 0 = 8 +
Ø
31.
Ø
m−4=3
74. 12 = 5 +
m
√
2h = 8
√
√
25
√
m−3=8
27.
9
64
√
54. 1 = 2 3x
9
20
n + 30 = 5
√
√
a=5
√
19. 3 x = 1
1
49
√
50. − 3n + 10 = 4
27
62.
69. 11 =
73.
46. 5 =
8
3
70
√
3.
√
15. 72 − 8 y = 0
4
w
1
=
3
8
w − 21
61. 7 =
Ø
√
13
5
38.
= +2 x
6
6
√
42. 0 = 3 + n Ø
Ø
5n + 6 = 1
√
16
√
34. 4 m − 7 = 13
4
9
225
3a − 1
√
53. 4 5y = 6
100
√
30. 2 x = −12
Ø
g+9=0
6h = 4
√
101—Give the restriction(s) on each variable, then solve.
√
26.
4
9
1
8
37. 3 p − =
5
5
√
h=4
22. 13 =
81
√
45.
y
√
18. 1 = 7 w
4
25
√
33. 10 = 3 k + 4
√
√
√
√
14. 7 n = 14
16
√
29. −5 n = 10
41.
10 =
√
10. 2 = − c
Ø
y−5
√
2
a
=
9
3
6.
256
√
13. 12 = 3 x
21. 4 =
16—Solve and check.
−13
11
4−c−7
80. −5 = 2 −
√
−140
2x + 15
17
√
81. 2 = − 8 − 2d
Ø
√
85. 3 7x − 6 = 18
6
82. 12 +
√
3+w =5
83. 4 −
Ø
√
86. 0 = −15 + 5 5a + 4
√
3k + 1 = 8
√
87. 10 = 2 4 − 3n
Ø
−7
1
√
89. 2 m − 2 − 3 = 0
17
4
84. −10 =
√
y−4−9
√
88. 32 − 10 3p + 4 = 12
0
√
90. 5 + 6 4h − 1 = 17
5
4
√
91. 0 = −24 + 8 1 + 3x
√
92. 6 = 1 + 2 w + 6
1
4
8
3
93.
r
y
=6
2
97. 8 −
r
94. 10 =
72
2d
= −2
7
350
98.
r
Ø
r
x
5
95. −7 =
500
2y
−5=1
5
90
99.
r
ALG catalog ver. 2.6 – page 276 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
r
a
− 10
7
8x
=4
5
10
63
96. 8 +
r
m
= 13
6
100. 0 = −9 +
r
3c
2
150
54
MI
101. 6 +
r
105. 2 =
p
109.
r
n
=0
10
102. 3 = 10 +
Ø
2(3 − x)
4
2w − 5
=1
2
−29
7
2
106. 4 =
p
110. 2 =
r
r
h
5
103.
Ø
3(m − 1)
3
4 − 3k
7
49
−8
r
7s
= −2
3
r
2y −
Ø
1
3 5
=
4
2 4
√
3 + 2h − 1
111. 4 =
2
107.
13
104.
r
3
+ 8 = −1
10
108.
r
r
1
2
+ =
3 9
3
112.
8−
Ø
1
√
5−n
= −1
3
−116
√
2p + 1
113. 6 =
+ 10
2
√
√
x
x
−
2
4
√
√
121. y = 2 3 12
129. x −
133.
137.
√
√
√
5=
Ø
√
w
w
+
=2
3
6
√
√
122. 3 c = 6 2 8
√
√
126. 10 k = 2 10
√
20
9
6
w= √
w
6
16
2
5
√
√
50 = n + 2 4 2
√
√
x
134. √ − 18 = 0 24
32
√
3 5
130.
√
138. √
4a − 1
=8
2
√
2 c 3
116. 1 =
+
Ø
3
2
√
√
2 d
d
120.
144
−2=
3
2
√
√
124. 0 = x − 5 3 75
Ø
√
118.
400
9
5
y
27 = √
3
115. 5 −
Ø
√
117. 5 =
√
√
125. 3 5 = 5 a
2
√
114. − 4 y = 2
3
r
√
4
= 3y − 5
3y − 5
√
3 n
n
=
5
10
√
√
123. 10 5 = 5 n 20
119. 1 +
100
√
√
127. 2 y − 7 2 = 0
49
2
131.
√
√
80 − h = 45
√
a
135. √ = 24
6
139.
√
√
√
128. 8 3 = 6 w
√
√
75 = r − 12
√
√
m
136. 12 = √
360
30
132.
5
12
√
7
x+2= √
x+2
16
3
5
√
10
140. √ = 5c
5c
√
7 3
2
3
8
141. √
−2=0 9
n+7
√
√
145. 10 = 2a2 ±5 2
√
149. 7 = 13 + d2 ±6
153. 3 =
p
(m − 2)2
√
√
146. 12 = 3w2
√
150. 5 − x2 = 1
154.
5, −1
165.
√
n2 + 5n − 6
5c2 − 2c = 4
y+6=
±2
155. 4 −
162. 6 =
166. 2 +
2, − 85
√
2y + 3
170.
3
√
p
√
144. 0 = √
21
2
p
y 2 − 9y
−4
√
±3 2
(3 − n)2 = 0
12, −3
√
159. 5 = ( y − 4 )2
163.
156.
5, −20
−3
±9
√
±4 2
p
(r + 7)2 = 10
√
160. 8 = ( 5 − k )2
9
√
x2 + 15x = 10
1
1
−
1−x 2
√
148. 9 − h2 = 0
√
152. c2 + 4 = 6
√
7, −1
√
158. ( c + 14 )2 = 10
2
12
2y − 5
k 2 ±6
p
151. 4 = y 2 − 2
147. 6 =
(p + 1)2 − 1 = 0
−5,
169.
√
±4 3
p
4, −9
√
143. 3 = √
2
−2, 0
√
157. ( x + 5 )2 = 7
161. 0 =
10
142. 5 = √
2p
164. 7 −
√
3, −17
−3
a2 − 6a = 3
8, −2
2w2 + 5w = 7
167. 12 = 9 +
5
2
√
5k 2 + 4k
168. 2 =
√
3r2 − r
−1,
4
3
1, − 95
3x + 10 =
√
5x
5
171.
√
√
n + 7 = 3n − 5
6
172.
√
√
10a − 7 = 5 − 2a
1
√
√
c + 2 = 2c + 5
√
177. k = k + 2 4
173.
181. c −
185.
√
√
3c = 6
√
189. 2 x + x = 8
p
5 + 7d =
178. 6 +
√
√
186. 4 c − 4 = c
√
190. 3 h − 2 = h
4
5
194. 4 = x +
2d
Ø
√
√
175. n − 2 = 3n + 4
179. n −
9
5y
√
24
y 2 + 11
√
a=a
182. 10 = y −
12
6a − a = −12
193. y = −1 +
174.
Ø
√
183.
20
n = 12
√
2w + 4 = w
√
187. y = 3 y + 10
4
√
191. 6 = 5 p − p
1, 4
√
x2 − 8
√
3
195. c +
ALG catalog ver. 2.6 – page 277 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
16
184.
8
√
2x + 12 = x
√
188. m = 2 m − 1
25
√
192. 8 = 6 k − k
4, 9
c2 + 3 = 3
Ø
√
√
176. 11h = h − 20
√
180. 6 = h + h 4
1
196.
Ø
18
1
4, 16
√
n2 − 13 + 1 = n
7
MI
197.
201.
√
√
h + 2 − h = −10
x2 + 5 + 1 = 2x
√
198. a +
14
202. 2y =
2
33 − 4a = 3
√
1 + 9y + 6
−4
7
199. r = 5 +
203.
√
4r + 1
200. −7 +
12
√
3a + 13 − 2a = −3
√
w+9=w
204. 3c = 1 +
√
−5
7c2 − 1
Ø
4
205.
p
2+
209. 5 =
213.
√
√
p
n=3
18 −
√
206. 2 =
49
a−1
210.
Ø
p√
p√
2k − 6
50
2m + 3 + 10 = 4
√
h
214.
4
√
2k + 4 −
√
k=2
219.
221.
223.
√
√
√
√
3y + 1 − 1 =
w−5−
√
√
y+4
w + 10 = −3
√
√
3p + 8 = 3 p − 2 2
16y + 1 =
√
2y + 1 +
√
√
√
225. 2 r − r − 3 = 5 + r
227.
√
23w + 4 −
√
0,
3
2
4
√
5x − 2 − 3 w = 0
√
√
4
229. √
− a− a−4=0
a−4
√
3
√
231. √ − y = y + 1
y
9
5
212. 6 =
0,
216.
16
3
2
p√
3w + 21 = 5
p
√
√
a − 5 − 2a − 3 = 0
√
√
2n + 11 = 2 + n + 2
14, 6
7, −1
3
2
224.
√
√
√
m+8− m−7= 5
226.
√
√
√
x + 1 + x + 2 = 2x + 3
−2, −1
228.
√
√
√
4 − y + y − 9 = y − 14
5, 9
√
3
1−2 n
√
230. √ − 5 =
n
n
232. √
ALG catalog ver. 2.6 – page 278 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
12
4
9
√
√
2
= x+ x+1
x+1
1
3
30 +
√
16
3
y+4
√
√
x+1= x+4
9
4
√
√
√
222. 2c − 3 + 4c + 9 = 15
0, 8
√
6y
20
√
√
5c + 1 = 1 − 3 c
220.
6
5x − 6 = 2
215.
218. 2 +
5
208.
p√
0, 16
217.
Ø
10 +
211.
33
2
h + 12 = 2 +
√
h
p
207. 4 =
32
9
4
MJ
Topic:
Using Pythagorean Theorem I. See also categories MK and ML (word problems involving radical
equations).
Directions:
120—If a, b, and c are sides of a right triangle (c is the hypotenuse), find the missing side.
1.
a = 4, b = 3
5.
b = 5, c = 13
9.
a = 2, b = 5
c=5
a = 12
√
2.
a = 5, b = 12
6.
a = 4, c = 5
c = 13
b=3
b = 15, c = 17
a=8
4.
a = 8, b = 15
c = 17
8.
a = 6, c = 10
b=8
√
13
12. a = 5, b = 1
c=
√
c = 2 10
15. a = 1, b = 7
√
c=5 2
16. a = 2, b = 4
√
c=2 5
18. b = 1, c = 9
√
a=4 5
19. a = 1, c = 3
√
b=2 2
20. b = 1, c = 5
√
a=2 6
22. a = 3, c = 7
√
b = 2 10
23. b = 8, c = 12
√
a=4 5
24. a = 4, c = 10
√
b = 2 21
√
a=4 6
27. a = 9, c = 11
√
b = 2 10
28. b = 7, c = 11
√
a=6 2
c=
13. a = 8, b = 4
√
c=4 5
14. a = 6, b = 2
17. a = 1, c = 7
√
b=4 3
21. b = 2, c = 6
√
a=4 2
25. a = 10, c = 12
7.
c = 10
c=
10. a = 1, b = 4
26. b = 2, c = 10
√
a = 8, b = 6
11. a = 3, b = 2
29
c=
3.
17
√
26
√
b = 2 11
29. a =
33. a =
√
√
5, b = 2
11, b =
√
c=3
30. a = 3, b =
5
34. a =
c=4
√
√
6, b =
7
c=4
√
10
31. a = 7, b =
35. a =
√
√
15
3, b =
√
c=8
6
c=3
32. a =
36. a =
√
√
11, b = 5
7, b =
√
c=6
2
c=3
c=4
37. b = 3, c =
√
√
√
√
13
a=2
38. a = 2, c =
13
b=3
39. b = 1, c =
10
a=3
40. a = 3, c =
10
b=1
√
41. a = 4 5, b = 1
c=9
√
42. a = 2 6, b = 1
c=5
√
43. a = 1, b = 2 2
c=3
√
44. a = 1, b = 4 3
c=7
√
45. a = 3, b = 3 2
√
46. a = 5, b = 5 2
√
c=3 3
√
49. a = 3 5, c = 7
b=2
√
53. a = 2, b = 2 6
6, b = 2 3
62. a =
√
a=2 3
√
5, b = 5 3
√
c=4 5
√
√
2, b = 3 2
√
√
c=2 5
71. a =
√
√
60. a = 2 6, b = 2 3
c=6
√
√
7, b = 2 5
64. a =
√
6, b = 3 2
√
√
68. a = 7 2, c = 5 6
√
b = 2 13
26
√
√
√
c=2 6
√
√
67. b = 2 7, c = 3 6
a=
a=5
√
c=2 6
√
√
59. a = 2 2, b = 2 7
63. a =
√
52. b = 4 6, c = 11
√
56. a = 2, b = 2 5
√
c=3 3
√
66
70. a =
b=4
c=6
√
2, b = 4 3
√
√
66. a = 4 2, c = 3 11
b=
√
51. a = 2 5, c = 6
√
c = 2 10
√
c=5 2
√
√
65. b = 4 3, c = 2 15
√
√
√
c=2 3
√
55. a = 4, b = 2 6
c=5
√
√
c=3 2
69. a =
a=3
√
√
58. a = 2 5, b = 5
c=5
61. a =
√
50. b = 2 10, c = 7
√
c=2 7
√
√
57. a = 3 2, b = 7
√
48. a = 2 2, b = 2
√
c=4 3
√
54. a = 4, b = 2 3
√
c=2 7
√
√
47. a = 4 2, b = 4
√
c=5 3
√
5, b = 2 10
√
c=3 5
72. a =
√
√
10, b = 3 2
√
c=2 7
Special triangles (30-60-90 and 45-45-90)
73. a = 1, b = 1
c=
√
77. a = 5, c = 5 2
81. a =
√
6, b =
√
b=5
√
6
√
c=2 3
√
√
85. a = 3 2, b = 3 2
c=6
2
√
c=5 2
74. a = 5, b = 5
78. b = 1, c =
82. a =
√
√
2
10, b =
a=1
√
10
√
c=2 5
√
√
86. a = 2 2, b = 2 2
c=4
75. a = 2, b = 2
√
c=2 2
√
79. a = 6, c = 6 2
83. a =
c=
√
√
3, b =
√
b=6
3
6
√
√
87. a = 5 2, b = 5 2
c = 10
ALG catalog ver. 2.6 – page 279 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
76. a = 7, b = 7
√
c=7 2
√
80. b = 4, c = 4 2
84. a =
c=
√
5, b =
√
a=4
5
√
10
√
√
88. a = 7 2, b = 7 2
c = 14
MJ
89. b =
a=
√
√
5, c =
√
10
90. a =
b=
5
93. a = 1, b =
√
3
√
3, c =
√
91. b =
6
√
3
a=
√
c=2
√
97. a = 2 3, b = 6
√
3, b = 3
105. a =
√
√
c=2 3
√
99. a = 9, b = 3 3
√
√
b=5 3
√
6, b = 3 2
√
15, c = 2 15
√
a=3 5
c=8
102. b = 4, c = 8
106. a =
√
√
a=4 3
√
15, b = 3 5
√
c = 2 15
110. a =
√
√
6, c = 2 6
√
b=3 2
103. a = 1, c = 2
b=
√
3
√
c=4 6
√
√
111. b = 3 3, c = 6 3
ALG catalog ver. 2.6 – page 280 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
a=
√
2
√
96. a = 2 3, b = 2
c=4
√
c = 10 3
√
√
107. a = 2 6, b = 6 2
a=9
√
2, c = 2
√
100. a = 15, b = 5 3
√
c=6 3
√
c=2 6
109. b =
92. b =
6
√
95. a = 4 3, b = 4
√
c=4 3
101. a = 5, c = 10
√
√
6, c = 2 3
c=6
94. a = 3, b = 3 3
98. a =
√
104. b = 2, c = 4
√
a=2 3
√
√
108. a = 3 6, b = 9 2
√
c=6 6
√
√
112. a = 2 3, c = 4 3
b=6
MK
Topic:
Using Pythagorean Theorem II. See also categories MJ and ML (word problems involving radical
equations).
Directions:
0—(No explicit directions.)
1.
Find c.
10
2.
Find a.
5
3.
Find c.
25
4.
Find a.
15
RT-TRI03.PCX
RT-TRI04.PCX
RT-TRI01.PCX
RT-TRI02.PCX
5.
Find x.
√
5 5
6.
Find x.
√
2 7
7.
Find y.
√
2 10
8.
Find y.
√
6 2
RT-TRI07.PCX
RT-TRI08.PCX
RT-TRI05.PCX
RT-TRI06.PCX
9.
Find a.
√
4 3
10. Find c.
√
6 2
11. Find y.
2
12. Find x.
√
5 2
RT-TRI10.PCX
RT-TRI09.PCX
RT-TRI12.PCX
RT-TRI11.PCX
13. Find x.
2
14. Find y.
√
5 3
15. Find a.
√
4 3
16. Find c.
8
RT-TRI16.PCX
RT-TRI15.PCX
RT-TRI14.PCX
RT-TRI13.PCX
ALG catalog ver. 2.6 – page 281 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
MK
17. Find x.
√
2 14
18. Find x.
3
19. Find y.
3
20. Find y.
√
2 5
TSHELL01.PCX
TSHELL03.PCX
TSHELL02.PCX
TSHELL04.PCX
21. Find the length GH.
22. Find the length CD.
√
5 2
√
2 14
DIABOX01.PCX
√
5 10
√
√
√
6, 4 2 and 26 is
√
√
√
If c = 4 2, a = 6 and b = 26, then a2 + b2 = c2 .
√
√
√
3, 4 3 and 3 5. Show
√
√
√
If c = 4 3, a = 3 and b = 3 5, then a2 + b2 = c2 .
24. Find the length EF.
√
2 22
DIABOX03.PCX
DIABOX02.PCX
25. Show that a triangle with sides
a right triangle.
27. The sides of a triangle are
that the triangle is right.
23. Find the length AB.
DIABOX04.PCX
√
√
26. Show that a triangle with sides 7, 26 and 5 3 is a
right triangle. If c = 5√3, a = 7 and b = √26, then a2 + b2 = c2 .
√
√
28. The sides of a triangle are 3 3, 5 and 2 13. Show
that the triangle is right.
√
√
If c = 2 13, a = 5 and b = 3 3, then a2 + b2 = c2 .
√
29. The sides of a right triangle are 4 and 4 2. Find
the hypotenuse. 4√3
30. The sides of a right triangle are
the hypotenuse. 5√2
31. How long is the hypotenuse of a right triangle whose
sides are 5 and 15 ? 5√10
32. How long is the hypotenuse of a right triangle whose
sides are 12 and 6 ? 6√5
33. One side of a right triangle is 8 and the hypotenuse
is 12. What is the other side? 4√5
34. One side of a right triangle is 10 and the hypotenuse
is 14. What is the other side? 4√6
35. In √
a right triangle, one side is
is 30. Find the other side.
√
6 and the hypotenuse
√
2 6
√
35 and
√
15. Find
√
36. In a right triangle, one side is 2 3 and the
hypotenuse is 10. Find the other side. 2√22
37. The sides of a right triangle are a and b, and the
hypotenuse is c. Find c, if a = 12 and b = 16. 20
38. The sides of a right triangle are a and b, and the
hypotenuse is c. Find c, if a = 20 and b = 15. 25
39. The sides of a right triangle are a and b, and the
hypotenuse is c. Find b, if a = 10 and c = 26. 24
40. The sides of a right triangle are a and b, and the
hypotenuse is c. Find a, if b = 30 and c = 34. 16
41. The sides of a right triangle are√a and b, and
√ the
hypotenuse is c. Find a, if b = 6 and c = 33.
42. The sides of a right triangle are √
a and b, and the
hypotenuse is c. Find a, if b = 4 5 and c = 12. 8
√
3 3
ALG catalog ver. 2.6 – page 282 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
MK
43. The sides of a right triangle are √
a and b, and√the
hypotenuse is c. Find c, if a = 2 7 and b = 2 2.
6
44. The sides of a right triangle are√a and b, and
√ the
hypotenuse is c. Find c, if a = 5 and b = 2 10.
√
3 5
45. In right triangle ABC, the hypotenuse AB is 9.
−−−
If BC is 7, how long is AC ? 4√2
46. In right triangle ABC, the hypotenuse AB is 8.
−−−
If AC is 6, how long is BC ? 2√7
47. In triangle DEF , 6 E is right. If DE and EF are
each 7, what is DF ? 7√2
48. In triangle
DEF , 6 E is right. If DE is 5 and EF
√
is 5 3, what is DF ? 10
49. The legs of a right triangle are 7.5 cm and 12.1 cm.
Approximately how long is the hypotenuse? ≈ 14.2 cm
50. The legs of a right triangle are 10.5 in. and 5.8 in.
Approximately how long is the hypotenuse? ≈ 12 in.
51. In a right triangle, the hypotenuse is 5.25 feet. If
one leg is 3.5 feet, approximately how long is the
other leg? ≈ 3.91 ft
52. In a right triangle, the hypotenuse is 16 meters. If
one leg is 11.8 meters, approximately how long is the
other leg? ≈ 10.8 m
ALG catalog ver. 2.6 – page 283 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
ML
Topic:
Word problems involving square roots. See also category NH (quadratic formula).
Directions:
0—(No explicit directions.) 16—Solve and check.
40—Write an equation and solve.
1.
The hypotenuse of a right triangle is 15 cm long.
2.
Find the lengths of the two legs if their sum is 21 cm.
9, 12 cm
39—Translate and solve.
The hypotenuse of a right triangle is 13 in. long.
Find the lengths of the two legs if their sum is 17 in.
5, 12 in.
3.
The perimeter of a right triangle is 200 inches. The
length of the hypotenuse is 85 inches. Find the
length of each leg. 40, 75 in.
4.
The perimeter of a right triangle is 270 centimeters.
The length of the hypotenuse is 117 centimeters.
Find the length of each leg. 45, 108 cm
5.
The hypotenuse of a right triangle is 25 ft. One leg
is 17 ft longer than the other. Find the lengths of
the legs. 7, 24 ft
6.
The hypotenuse of a right triangle is 30 m. One leg
is 6 m longer than the other. Find the lengths of the
legs. 18, 24 m
7.
One leg of a right triangle is twice as long as the
other.
√ How long is each leg, if the hypotenuse
is 2 10 ? 2√2, 4√2
8.
One leg of a right triangle is three times as long as
the√other. How long is each leg, if the hypotenuse
is 30 ? √3, 3√3
9.
In a right triangle, the hypotenuse√is twice as long as
the first leg. If the second leg is 15, how long is
the hypotenuse? 2√5
10. In a right triangle, the hypotenuse√is twice as long as
the first leg. If the second leg is 6, how long is the
hypotenuse? 2√2
11. The two legs in a right triangle are the same
√ length.
How long is each leg if the hypotenuse is 22? √11
12. The two legs in a right triangle are the same
√ length.
How long is each leg if the hypotenuse is 6 3? 3√6
13. In a right triangle, the long leg is 2 more than twice
the short leg, and the hypotenuse is 8 more than the
short leg. Find the short leg. 5
14. In a right triangle, the long leg is 1 less than twice
the short leg, and the hypotenuse is 9 more than the
short leg. Find the short leg. 8
15. In a right triangle, the hypotenuse is 2 less than
twice the short leg, and the long leg is 2 more than
the short leg. Find the short leg. 6
16. In a right triangle, the hypotenuse is 1 more than
twice the short leg, and the long leg is 1 more than
the short leg. Find the short leg. 3
17. In a square whose side is 8, what is the length of
each diagonal? 4√2
√
19. In a square whose side is 7 2, what is the length of
each diagonal? 14
18. In a square whose side is 10, what is the length of
each diagonal? 5√2
√
20. In a square whose side is 3 2, what is the length of
each diagonal? 6
21. A rectangle has dimensions 9 × 12. What is the
length of each diagonal? 15
22. A rectangle has dimensions 8 × 15. What is the
length of each diagonal? 17
23. A rectangle has dimensions 2 × 8. What is the length
of each diagonal? 2√17
24. A rectangle has dimensions 5 × 10. What is the
length of each diagonal? 5√5
25. The side of a square is 8.6 in. What is the length of
a diagonal? (Round to nearest tenth.) 12.2 in.
26. The side of a square is 11.2 cm. What is the length
of a diagonal? (Round to nearest tenth.) 15.8 in.
27. The length of a rectangle is 6 ft and the width
is 4.6 ft. What is the length of a diagonal? (Round
to the nearest hundredth.) 7.56 ft
28. The length of a rectangle is 1.9 m and the width
is 0.9 m. What is the length of a diagonal? (Round
to the nearest hundredth.) 2.10 ft
29. One dimension of a rectangle is 8. Find the other
dimension, if the diagonal is 12. 4√5
30. One dimension of a rectangle is 6. Find the other
dimension, if the diagonal is 9. 2√5
31. In a rectangle, each diagonal is 13. If the width of
the rectangle is 5, what is the length? 12
32. In a rectangle, each diagonal is 10. If the length of
the rectangle is 8, what is the width? 6
33. How long should the side of a square be so that it
has the same area as a 5 × 8 rectangle? 2√10
34. How long should the side of a square be so that it
has the same area as a 6 × 9 rectangle? 3√13
35. The base and altitude in a triangle are equal. The
area of the triangle is 100 cm2 . Find the lengths of
the base and altitude. 10√2
36. The base and altitude in a triangle are equal. The
area of the triangle is 64 in2 . Find the lengths of the
base and altitude. 8√2
ALG catalog ver. 2.6 – page 284 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
ML
37. In a right triangle, the hypotenuse is 10 and the two
legs are the same length. What is the area of the
triangle? 25
39. In a√right triangle, the hypotenuse is 6 and one leg
is 3 2. What is the area of the triangle? 9
38. In a right triangle, the hypotenuse is 16 and the two
legs are the same length. What is the area of the
triangle? 64
√
40. In a right triangle, the hypotenuse is 2 13 and one
leg is 4. What is the area of the triangle? 12
41. Each side of an equilateral triangle is 10 ft. Find the
length of an altitude in the triangle. 5√3
42. Each side of an equilateral triangle is 14 m. Find the
length of an altitude in the triangle. 7√3
43. Find the area
√ of an equilateral triangle whose sides
are each 4 3. 6√3
44. Find the area
√ of an equilateral triangle whose sides
are each 8 3. 24√3
45. The dimensions of a room are 7.5 × 8 × 6 m. Find
the length of a wire that stretches from a corner of
the floor to the far corner of the ceiling. 12.5 m
46. The dimensions of a garage are 9 × 12 × 8 ft. Find
the length of a wire that stretches from a corner of
the floor to the far corner of the ceiling. 17 ft
47. The dimensions of a box are 4 × 5 × 6 in. What is
the length of a diagonal? (Round answer to nearest
tenth). 8.8 in.
48. The dimensions of a box are 5 × 10 × 20 cm. What is
the length of a diagonal? (Round answer to nearest
tenth). 23.0 cm
49. Cornwall is approximately 36 km south of Alexandria
and 48 km east of Chesterville. What is the
straight-line distance between Alexandria and
Chesterville? ≈ 60 km
50. Porterville is approximately 48 mi north of Bakersfield
and 20 mi east of Tipton. What is the straight-line
distance between Bakersfield and Tipton? ≈ 52 mi
51. Cities A and B are joined by a direct highway. They
are also connected by a highway that runs 70 mi east
from City A and then 24 mi north to City B. How
much shorter is the direct route than the east-north
route? 20 mi
52. Cities A and B are joined by a direct highway. They
are also connected by a highway that runs 21 km
south from City A and then 72 km west to City B.
How much shorter is the direct route than the
east-north route? 18 km
53. The diagonals of two different rectangles are 65 ft.
The width of one rectangle is 25 ft and the width of
the other rectangle is 39 ft. How much longer is one
rectangle than the other rectangle? 8 ft
54. The diagonals of two different rectangles are 75 m.
The width of one rectangle is 45 m and the width of
the other rectangle is 27 m. How much longer is one
rectangle than the other rectangle? 12 m
DIAREC1.PCX
DIAREC1.PCX
56. The diagonals of two different rectangles are 85 in.
The length of one rectangle is 77 in. and the length
of the other rectangle is 84 in. How much wider is
one rectangle than the other rectangle? 23 in.
55. The diagonals of two different rectangles are 100 cm.
The length of one rectangle is 96 cm and the length
of the other rectangle is 80 cm. How much wider is
one rectangle than the other rectangle? 32 cm
DIAREC2.PCX
DIAREC2.PCX
ALG catalog ver. 2.6 – page 285 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
ML
57. A wire runs from the top of a 12 m flagpole to a
point on the ground that is 9 m from the base of the
pole. How long is the wire? 15 m
58. A wire runs from the top of a 30 foot flagpole to a
point on the ground that is x feet from the base of
the pole. If the wire is 34 feet long, what is x ? 16
FLGPOL1.PCX
FLGPOL1.PCX
59. A wire runs from the top of a 6 meter telephone
pole to a point on the ground that is x meters from
the base of the pole. If the wire is 6.25 meters long,
what is x ? 1.75
60. A wire runs from the top of an 18 ft telephone pole
to a point on the ground that is 7.5 ft from the base
of the pole. How long is the wire? 19.5 ft
TELPOL1.PCX
TELPOL1.PCX
61. One end of a ramp is raised to the back of a truck,
3 feet above the ground (see figure). The other end
rests on the ground, 9 feet behind the truck. What
is the approximate length of the ramp? ≈ 9.5 ft
62. One end of a ramp is raised to the back of a truck,
1.5 meters above the ground (see figure). If the ramp
is 4 meters long, approximately how far behind the
truck is the other end of the ramp? ≈ 3.7 m
RAMPTR1.PCX
RAMPTR1.PCX
63. A ladder leans against a second-story window, as
shown in the figure. If the ladder is 5 meters long,
and the base of the ladder is 1.75 meters from the
house, how high is the window? ≈ 4.7 m
64. A ladder leans against a second-story window, as
shown in the figure. If the window is 16.5 feet above
the ground, and the base of the ladder is 4 feet from
the house, how long is the ladder? ≈ 17 ft
LADDER1.PCX
LADDER1.PCX
ALG catalog ver. 2.6 – page 286 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
ML
65. A traffic light is suspended between two poles, as
shown in the figure. The poles are 13 m apart and
8 m tall. If 14 m of wire is used to suspend the light,
approximately how far above the street is it? ≈ 7.4 m
66. A traffic light is suspended between two poles, as
shown in the figure. The poles are 38 ft apart and
25 ft tall. If 40 ft of wire is used to suspend the light,
approximately how far above the street is it?
≈ 18.8 ft
TLIGHT1.PCX
TLIGHT1.PCX
68. The roof of a house is supported by two angled
beams, as shown in the figure. Each beam is
8 m long. If the house is 15 m wide and 6 m tall
(measured to the top of the roof), what is the
approximate ceiling height? ≈ 3.2 m
67. The roof of a house is supported by two angled
beams, as shown in the figure. Each beam is
34 ft long. If the house is 65 ft wide and 25 ft tall
(measured to the top of the roof), what is the
approximate ceiling height? ≈ 15 ft
ROOFBM1.PCX
ROOFBM1.PCX
69. A 6 12 meter ladder leans against the side of a
building, as shown in the figure. The distance x is
3 12 meters less than the distance y, which is the
highest point reached by the ladder. Find x. 2.5 m
70. A 12 12 foot ladder leans against the side of a building
(refer to the previous figure). The distance x is
8 12 feet less than the distance y, which is the highest
point reached by the ladder. Find x. 3.5 ft
LADDER3.PCX
LADDER3.PCX
71. One end of a cable is attached to the top of an
antenna, as shown in the figure. The other end
is staked into the ground a distance d from the
bottom of the antenna. If the height of the antenna
is 12 meter less than the length of the cable, and
d = 4 meters, find the length of the cable. 16.25 m
72. One end of a cable is attached to the top of an
antenna (refer to the previous figure). The other
end is staked into the ground a distance d from the
bottom of the antenna. If the height of the antenna
is 1 12 meters less than the length of the cable, and
d = 6 meters, find the length of the cable. 12.75 m
ANTENNA1.PCX
ALG catalog ver. 2.6 – page 287 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
ANTENNA1.PCX
ML
73. Town C is south of Town A and east of Town B.
The distance from A to C is 12 km. The distance
from A to B is 4 km more than the distance from C
to B. How far apart are Towns A and B ? 20 km
ABC-TRI5.PCX
75. Cloverdale is north of Bridgeport and 15 miles west
of Albertsville. The distance between Cloverdale and
Bridgeport is 9 miles less than the distance between
Bridgeport and Albertsville. What is the distance
between Bridgeport and Albertsville? 17 mi
74. Town C is south of Town B and west of Town A.
The distance from A to C is 24 miles. The distance
from B to C is 16 miles less than the distance from
B to A. How far apart are Towns B and A ? 26 mi
ABC-TRI6.PCX
76. Egleston is north of Franklin and 20 miles east of
Grantsville. The distance between Franklin and
Grantsville is 10 km more than the distance between
Franklin and Egleston. What is the distance between
Franklin and Egleston? 15 km
Time, distance, rate
77. A ship navigated a course due west for 26 miles, and
then went a certain distance to the north. The ship
ended up 52 miles from where it started. How far
did it go to the north? ≈ 45 mi
78. An airplane flew a route to the north for 240 km, and
then went a certain distance to east. The airplane
ended up 410 km from where it started. How far did
it fly to the east? ≈ 332 km
79. At noon, a train leaves the station and travels
south at 35 mph. Two hours later it turns east
and increases its speed to 40 mph. At 5:30 that
afternoon, approximately how far is the train from
the station? ≈ 156.5 mi
80. A ship leaves port at 1:30 pm and sails west
at 20 km/hr. At 4 pm, it begins sailing north at a
rate of only 12 km/hr. Approximately how far is the
ship from its port at 7 pm? ≈ 61.6 km
81. A boy and a girl walk apart from each other, starting
at the same time and place. The boy walks north at
3 mph, and the girl walks east at 4 mph. How far
apart are they after 2 hours? 10 miles
82. Two cars pull out of an intersection at the same
time. One goes west at 24 km/hr and the the other
goes south at 18 km/hr. How far apart are the cars
after 10 minutes? 5 km
83. Two boats speed apart from each other, one going
north at 24 mph, and the other going west at 18 mph.
After how many hours will they be 75 miles apart?
84. Two cars leave a highway intersection at the same
time. One heads east at 60 kph and the other heads
north at 80 kph. After how many hours will they be
120 km apart? 1.2 hrs
2.5 hrs
85. It takes Paul 10 minutes to walk halfway around a
city block. The block is 500 m long and 375 m wide.
If Paul could walk straight from one corner to the
opposite corner, approximately how long would it
take? ≈ 7.1 min
86. It takes Jeanine 5 minutes to walk halfway around
the school playground. The dimensions of the
playground are 600 × 400 ft. If Jeanine could walk
straight from one corner to the opposite corner,
approximately how long would it take? ≈ 3.6 min
87. Nathan can run around the perimeter of the park in
12 minutes. If the park is square, how long should
it take him to run from one corner to the opposite
corner? ≈ 8.5 min
88. It takes Ms. Adams 14 minutes to jog from one
corner of the park to the opposite corner. If the park
is square, how long should it take her to run around
the perimeter? ≈ 39.6 min
89. Two motorists left the gas station at the same time.
One drove east on the highway, and the other south
on the expressway. The driver on the highway went
10 km/hr faster than the other driver. After one
hour they were 100 km apart. Find the speed of each
driver. 60, 80 km/hr
90. Two motorists left a highway toll plaza at the same
time. One continued west on the highway. The other
got on a backroad going north. The driver going
north went 28 mph slower than the other driver.
After one hour they were 68 mi apart. Find the
speed of each driver. 32, 60 mph
91. A car starts traveling due east along a road. At the
same time and place, another car starts traveling
north at a speed that is 8 mph faster than that of
the first car. After 2 hours, the cars are 80 mi apart.
At what speeds are they traveling? 24, 32 mph
92. From the corner of A Street and Ninth Avenue, Ken
rides his bike north and Karen rides her bike east.
Karen goes 3 km/hr faster than Ken. After 2 hours,
they are 30 km apart. How fast each going?
9, 12 km/hr
ALG catalog ver. 2.6 – page 288 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
NA
Topic:
Perfect square trinomials.
Directions:
78—Find the missing term that makes the trinomial a perfect square.
1.
n2 + ( ? ) + 9
6n
2.
64 − ( ? ) + c2
5.
1 − ( ? ) + x2
2x
6.
w2 + ( ? ) + 16
9.
r4 − ( ? ) + 121
22r2
13. 16 + ( ? ) + 49y 2
17. m4 + ( ? ) + n2
56y
2m2 n
16c
3.
a2 − ( ? ) + 81
18a
4.
4 + ( ? ) + y2
8w
7.
36 + ( ? ) + h2
12h
8.
n2 − ( ? ) + 25
10. 100 + ( ? ) + k 6
14. 144u2 − ( ? ) + 9
18. a2 − ( ? ) + b6
11. c10 + ( ? ) + 144
20k 3
24c5
15. 25 − ( ? ) + 121a2
72u
110a
19. w4 y 8 + ( ? ) + z 2
2ab3
22. k 4 + ( ? ) + 81n2
6xy
18k 2 n
10n
12. 169 − ( ? ) + z 8
26z 4
16. 4w2 + ( ? ) + 81
36w
20. k 2 + ( ? ) + m10 n6
2w2 y 4 z
21. 9x2 − ( ? ) + y 2
4y
2km5 n3
23. 121a2 − ( ? ) + y 8
24. u2 + ( ? ) + 64x4
16ux2
22ay 4
25. 25a6 − ( ? ) + 4b2
26. 9w2 + ( ? ) + 49z 4
20a3 b
27. 144a2 − ( ? ) + 25b2
120ab
42wz 2
29. c2 + 8c + ( ? )
30. r2 − 18r + ( ? )
16
33. ( ? ) − 70a + 49
180x2 y 5
31. m2 − 2m + ( ? )
81
34. ( ? ) + 60y + 9y 2
100
35. ( ? ) − 2k + 1
49
38. 4p2 + 12p + ( ? )
9
39. 4y 2 − 44y + ( ? )
41. ( ? ) + 12ax + 4x2
9a2
42. ( ? ) − 2by + y 2
k2
43. ( ? ) − 4pr + r2
b2
32. u2 + 24u + ( ? )
1
25a2
37. 9x2 − 42x + ( ? )
28. 81x4 + ( ? ) + 100y 10
121
144
36. ( ? ) + 20m + 25
4m2
40. 9k 2 + 78k + ( ? )
169
44. ( ? ) + 28a2 b + 4b2
4p2
49a4
45. a2 b2 + 4ab + ( ? )
46. 25c4 − 20c2 y + ( ? )
4
47. 4p4 q 2 − 4p2 q + ( ? )
1
48. 81r10 + 126r5 y 3 + ( ? )
4y 2
49. ( ? ) − 140x2 y + 100y 2
50. ( ? ) + 72a2 c2 + 16c4
49x4
51. ( ? ) − 30h3 k + k 2
81a4
53. 49k 2 + 112km3 + ( ? )
100c4 y 2
61. a2 + a + ( ? )
1
4
1
65. ( ? ) − r + r2
2
1
16
69. 4w2 + 3w + ( ? )
1
4
1
1
77. ( ? ) + x + x2
3
16
4
9
59. ( ? ) − 182rp3 w + 169r2
60. ( ? ) + 260knp4 + 100n2
169k 2 p8
62. y 2 − 3y + ( ? )
9
4
63. ( ? ) + 7z + z 2
1
66. ( ? ) + c + c2
3
1
36
4
67. x2 − x + ( ? )
5
4
25
3
68. m2 + m + ( ? )
4
9
64
71. 9b2 − 2b + ( ? )
1
9
72. ( ? ) + 4x + 16x2
1
4
74. c2 − ( ? ) +
a
81y 6
49p6 w2
70. ( ? ) − 10m + 16m2
9
16
56. 100x4 − 180x2 y 3 + ( ? )
36c2
58. ( ? ) − 240c2 dy + 144d2
121a2 x6
144u8
55. 49a6 − 84a3 c + ( ? )
25z 4
57. ( ? ) + 132ab2 x3 + 36b4
52. ( ? ) + 120u4 + 25
225h6
54. 64y 4 + 80y 2 z 2 + ( ? )
64m6
73. a2 + ( ? ) +
49y 6
78.
4
25
4 4
− p + (?)
9 5
25
16
64. ( ? ) − 15t + t2
49
4
4x
76.
9 2
y + (?) + 4
16
4
16
79. ( ? ) − b +
5
25
1 2
4b
80.
3
1 2
a + a + (?)
4
10
4
5c
75. 36x2 − ( ? ) +
9 2
25 p
1
9
225
4
3y
9
100
81. w2 + ( ? ) + 0.64
1.6w
82. 0.01 − ( ? ) + x2
0.2x
83. 0.36 + ( ? ) + c2
1.2c
84. z 2 − ( ? ) + 0.09
0.6z
85. a2 + 1.4a + ( ? )
0.49
86. ( ? ) − 0.8n + n2
0.16
87. ( ? ) + 2.6p + p2
1.69
88. r2 − 2.2r + ( ? )
1.21
89. 0.25y 2 − ( ? ) + 100
10y
90. 0.04 + ( ? ) + 25u2
2u
91. 0.09 − ( ? ) + 0.16a2
0.24a
ALG catalog ver. 2.6 – page 289 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
92. 1.44b2 + ( ? ) + 0.04
0.48b
NB
Topic:
Solving equations with perfect squares. See also categories JB–JD (factorable equations).
Directions:
15—Solve.
1.
x2 = 16
y2
±4
− 49
5.
0=
9.
−5h2 = −20
±7
17. y 2 =
9
16
±
Ø
3
4
8
7
±
45.
49.
k2
5
w2
4
=3
−
14. −36 = c2
Ø
100
= k2
81
±
2
7
26.
√
±5 3
√
± 15
46.
√
24
=0
64
6
2
±
50.
√
√
3
±
2
±3
62. −3 =
73. −
c2
+ 41 = 37
5
√
±2 5
77. (p + 4)2 = 36
2, −10
81. 100 = (x − 15)2
5, 25
1
4
3 1
,
2 2
89. (r + 2)2 = 6
−2 ±
85. (a − 1)2 =
66. 3d2 + 22 = 52
70. 13 = 10 − x2
Ø
90. 15 = (m − 7)2
5
3
28.
√
± 23
97. 3(b + 5)2 = 75
√
6 ± 2 14
0, −10
25
49
±
√
± 10
Ø
√
±4 3
√
5
±
3
36. 72 = w2
√
± 15
√
±6 2
√
± 3
40. 0 = 42 − 14t2
44. 4n2 = −64
48. 15 =
x2
6
Ø
√
±3 10
52. 0 = 10a2 −
√
4
5
±7
±6
63. −5 + 3a2 = −2
√
± 10
67. 6w2 − 9 = 33
√
± 7
68. 8k 2 + 17 = 41
√
± 3
71. 25 = 20 − 5b2
Ø
72. 29 − 4y 2 = 31
Ø
Ø
75. 85 =
14, −8
−9, −15
3
9
− ,−
2
2
7±
√
15
√
2
5
±
56. −40 + k 2 = 9
±5
5
7
4
7
±
32. 0 = u2 − 15
√
±3 5
Ø
±0.1
48
= 3a2
49
±2 3
1 2
n + 10
2
79. (b − 8)2 = 64
√
±5 6
16, 0
83. 25 = (w + 8)2
87.
±1
−3, −13
1
= (r − 2)2
9
7 5
,
3 3
91. 2 = (3 − h)2
3±
60. 73 = u2 + 10
94. 18 = (k + 5)2
95. (a + 15)2 = 44
√
−5 ± 3 2
√
−15 ± 2 11
98. −6(1 − y)2 = −6
0, −2
99. 80 = 5(c − 2)2
80. 81 = (u + 9)2
84. (y − 11)2 = 49
16
= (d + 1)2
25
√
±3 2
0, −18
4, 18
1
9
− ,−
5
5
92. (x + 12)2 = 11
√
11
96. (d − 9)2 = 28
6, −2
±10
76. 63 = 10x2 − 117
88.
√
2
√
±3 7
1
64. 76 − m2 = 51
4
−12 ±
93. 56 = (f − 6)2
±1
59. w2 + 9 = 21
78. 121 = (x − 3)2
9
4
±
2 2
m − 32
3
1
1
= z2
9
5
±11
√
±4 2
√
±2 2
82. (a + 12)2 = 9
24. 0.01 = r2
55. 20 = h2 − 5
74. −25 + 2x2 = −9
86. (t + 3)2 =
√
6
51.
±11
k2
− 15
3
±0.8
c2
5
− =0
5
9
47. 0 =
±9
− 121 = 0
20. 0 = u2 −
43. 100 + 10x2 = 0
√
± 6
8.
k2
1
2
±
39. −60 = −6c2
Ø
a2 = 81
16. 0 = −18 − w2
Ø
1
− p2 = 0
4
4.
12. 7 = 7w2
±3
35. −45 + y 2 = 0
√
±4 5
61. 23 = 2r2 + 5
±6
31. −x2 = −23
√
±2 6
58. 17 = −15 + p2
69. y 2 + 32 = 10
27.
√
± 6
±2 11
√
± 2
− t2
23. x2 = 0.64
12
5
±
57. x2 − 16 = 28
65. 7 − 9x2 = −11
19.
±1.5
54. b2 + 11 = 132
±4
0 = 36
±12
11. 4r2 − 36 = 0
10
9
3 2
a −9=0
2
9
3 2
x =
2
8
7.
144 = r2
15. a2 = −12
34. 0 = 24 − p2
38. 400 = 5s2
±15
±4
42. 0 = 7a2 + 42
Ø
53. 20 = n2 + 4
24
1 2
x =
6
25
30. 6 − p2 = 0
37. 2k 2 − 150 = 0
41. −32 = 8y 2
+ 225 = 0
22. t2 − 2.25 = 0
√
±2 10
33. −d2 = −40
6.
−y 2
3.
±3
±1.1
√
± 11
29. 11 = m2
w2 = 9
18.
21. 0 = −n2 + 1.21
25. 0 = 14z 2 −
2.
26—Solve by any method.
10. 0 = 3y 2 − 48
±2
13. 64 + n2 = 0
23—Solve by factoring
√
9±2 7
100. −63 = −7(x + 8)2
−5, −11
101. 3(r + 2)2 =
1
3
5
7
− ,−
3
3
102.
9
= 2(n − 3)2
2
9 3
,
2 2
103. 15(a + 1)2 =
20
3
104.
24
= 6(y − 2)2
49
16 12
,
7 7
1
5
− ,−
3
3
105. −38 = (3 − x)2
Ø
106. −6(y − 4)2 = 24
Ø
107. −3(n − 2)2 = 18
ALG catalog ver. 2.6 – page 290 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
Ø
108. −68 = 2(15 − x)2
Ø
NB
109. −4(h − 2)2 = −40
2±
√
110. 7(x + 5)2 = 77
−5 ±
10
113. 30 = (r − 1)2 + 5
√
4 ± 14
11
5±
6
121. 14 = (b − 7)2 + 18
20, −4
118. −12 = 19 − (5 − k)2
Ø
√
−3 ±
122. (x + 10)2 + 23 = 13
125. 47 = (4 + y)2 − 5
−4 ± 2 13
129.
1
(1 − w)2 + 7 = 10
3
2±
√
14
134.
−7 ±
137. 24 + 3(w − 12)2 = 120
√
√
135. 9(r + 9)2 − 22 = 23
−9 ±
10
138. 6 − 2(k + 7)2 = −18
1
7
− ,−
2
2
142. 45 = 25(b − 3)2 − 4
−
146. 25(r − 3)2 − 28 = −11
√
3±
149. 50 = 45 − 4(3 − x)2
157. (2x − 7)2 = 36
17
5
147. 4(x − 8)2 − 7 = 15
8±
150. (5 − p)2 + 17 = −7
Ø
154. 8(a + 5)2 + 9 = 63
158. 25 = (3w − 2)2
2
136. 24 = 30 − (2 − n)2
7
√
21
140. 58 = 4(y + 6)2 − 38
144. 1 − 9(x + 1)2 = −24
148. 15 = 7 + 36(c + 2)2
√
22
2
−2 ±
151. 2(x − 11)2 + 30 = 19
√
4 6
−5 ±
3
−1,
7
3
2
3
152. −24 = (a + 4)2 − 12
Ø
155. 35 = 3(y + 5)2 + 3
√
3 3
−5 ±
2
1 13
,
2 2
0, 20
2
8
,−
3
3
Ø
√
2 5
11 ±
3
132. 8(x − 10)2 − 650 = 150
√
Ø
153. 33 = 9(d − 11)2 + 13
128. 23 = 55 − (7 − k)2
√
−6 ± 2 6
25
15
,−
4
4
√
5
9
124. 40 = 25 − 5(x + 3)2
2±
143. 16(a + 5)2 − 5 = 20
12 8
,
5 5
145. 10 = 81(h + 7)2 + 5
5
√
4 ± 2 19
−7 ± 2 3
141. 13 = 4(m + 2)2 + 4
√
139. 65 = 8(c − 4)2 − 11
√
12 ± 4 2
−7 ±
2, −14
3
(p + 7)2 − 7 = 8
2
√
21
√
7±4 2
3
(m + 6)2 − 11
4
131. 37 =
13, 3
4, −2
133. 84 = 5(y − 2)2 + 14
√
−7 ± 3 11
130. 65 = 15 + 2(a − 8)2
120. 35 = 14 + (d + 6)2
Ø
127. 125 = 26 + (c + 7)2
√
3±2 5
116. 85 − (3 − x)2 = 60
−6 ±
3
Ø
126. 5 − (3 − y)2 = −15
√
√
123. 12 − 7(5 − u)2 = 40
Ø
√
7
8, −2
119. 7 − (p + 3)2 = 4
31
112. 35 = 5(y + 1)2
−1 ±
115. 12 + (m − 8)2 = 156
3, −9
117. (h − 9)2 + 5 = 11
9±
111. −28 = −2(a − 4)2
114. 28 = (w + 3)2 − 8
6, −4
√
√
156. 18 − 7(p − 1)2 = 10
1±
159. 144 = (5b + 11)2
√
2 2
7
160. (4k − 9)2 = 49
4,
1
2
23
1
,−
5
5
161. (2m + 15)2 = 11
√
−15 ± 11
2
165. 4(7x − 4)2 = 48
√
4±2 3
7
169. 15 = (3x − 4)2 + 5
√
4 ± 10
3
173. 80 − 9(2x + 15)2 = 35
√
−15 ± 5
2
162. (4a − 9)2 = 68
163. 5 = (6x − 2)2
√
9 ± 2 17
4
166. 30 = 5(2u + 7)2
√
−7 ± 6
2
170. −8 = (12c − 3)2 − 20
√
3±2 3
12
174. 4(5b − 1)2 + 3 = 15
√
1± 3
5
√
2± 5
6
167. 4(10h + 1)2 = 180
√
−1 ± 3 5
10
171. 25 − (5k + 2)2 = 10
√
−2 ± 15
5
175. 29 = 5(12y + 8)2 − 11
√
−4 ± 2
6
ALG catalog ver. 2.6 – page 291 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
164. 45 = (7n + 6)2
√
−6 ± 3 5
7
168. 35 = 7(3p − 10)2
10 ±
3
√
5
172. −11 − (7r + 8)2 = −59
√
−8 ± 4 3
7
176. 20 = 2(3a − 7)2 + 8
√
7± 6
3
NC
Topic:
Solving equations by completing the square.
Directions:
24—Solve by completing the square.
1.
m2 + 2m = 15
3, −5
2.
33 = a2 − 8a
15—Solve.
3.
11, −3
26—Solve by any method.
−28 = c2 + 16c
4.
x2 − 22x = −40
8.
0 = y 2 + 10y − 24
20, 2
−14, −2
5.
0 = a2 − 4a − 77
6.
r2 + 6r − 27 = 0
3, −9
7.
x2 − 2x − 35 = 0
7, −5
−7, 11
9.
2, −12
u2 − 10u = −25
10. r2 + 6r = −9
5
13. w2 + 18w + 80 = 0
14. 0 = k 2 + 20k − 44
−8, −10
22. −4 = n2 + 5n
6, −5
25. 44 = x2 + 7x
4, −11
23. r2 − 3r = 10
−1, −4
26. a2 − 9a = 36
4, 20
19. −18 = r2 − 8r
Ø
20. w2 − 10w + 26 = 0
Ø
24. 6 = p2 + p
5, −2
27. −30 = k 2 + 11k
12, −3
−2 ±
√
15
30. x2 − 8x = −10
4±
√
6
−3 ±
33. u2 − 12u = −8
34. p2 + 18p = 9
√
2±2 7
Ø
41. y 2 − 16y + 16 = 0
38. y 2 + 12y = −56
−4 ±
8±4 3
45. d2 + 5d = 8
√
−5 ± 57
2
49. w2 − 9w − 6 = 0
√
√
−11 ± 101
21
3±
46. −1 = r2 − 3r
√
13
2
√
−7 ± 41
2
36. g 2 − 2g = 44
√
1±3 5
40. v 2 + 22 = 4v
Ø
44. a2 − 14a − 3 = 0
√
7 ± 2 13
51. k 2 + 9k + 3 = 0
52. 0 = g 2 − 5g − 12
√
−7 ± 13
2
5±
−4,
58. 0 = 3y 2 − 2y − 1
1
2
61. 4h2 + 16h = −20
65. 6k 2 − k − 1 = 0
1, −
Ø
1
1
,−
2
3
69. 2x2 − 3x − 3 = 0
−1, −
62. 3z 2 + 12z + 15 = 0
66. 4r2 + 4r = 1
−
1
2
56. 0 = 6x2 + 6x − 72
Ø
2
1
,−
3
2
74. 0 = 6y 2 − 3y + 1
5
1 2 2
d − d=−
15
3
3
5
78.
Ø
8, −2
75. −3 = 8t2 − 6t
79.
1 2 2
p − p = 10
6
3
2
3
64. 0 = 5h2 − 10h + 20
68. 0 = 9y 2 − 6y + 1
−5 ±
4
Ø
1
3
10, −6
√
5
76. 0 = 9c2 − 6c + 5
80.
Ø
1 2
12
h −h+
=0
10
5
6, 4
ALG catalog ver. 2.6 – page 292 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
3, −
72. 4r2 + 10r = −5
√
5 ± 17
4
Ø
1 2 3
a − a=2
8
4
63. −6 = 2x2 + 4x
71. 2a2 − 5a + 1 = 0
√
−1 ± 13
6
73. 10m2 − 5m = −2
77.
60. 6 = 3p2 − 7p
5
2
67. 6x2 − x = 2
70. 1 = 3m2 + m
√
3 ± 33
4
Ø
73
3, −4
59. −5 = 2w2 + 7w
1
3
√
2
4, −1
57. 2m2 + 7m − 4 = 0
√
−1 ± 33
2
48. 8 = p2 + p
55. 4k 2 − 12k − 16 = 0
7, −2
√
11
47. k 2 − 7k = −9
√
−9 ± 69
2
54. 2y 2 − 10y = 28
−1
Ø
43. 0 = g 2 + 22g + 20
50. 0 = z 2 + 7z + 2
√
9 ± 105
2
53. 3a2 + 6a = −3
39. 0 = s2 + 10s + 30
Ø
42. 0 = k 2 + 8k − 5
√
5±
2
√
12 ± 2 26
−9 ± 3 10
37. x2 − 8x + 22 = 0
32. t2 − 10t = −14
35. w2 − 24w = −40
√
2, −3
−7, −8
31. m2 + 6m = −7
√
Ø
28. v 2 + 15v = −56
−5, −6
29. h2 + 4h = 11
2
16. y 2 − 24y + 80 = 0
1, −15
18. 0 = n2 − 4n + 20
Ø
12. −4 = k 2 − 4k
−6
15. 0 = m2 + 14m − 15
−22, 2
17. p2 + 6p = −12
21. y 2 − y = 30
11. −36 = t2 + 12t
−3
Ø
NC
3
7
81. 0 = b2 − b −
4
4
1, −
7
4
82.
1 2 7
1
x + x=−
2
6
3
5
2
83. 0 = d2 − d +
3
3
1,
2
3
84.
3
5
= t2 − t
2
4
2, −
3
4
1
− , −2
3
2
7
85. r2 − r =
5
25
89.
√
1±2 2
5
√
3 ± 14
11
3 2
9
x − 6x − = 0
2
2
2±
√
93. h2 + 1.8h = −0.32
−0.2, −1.6
1.7 ±
√
−10, 0.2
91.
5 2
25
a − 20a =
2
2
√
21
95. w2 − 1.4w = −0.24
99. v 2 + 2v − 33.4 = 0
−1 ±
102. t2 − 2.8t − 5 = 0
6.96
106. g 2 + 4.6g = 54
√
−2 ± 11
9
−1.2, −0.2
30.9
√
4
7
87. y 2 + y =
9
81
4±
6
98. w2 − 6w − 21.9 = 0
1.4 ±
8.89
105. z 2 + 9.8z = 2
√
94. r2 + 2.6r = −1.05
3±
119.8
101. b2 − 3.4b − 6 = 0
√
4 2
y + 8y + 4 = 0
3
−0.5, −2.1
97. n2 + 20n − 19.8 = 0
−10 ±
90.
−3 ±
7
√
6
5
x−
=0
11
121
86. x2 −
√
34.4
103. h2 − 1.8h = −0.65
1.3, 0.5
−10, 5.4
107. 0.2x2 + 3.1x = 28.6
6.5, −22
ALG catalog ver. 2.6 – page 293 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
4
1
88. 0 = p2 + p +
7
49
−2 ±
7
92.
√
3
3 2
21
m + 6m = −
5
5
√
−5 ± 3 2
96. k 2 − 3.2k = −1.12
2.8, 0.4
100. y 2 − 10y − 8.7 = 0
5±
√
33.7
104. n2 + 2.6n = −1.68
−1.2, −1.4
108. 0.2r2 + 3.7r = 1.9
0.5, −19
ND
Topic:
Solving equations with the quadratic formula.
Directions:
25—Solve using the quadratic formula.
1.
x2 − 4x + 3 = 0
5.
0 = 2w2 − 3w + 1
3, 1
1,
1
2
2.
t2 + 5t + 6 = 0
6.
0 = 2x2 + 3x − 5
1, −
9.
3y 2 − 27 = 0
13. 0 = 5r2 − 20r
−2, −3
17. 3b2 − 14b + 15 = 0
0, −3
3
−1, −
7
21. 5x2 = 13x − 12
25. 21 − 2v = 3v 2
−3,
7
3
26. 5c2 = 4 + 8c
29. 3x = 10 − 4x2
−2,
5
4
30. 6 − 6d2 = 5d
33. x2 + 25 = 10x
5
37. 9z 2 + 1 = −6z
−
1
3
38. 9 + 16a2 = 24a
−1,
1
3
42. 15y 2 − 1 = 2y
41. 3x2 = 1 − 2x
2±
√
1±
√
3±
15
53. 2h2 − 6h + 1 = 0
2
3
,−
3
2
−7
3
4
1
1
,−
3
5
65. 0 = 3r2 + 5r − 1
Ø
35. v 2 + 16 = 8v
43. 5r − 3 = 2r2
−3, −
1
2
1
1
,−
3
5
√
11
−
3
7
3
2
Ø
28. 11z − 12 = 2z 2
4,
32. 3 − k = 4k 2
3
4
−1,
40. 4b = 4b2 + 1
−10
1
2
1 3
,
2 2
52. −v 2 + 8v − 10 = 0
4±
√
6
56. 0 = 5t2 − 2t − 4
1±
√
21
5
59. 0 = 2z 2 + 20z + 1
60. 5k 2 − 10k + 3 = 0
√
−10 ± 7 2
2
5±
√
10
5
63. 0 = 4d2 + 6d + 3
64. 9h2 − 3h + 2 = 0
Ø
67. 4w2 + 3w − 2 = 0
Ø
68. 0 = 2f 2 − f − 12
√
−3 ± 41
8
1±
√
97
4
69. −4k 2 − k + 4 = 0
70. 0 = −3m2 + 5m − 1
71. −2z 2 − 7z − 2 = 0
72. 0 = −4h2 + 3h + 9
73. 0 = 6y 2 − 3y − 5
74. 12x2 + 11x + 1 = 0
75. 0 = 3w2 − 11w + 9
76. 4d2 + 3d − 8 = 0
√
−1 ± 65
8
√
5 ± 13
6
√
−11 ± 73
24
√
3 ± 129
12
77. x2 = 6x − 7
81. 3c + 2 = 6c2
3±
√
2
√
3 ± 57
12
78. p2 + 8p = 6
82. 7 = 2h2 − 3h
−4 ±
√
22
3±
√
65
4
√
3 ± 3 17
8
√
−7 ± 33
4
√
11 ± 13
6
√
−3 ± 137
8
√
79. −4t = t2 − 6
−2 ±
83. 6y 2 = 2 + 5y
√
5 ± 73
12
ALG catalog ver. 2.6 – page 294 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
3
2
√
−3 ± 2 2
7
√
√
1 ± 89
4
1
6
48. 0 = x2 + 6x + 1
55. 9y 2 − 10y − 3 = 0
Ø
0, −5
44. 3 − 8w = −4w2
5 ± 2 13
9
62. 0 = 8r2 + 2r + 3
±3
36. 20x + 100 = −x2
4
1,
3
4
24. n2 + 8 = −5n
Ø
√
5 ± − 42
66. 2t2 − t − 11 = 0
√
−5 ± 37
6
4p2 + 11p + 6 = 0
1, −3
20. 6y 2 + 11y − 2 = 0
51. −g 2 + 10g + 17 = 0
√
6 ± 33
3
61. 20m2 − 2m + 7 = 0
8.
−2,
31. 2g + 1 = 15g 2
−2 ±
58. 3p2 − 12p + 1 = 0
√
−2 ± 7
3
19. 5x2 − 13x − 6 = 0
47. 0 = c2 + 4c − 3
√
−1 ± 71
7
57. 0 = 3x2 + 4x − 1
0 = k 2 + 2k − 3
16. 0 = 3q 2 + 15q
39. −42w = 9 + 49w2
54. 0 = 7y 2 + 2y − 10
√
3± 7
2
4.
12. 9d2 − 81 = 0
0, 2
27. 7h + 3 = −2h2
50. 0 = −x2 + 6x + 2
√
15. 4c2 − 8c = 0
2
5
2, −
√
1±2 3
49. 0 = −z 2 + 2z + 14
2
1
,−
3
2
±4
23. 10y + 30 = −y 2
46. z 2 − 2z − 11 = 0
5
6z 2 − z − 2 = 0
Ø
34. −49 = m2 + 14m
45. a2 − 4a − 1 = 0
7.
7, 2
2
3, −
5
22. 6t2 + 9t = −10
Ø
0 = m2 − 9m + 14
11. 0 = v 2 − 16
±2
18. 7m2 + 10m + 3 = 0
5
3,
3
3.
−2, −
14. 4x2 + 12x = 0
0, 4
26—Solve by any method.
5
2
10. 0 = 5j 2 − 20
±3
15—Solve.
10
80. 4y + 3 = y 2
84. 1 − 8r2 = 3y
2±
√
7
√
−3 ± 41
16
ND
85. b − 1 = −4b2
√
−1 ± 17
8
87. 5g 2 + g = 5
90. k = 5k 2 + 2
91. 2d = −7 − 4d2
√
−7 ± 181
22
89. y − 4 = 11y 2
Ø
93. 3r2 + 2r = 4
√
−1 ± 13
3
Ø
√
94. 10g 2 − 2g = 7
97. 8n2 = 5 − 2n
98. 3v 2 = 6 − 4v
101. −4x = 7 − 2x2
102. 4g = −3 + 5g 2
√
−1 ± 41
8
√
2±3 2
2
105. 6y + 5 = −4y 2
1 ± 71
10
√
−2 ± 22
3
106. 8t − 10 = 3t2
111. (p − 1)2 = 8(7 − p)
1
5
115. x + 4 = (x − 2)(x − 3)
3±
√
√
7
√
41
3± 5
4
√
−5 ± 97
18
108. −5p = 10p2 + 4
Ø
5±
124.
√
1
5
4±
131. 3a(a − 2) = a(4 + a) − 15
Ø
133. 0 =
3±
137.
1 2
h − 2h + 1
3
√
√
−2 ± 10
3
√
√
− 3 ± 15
2
√
Ø
1±
√
10
2 2 3
x − x+1
5
2
5±
3
1
n + = −n2
5
2
√
150. y 2 − y 6 = 3
146.
√
6±3 2
2
3
65
8
1±
√
26
5
2
2
136. 0 = c2 − c −
3
3
139. 0 =
√
1± 7
3
5
3 2 1
x − x−
8
2
4
140.
√
34
3
147. −2y 2 = 1.8y + 0.5
√
151. n2 = 3 + 2n 5
√
ALG catalog ver. 2.6 – page 295 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
5±2 2
1
5 2 2
y − y− =0
6
3
2
2±
√
19
5
144. 0.1w2 = 0.6w − 0.3
√
5 ± 13
3
Ø
√
2
4
135. d2 + d − = 0
5
5
143. u − 0.4 = 0.3u2
√
−9 ± 41
20
√
Ø
2±
142. 0.9s + 0.1 = −s2
√
−1 ±
132. 10y(y − 1) = −2(y + 12)
√
−1 ± 21
5
10
√
15 ± 65
8
√
13 ± 89
20
1
3
m = m2 + 1
2
4
√
149. 3 − a2 = a 3
1 2
3
a −a− =0
4
2
138. 0 =
141. x2 − 1.3x + 0.2 = 0
2
3
√
22
3
130. (5a + 2)(a − 1) = 3 − a
2±
6
145.
128. (4y + 1)(2y − 3) = 2
√
43
6
134.
3 2
1
y +y− =0
4
2
−
√
−5 ± 89
16
√
3 ± 17
4n2 = (5n + 2)(n − 1)
2
√
5 ± 13
6 − 2d = (2d − 3)2
4
126. (3w + 1)2 = 6w2 + 7
23
3± 5
4
1±
Ø
9
2
0,
√
129. 5 + n = (2n + 1)(3n − 2)
√
23
3
√
−1 ± 7
3
5±
107. 2c = 5c2 + 7
122.
125. 7p2 − 11p = (2p + 1)(3p − 2)
127. −5 = (2x − 4)(2x + 1)
Ø
120. −1 = (8c − 3)(c + 1)
4
123. (3x + 4)(3x − 2) = x − 6
57
√
−3 ± 4 2
2
116. (t + 4)(t − 4) − 13 = 2(t − 10)
5
√
8
104. 4r2 − 23 = −12r
√
121. (2y − 1)2 = 2y
100. 2 − 2z = 3z 2
3±
103. 10h − 2 = 11h2
118. d(3d − 8) − 2 = 0
7±
96. 9b2 = 30b − 2
114. 0 = (3b − 2)(3b + 4) + 9
10
119. (2x + 1)(x − 4) = −5
√
2 ± 46
7x2 = 6 + 4x
7
√
1 ± 31
3 + 2q = 10q 2
10
112. 5y + 1 = 5y(3 − 5y)
−
5±
99.
92. c = 5 + 20c2
Ø
110. r(3r − 5) = r2 + 4r
5, −11
113. (5p + 3)(5p − 1) = −4
117. 5k(k − 2) = −3
Ø
2
3
109. 3k(2 − 3k) = 4(1 − k) − 2k
95.
88. 4v 2 − 3v = 3
√
5± 3
11
√
2 ± 19
5
Ø
√
−1 ± 101
10
86. 7a − 3 = −11a2
3±
Ø
√
6
148. 0.5x2 + 1.6 = 1.2x
√
√
152. p2 2 − p = 2 2
√
√
2 ± 34
4
Ø
NE
Topic:
Solving for other variables (quadratic equations and square roots).
Directions:
19—Solve for the indicated variable.
r
1.
W = I 2 R; for I
r
W
R
2.
d = 16t2 ; for t
3.
F =
mv 2
; for v
gr
r
F gr
m
4.
V = πr2 h; for r
r
5.
s=
6.
A = 4πr2 ; for r
r
7.
V =
1 2
πr h; for r
3
r
3V
πh
8.
L=
9.
h=
r
; for p
1 − p2
r
h−r
h
10. T =
1 2
gt ; for t
2
r
2s
g
z
√
11. v 2 = w2 + ; for z 2 v2 − w2
4
r
A
13. r =
; for A πr2
π
r
`
t2 g
15. t = π
; for `
π2
g
17. t = 2π
r
`
; for `
g
23. s =
h2 +
√
25. x2 − 2x + c = 0; for x
27. 2x2 + 2x − c = 0; for x
4s2 − e2
2
1±
√
1−c
−1 ±
29. ax2 − x + 2 = 0; for x
1±
31. ax2 − x + c = 0; for x
1±
2
; for n
n2 − w
r
R2 − r 2
; for r
t
r
2K
; for K
14. V =
m
√
2
+w
T
R2 − kt
V 2m
2
1
1
√
; for L
4Cf 2 π 2
2π LC
r
3h
2m2
; for h
18. m =
3
2
gt2
4π 2
e2
; for h
4
A
4π
16. f =
r
r
V
πh
t2 √
L
5
12. k =
Fgr
wy 2
; for F
gr
w
p
p
21. w = x2 + y 2 ; for x
w2 − y2
19. y =
25`2
; for `
t4
d
16
√
1 + 2c
2
20. n =
22. b =
r
√
24. K =
πV
; for L
5L
c2 − a2 ; for a
r
πV
5n2
√
c2 − b2
d2
− 3p2 ; for p
3
26. y 2 + 2y − c = 0; for y
√
d2 − 3K 2
3
−1 ±
√
1+c
28. ay 2 − 6y + 1 = 0; for y
3±
1 − 8a
2a
30. n2 + bn − 2 = 0; for n
−b ±
√
1 − 4ac
2a
32. y 2 + by − b2 = 0; for y
√
ALG catalog ver. 2.6 – page 296 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
9−a
a
√
b2 + 8
2
√
−b ± b 5
2
NF
Topic:
Finding a quadratic equation from roots.
Directions:
79—Find a quadratic equation with the given root(s).
1.
0, 6
5.
0, −
9.
4, 7
x2 − 6x = 0
2
3
3x2 + 2x = 0
x2 − 11x + 28 = 0
2.
0, 20
6.
0, −
x2 − 20x = 0
3.
0, −5
5x2 + x = 0
7.
0,
1
5
10. 2, −3
x2 + x − 6 = 0
7
4
17. 9
x2 − 9x − 22 = 0
x2 − 18x + 81 = 0
21. ±8
25. −5,
x2 − 64 = 0
2
3
3x2 + 13x − 10 = 0
14. −9, −6
x2 + 15x + 54 = 0
4.
0, −8
4x2 − 7x = 0
8.
0,
11. −1, −10
x2
13. −2, 11
x2 + 5x = 0
1
6
x2 + 8x = 0
6x2 − x = 0
12. −5, 9
x2 − 4x − 45 = 0
16. 12, 1
x2 − 13x + 12 = 0
+ 11x + 10 = 0
15. −5, 15
x2 − 10x − 75 = 0
18. −12
x2 + 24x + 144 = 0
19. −7
x2 + 14x + 49 = 0
20. 5
22. ±12
x2 − 144 = 0
23. ±6
x2 − 36 = 0
24. ±15
26. −1, −
7
4
27. 11,
1
3
3x2 − 34x + 11 = 0
x2 − 10x + 25 = 0
28. 7, −
x2 − 225 = 0
5
2
2x2 − 9x − 35 = 0
4x2 + 11x + 7 = 0
1 6
29. − , −
4 5
30.
2 1
,
3 3
9x2 − 9x + 2 = 0
31.
20x2 + 29x + 6 = 0
33.
4
7
37. ±
49x2 − 56x + 16 = 0
3
8
√
64x2 − 9 = 0
41. ± 7 x2 − 7 = 0
√
45. ±2 11 x2 − 44 = 0
√
49. 4 ± 3 x2 − 8x + 13 = 0
7 4
,−
2 5
1 8
32. − ,
2 9
10x2 − 27x − 28 = 0
34. −
38. ±
3
8
7
6
√
64x2 + 48x + 9 = 0
35. −
36x2 − 49 = 0
39. ±
42. ± 10 x2 − 10 = 0
√
46. ±6 3 x2 − 108 = 0
√
50. −1 ± 11
5
2
1
4
√
√
54. −3 ± 2 6
x2 − 2x − 49 = 0
57.
−2 ±
3
√
3
58.
9x2 + 12x + 1 = 0
61.
√
3±2 2
5
36.
16x2 − 1 = 0
40. ±
43. ± 5 x2 − 5 = 0
√
47. ±3 15 x2 − 135 = 0
√
51. 2 ± 2 x2 − 4x + 2 = 0
25x2 − 30x + 1 = 0
9x2 − 60x + 100 = 0
5
9x2 − 25 = 0
3
√
44. ± 13 x2 − 13 = 0
√
48. ±2 6 x2 − 24 = 0
√
52. −3 ± 10
x2 + 6x − 1 = 0
√
55. 5 ± 3 3
√
56. −2 ± 5 2
x2 + 6x − 15 = 0
x2 − 10x − 2 = 0
x2 + 4x − 46 = 0
√
5± 2
2
√
3± 5
2
−1 ±
3
59.
2x2
− 6x + 2 = 0
60.
4x2 − 20x + 23 = 0
62.
10
3
4x2 + 20x + 25 = 0
x2 + 2x − 10 = 0
√
53. 1 ± 5 2
18x2 − 7x − 8 = 0
√
−2 ± 5 3
3
9x2 + 12x − 71 = 0
√
7
3x2 + 2x − 2 = 0
63.
√
−1 ± 3 7
4
8x2 + 4x − 31 = 0
ALG catalog ver. 2.6 – page 297 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
64.
√
5±3 2
2
4x2 − 20x + 7 = 0
NG
Topic:
Understanding the roots of an equation. See also category LG (statements about real numbers).
Directions:
37—Solve for k so that the equation has the given number of roots.
1.
x2 + 10x + k = 0 ; one root
k = 25
2.
x2 + 4x − k = 0 ; one root
3.
x2 − 12x − k = 0 ; one root
k = −36
4.
x2 − 20x + k = 0 ; one root
k = 100
5.
x2 + kx + 9 = 0 ; one root
6.
x2 − kx + 16 = 0 ; one root
k = ±8
7.
x2 + kx + 18 = 0 ; one root
√
k = ±6 2
8.
x2 − kx + 3 = 0 ; one root
9.
kx2 − 6x − 2 = 0 ; one root
k = − 92
10. kx2 + 5x + 10 = 0 ; one root
11. 3x2 + 2x − k = 0 ; one root
k = − 13
12. 6x2 − 8x + k = 0 ; one root
k = ±6
13. x2 + 10x + k = 0 ; two roots
15. x2 + 4x − k = 0 ; two roots
k < 25
k > −4
17. kx2 − 6x − 2 = 0 ; no real roots
19. 3x2 + 2x − k = 0 ; two roots
k < − 92
k > − 13
k = −4
√
k = ±2 3
k=
k=
14. x2 + 10x + k = 0 ; no real roots
16. x2 + 4x − k = 0 ; no real roots
18. kx2 + 5x + 10 = 0 ; two roots
5
8
8
3
k > 25
k < −4
k<
20. 6x2 − 8x + k = 0 ; no real roots
5
8
k>
8
3
special
21. x2 + kx + 9 = 0 ; two roots
k > 6 or k < −6
23. x2 − kx + 16 = 0 ; no real roots
25. kx2 − 12 = 0 ; one root
27. 6x2 − k = 0 ; one root
k>0
k<0
30. 3x2 + kx = 0 ; two roots
k > 8 or k < −8
k = 0 or k = 4
k 6= 0
32. x2 + kx + k = 0 ; no real roots
ALG catalog ver. 2.6 – page 298 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
−6 < k < 6
k=0
28. x2 + kx + k = 0 ; one root
k=0
31. 6x2 − k = 0 ; no real roots
24. x2 − kx + 16 = 0 ; two roots
26. 3x2 + kx = 0 ; one root
Ø
29. kx2 − 12 = 0 ; two roots
−8 < k < 8
22. x2 + kx + 9 = 0 ; no real roots
0<k<4
NH
Topic:
Word problems involving the quadratic formula. See also categories JF (factoring) and NG (square roots).
Directions:
0—(No explicit directions.) 16—Solve and check.
40—Write an equation and solve.
1.
3.
39—Translate and solve.
A number is 32 less than its reciprocal. Find the
number. −1±√10
A number is 4 more than its reciprocal. Find the
number. 2 ± √5
2.
The sum of a number and its reciprocal is 3. What
is the number? 3±√5
4.
The difference between a number and its reciprocal
is 10. What is the number? 5 ± √26
6.
Find a negative number whose square is 1 more than
the number itself. 1−√5
8.
Find a positive number whose square is 31 more
than the number itself. 1+5√5
3
2
5.
Find a positive number whose square is 3 more than
the number itself. 1+√13
7.
Find a negative number whose square is 29 more
than the number itself. 1−3√13
9.
Find two numbers whose sum is 16 and product
is 50. 8 ± √14
10. Find two numbers whose sum is 20 and product
is 80. 10 ± 2√5
11. Find two numbers whose sum is 10 and product
is 18. 5 ± √7
12. Find two numbers whose sum is 16 and product
is 30. 8 ± √34
2
2
2
2
Area, perimeter
13. A square has an area of 25 cm2 . By how much
should each of its sides be increased in order to
double its area? ≈ 2.1 cm
14. A square has an area of 18 ft2 . By how much should
each of its sides be increased in order to double its
area? ≈ 1.8 ft
15. A square of an area of 200 in2 . By how much should
each of its sides be decreased in order to form a
square with half the area of the original? ≈ 4.1 in.
16. A square of an area of 144 m2 . By how much should
each of its sides be decreased in order to form a
square with half the area of the original? ≈ 3.5 m
17. Each side of a square is decreased by 8 ft in order to
make a square with an area of 18 sq ft. What is the
length of each side of the original square? ≈ 12.2 ft
18. Each side of a square is decreased by 2 m in order to
make a square with an area of 72 m2 . What is the
length of each side of the original square? ≈ 10.5 m
19. Each side of a square is increased by 5 inches in order
to make a square with an area of 60 sq in. What is
the length of each side of the original square?
20. Each side of a square is increased by 3 cm in order
to make a square with an area of 80 cm2 . What is
the length of each side of the original square?
≈ 2.7 in.
≈ 5.9 cm
21. A painting is 12 cm longer than it is wide. Its
area is 240 sq cm. What are the dimensions of the
painting? about 10.6 × 22.6 cm
22. A chalkboard is 4 ft wider than it is tall. Its
area is 36 sq ft. What are the dimensions of the
chalkboard? about 4.3 × 8.3 ft
23. The length of a rectangle is 10 m more than twice the
width. The area is 120 m2 . What are the dimensions
of the rectangle to the nearest tenth? 5.6 × 21.2 m
24. The length of a rectangle is 8 in. less than twice the
width. The area is 210 in2 . What are the dimensions
of the rectangle to the nearest tenth? 12.4 × 16.8 in.
25. The perimeter of a building is 62 meters. It covers
an area of 200 m2 . Find the length and width of the
building (round to the nearest tenth). 9.2, 21.8 m
26. The perimeter of a garden is 38 feet. It covers an
area of 75 ft2 . Find the length and width of the
building (round to the nearest tenth). 5.6, 13.4 ft
27. A picture frame has an area of 340 cm2 and a
perimeter of 80 cm. What are the dimensions of the
picture frame? about 12.3 × 27.7 cm
28. A bulletin board has an area of 450 in2 and a
perimeter of 106 in. What are the dimensions of the
bulletin board? about 10.6 × 42.4 in.
29. Find the sides of a rectangle whose perimeter is 56
and diagonal is 22. 14 ± √46
30. Find the sides of a rectangle whose perimeter is 40
and diagonal is 18. 10 ± √62
31. A rectangular sandbox is made from a 19 ft piece of
wood (the wood is cut into four pieces). If the area
of the sandbox is 14 sq ft, what are its dimensions to
the nearest hundredth? 1.82 × 7.68 ft
32. A rectangular picture frame is made from a 3 m piece
of wood (the wood is cut into four pieces). If the
area of the frame is 0.55 m2 , what are its dimensions
to the nearest hundredth? 0.64 × 0.86 m
ALG catalog ver. 2.6 – page 299 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
NH
33. A rectangular patio is surrounded on three sides by
a fence (the remaining side is up against the house).
If the area of the patio is 38 m2 , and the total length
of fence is 18 m, what is the length and width of the
patio? about 11.2 × 3.4 m
34. A rectangular patio is surrounded on three sides by a
fence (the remaining side is up against the house). If
the area of the patio is 260 sq ft, and the total length
of fence is 48 ft, what is the length and width of the
patio? about 31.4 × 8.3 ft
FENCE1.PCX
FENCE1.PCX
35. A rectangular lot, whose area is 425 sq ft, has one of
its longer sides next to a river. The total length
of fence surrounding the lot is 70 ft. Find the
dimensions of the lot. about 54.4 × 7.8 ft
36. A rectangular lot, whose area is 210 m2 , has one of
its longer sides next to a river. The total length
of fence surrounding the lot is 42 m. Find the
dimensions of the lot. about 25.6 × 8.2 m
FENCE2.PCX
FENCE2.PCX
37. A rectangular flower bed, whose dimensions are
7 × 14 ft, has one of its longer sides against a house.
The remaining three sides are to be increased by a
strip of uniform width, so that the area of the garden
is increased by 50%. How wide should that strip be?
≈ 1.6 ft
38. A rectangular flower bed, whose dimensions are
4 × 10 m, has one of its longer sides against a house.
The remaining three sides are to be increased by a
strip of uniform width, so that the area of the garden
is increased by 75%. How wide should that strip be?
≈ 1.4 m
GARDEN1.PCX
GARDEN1.PCX
39. A rectangular deck, whose dimensions are 6 × 8 m,
has one of its shorter sides up against a wall. The
remaining three sides are to be increased by a
uniform strip of wood, so that the area of the deck is
tripled. How wide should that strip be? ≈ 3.3 m
40. A rectangular deck, whose dimensions are 9 × 12 ft,
has one of its shorter sides up against a wall. The
remaining three sides are to be increased by a
uniform strip of wood, so that the area of the deck is
doubled. How wide should that strip be? ≈ 2.8 ft
DECKHS1.PCX
41. The length of a rectangular garden is twice the
width. When a 5 ft strip is cut away from all sides
of the garden, the area is decreased by half. What
are the original dimensions of the garden?
DECKHS1.PCX
42. The length of a rectangular lawn is 5 m more than
the width. When a 2 m strip is added to all sides of
the lawn, the area is doubled. What are the original
dimensions of the lawn? about 7.7 × 12.7 m
about 26.2 × 52.4 ft
43. A rectangular swimming pool is 12 meters long and
8 meters wide. It is surrounded by a cement walkway
of uniform width. The area of the walkway is twice
the area of the pool. How wide is the walkway?
44. A rectangular parking lot is 60 feet long and 45 feet
wide. Its area is doubled when a strip of uniform
width is added to all four sides. How wide is the
strip? ≈ 10.7 ft
≈ 3.5 m
ALG catalog ver. 2.6 – page 300 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
NH
Right triangles
45. The hypotenuse of a right triangle is 8 meters. Find
the lengths of the legs, if one leg is 2 meters longer
than the other. about 4.6 and 6.6 m
46. The hypotenuse of a right triangle is 14 ft long. Find
the lengths of the legs, if one leg is 4 ft longer than
the other. about 7.7 and 11.7 ft
47. The hypotenuse of a right triangle is 5 cm long. Find
the lengths of the two legs if their sum is 6 cm.
48. The hypotenuse of a right triangle is 10 in. long.
Find the lengths of the two legs if their sum is 13 in.
about 1.1 and 4.9 cm
about 3.7 and 9.3 in.
49. In a right triangle, the two legs are each 5 less than
the hypotenuse. What is the hypotenuse? ≈ 17.1
50. In a right triangle, the two legs are each 8 less than
the hypotenuse. What is the hypotenuse? ≈ 27.3
51. In a right triangle, the first leg is 6 less than the
hypotenuse, and and the second leg is 8 less than the
hypotenuse. What is the length of the hypotenuse?
52. In a right triangle, the first leg is 2 less than the
hypotenuse, and and the second leg is 3 less than the
hypotenuse. What is the length of the hypotenuse?
≈ 23.8
≈ 8.5
53. The diagonal of a square is 5 in. longer than its sides.
How long is each side? ≈ 12.1 in.
54. The diagonal of a square is 8 cm longer than its
sides. How long is each side? ≈ 19.3 cm
55. In a square, the length of a side is 6 cm less than a
diagonal. How long is each diagonal? ≈ 20.5 cm
56. In a square, the length of a side is 2 in. less than a
diagonal. How long is each diagonal? ≈ 6.8 in.
57. A 14 ft ladder leans against the side of a building, as
shown in the figure. Find out how high the ladder
reaches, if the distance between the bottom of the
ladder and the building is 11 ft less than the distance
between the top of the ladder and the ground.
58. A 5 m ladder leans against the side of a building
(refer to the previous problem). Find out how high
the ladder reaches, if the distance between the
bottom of the ladder and the building is 2 m less
than the distance between the top of the ladder and
the ground. 2+√46 or about 4.4 m
√
11+ 271
2
or about 13.7 ft
2
LADDER2.PCX
LADDER2.PCX
59. A ladder leans against the side of a building, as
shown in the figure. The distance x is 4 feet less
than the distance y, which is 1 foot less than the
length of the ladder. How long is the ladder?
6+
√
10 or about 9.2 ft
60. A ladder leans against the side of a building (refer to
the previous problem). The distance x is 3 meters
less than the distance y, which is 1 meter less than
the length of the ladder. How long is the ladder?
4+
√
6 or about 6.4 m
LADDER3.PCX
LADDER3.PCX
ALG catalog ver. 2.6 – page 301 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
NH
Formulas
61. A prism, whose height is h, has a square base of
side s. Its surface area is given by the formula
S = 2s2 + 4hs. Find s, if the surface area is 200
square units and the height is 10. −10 + 10√2
62. A prism, whose height is h, has a square base of
side s. Its surface area is given by the formula
S = 2s2 + 4hs. Find s, if the surface area is 98
square units and the height is 7. −7 + 7√2
PRISM1.PCX
PRISM1.PCX
63. The surface area of a square prism is given by the
formula S = 2s2 + 4hs, where h is the height and s is
a side of the base. If a square prism has a surface
area of 132 in2 and a height of 11 in., what is s to
the nearest tenth of an inch? 2.7
64. The surface area of a square prism is given by the
formula S = 2s2 + 4hs, where h is the height and s is
a side of the base. If a square prism has a surface
area of 192 cm2 and a height of 8 cm, what is s to
the nearest tenth of an inch? 4.6
65. A boy throws a rock over the edge of a cliff. The
rock is h feet above the boy after t seconds, as given
by the formula h = 48t − 16t2 .
66. Frank throws a rock over the edge of a cliff. The
rock is h meters above him after t seconds, as given
by the formula h = 10t − 5t2 .
a) What is the height of the rock after 1 second?
3 seconds? 5 seconds?
a) What is the height of the rock after 1 second?
2 seconds? 4 seconds?
b) At what time is the rock at its maximum
height?
b) At what time is the rock 3 meters above Frank?
c) At what time is the rock 8 feet above the boy?
d) Can you figure out how long the rock is in the
air? Explain.
c) At what time is it 10 meters above him?
d) At what time is the rock 24 feet below the boy?
32, 0, −160 ft; 1.5 sec; about 0.2 and 2.8 sec; about 3.4 sec
5, 0, −40 m; about 0.4 and 1.6 sec; never; no, the height of the cliff is
unknown.
CLIFFR1.PCX
67. Jeanine jumps off a diving board into a swimming
pool. Her height is h meters above the board after
t seconds, as given by the formula h = 8t − 5t2 .
a) At what time is Jeanine 2 meters above the
diving board?
CLIFFR1.PCX
b) At what time is she 4 meters above the diving
board?
68. A woman jumps off a diving board into a swimming
pool. Her height is h feet above the board after
t seconds, as given by the formula h = 32t − 16t2 .
a) At what time is the woman 8 feet above the
diving board?
c) If the diving board is 3 meters above the water,
approximately how long is Jeanine in the air?
about 0.3 and 1.3 sec; never; about 1.9 sec
b) At what time is she 4 feet below the diving
board?
c) Can you figure out how long the woman is in
the air? Explain.
about 0.3 and 1.7 sec; about 2.1 sec; no, the height of the diving board
is unknown.
DIVER2.PCX
ALG catalog ver. 2.6 – page 302 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
DIVER2.PCX
NH
69. A football is thrown along a path which can be
described by the equation
1 2
x .
64
In the equation, y is the height of the football above
the ground at a horizontal distance x. Find the
distance of the ball when y = 19 feet (round to the
nearest tenth). 20.7, 43.3 ft
y =5+x−
70. A football is thrown along a path which can be
described by the equation
1 2
x .
20
In the equation, y is the height of the football above
the ground at a horizontal distance x. Find the
distance of the ball when y = 4 meters (round to the
nearest tenth). 2.3, 17.7 m
y =2+x−
FTBALL2.PCX
71. When an arrow is shot at a 45 ◦ angle, its path may
be described by the equation
1 2
x .
75
In the equation, y is the height of the arrow at a
horizontal distance x. (Note: the height is measured
above the shoulders.) Find the distance of the arrow
when the height is 10 meters. 11.9, 63.1 m
y =x−
FTBALL2.PCX
72. When an arrow is shot at a 45 ◦ angle, its path may
be described by the equation
1 2
x .
200
In the equation, y is the height of the arrow at a
horizontal distance x. (Note: the height is measured
above the shoulders.) Find the distance of the arrow
when the height is 45 feet. 68.4, 131.6 ft
y =x−
ARROW1.PCX
ARROW1.PCX
73. A rocket is shot straight up in the air at 200 meters
per second. The rocket will be at a height h after
t seconds, as given by the formula h = 200t − 5t2 .
At what time(s) is the ball at a height of 500 m?
about 2.7 and 37.3 sec
74. A baseball is hit straight up in the air at 128 feet
per second. The ball will be at a height h after
t seconds, as given by the formula h = 128t − 16t2 .
At what time(s) is the ball at a height of 144 ft?
about 1.4 and 6.6 sec
vt − 16t2
75. Use the formula h =
to find out how long it
takes an object to reach a height of 200 ft, if it is
shot upward with an initial velocity v of 128 ft/sec.
(Round answer to the nearest hundredth.) 2.13 sec
76. Use the formula h = vt − 5t2 to find out how long
it takes an object to reach a height of 50 m, if it is
shot upward with an initial velocity v of 60 m/sec.
(Round answer to the nearest hundredth.) 0.90 sec
77. The stopping distance d of a car traveling at
x mph may be approximated by the formula
d = 0.05x2 + 0.1x (d is measured in feet).
78. The stopping distance d of a car traveling at
x km/hr may be approximated by the formula
d = 0.006x2 + 0.02x (d is measured in meters).
a) What is the stopping distance of a car whose
speed is 30 mph? 50 mph?
a) What is the stopping distance of a car whose
speed is 40 km/hr? 80 km/hr?
b) The stopping distance of a certain car was 80 ft.
How fast was it going? 48, 130 m; about 39 mph
b) The stopping distance of a certain car was 25 m.
How fast was it going? 10, 39 m; about 63 km/hr
79. The density of smoke from a diesel truck is related
to its engine speed, and may be approximated by
the formula D = 2r2 − 14r + 30. In the formula,
D is measured in millions of particles per cubic foot,
and r is hundreds of revolutions per minute. If the
density of smoke from a truck is 14, what is its
engine speed (rpm)? about 144 or 556 rpm
80. The density of smoke from a diesel truck is related
to its engine speed, and may be approximated by the
formula D = 50r2 − 400r + 800. In the formula, D is
measured in millions of particles per cubic meter,
and r is hundreds of revolutions per minute. If the
density of smoke from a truck is 150, what is its
engine speed (rpm)? about 227 or 573 rpm
ALG catalog ver. 2.6 – page 303 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
OA
Topic:
Simple inequalities. See also category OB.
Directions:
15—Solve. 31—Solve by graphing. 34—Solve each inequality and graph.
35—Solve and graph. 88—Graph the solution.
1.
x + 7 ≥ 12
5.
6 < m + 15
9.
3
2
+ a < − 52
13. 0 ≥ k + 6
3
5
<
11 + y > 17
m > −9
6.
3≥9+p
a < −4
10. y +
k ≤ −6
17. y − 9 6= 13
21. y −
2.
x≥5
y=
6 22
7
5
p ≤ −6
≥ − 14
y ≥ −2
3.
3
4
7.
2 > y + 10
+a≤
7
8
a≤
1
8
y < −8
11. r + 5 ≤ −11
r ≤ −16
8.
18 ≤ 21 + x
k<
m < 12
16. 0 < a − 15
18. 5 6= x − 8
x 6= 13
19. 33 6= 18 + r
r 6= 15
20. 14 + p 6= 23
c≤1
23. 8 < k − 14
3
7
≥c−
4
7
k > 22
26. x − 21 ≤ −25
x ≤ −4
27. y − 4 31 ≥ −5
y ≥ − 23
29. −12 + x > −6
x>6
30. −4 ≤ −17 + p
p ≥ 13
31. −14 > r − 24
r < 10
35. x − 8 6= −26
x=
6 −18
34. a + 35 6= 19
a 6= −16
1
10
x ≥ −3
12. 7 + p > −13
15. m − 12 < 0
m < −5
y 6= −7
<
3
10
k+
c ≥ −7
25. −15 > m − 10
33. 17 6= 24 + y
1
5
4.
14. c + 7 ≥ 0
22.
y<2
7
4
y>6
p > −20
a > 15
p=
6 9
24. a − 8 ≥ 19
a ≥ 27
28. −4 < c − 2 21
c > − 32
32. c − 19 ≤ −11
c≤8
36. −20 6= −12 + m
m 6= −8
37. 3k ≥ 21
k≥7
38. 42 ≤ 6x
41. 10 ≤ 12 y
y ≥ 20
42.
45. 8y ≥ 30
y≥
49. −18 > 3a
53. 6p > 6
54. 4 ≤ a
c 6= −3
69. −35 < −5y
73. 3y ≤ −20
y ≤ − 20
3
77. −36 > −10c
81. −45x < 9
c>
18
5
x > − 15
74. 22 ≥ −4a
k ≤ −11
a<
5
3
r ≤ − 16
86. 108 6= −18a
89. 4 − c < 16
c > −12
90. 2 − x ≥ 10
a 6= −6
x ≤ −8
y > −3
52. −24 ≥ 8c
y 6= 4
67. −2c ≤ 26
c ≥ −13
68. −4x < 32
102. 3 > −a − 12
105. 3k + 8 ≥ 17
k≥3
106. 5 + 2x > 19
109. 15 ≤ 4c − 9
c≥6
110. 21 < −9 + 5r
a > −15
x>7
r>6
x > −8
72. −27 ≤ −9p
m > − 15
4
79. −9y ≤ −24
y≥
83. −10 > 15c
c < − 23
8
3
76. 3x > −14
p≤3
x > − 14
3
80. −33 ≥ −6k
84. −35p ≤ 14
k≥
11
2
p ≥ − 25
87. −90 6= −15y
y=
6 6
88. −13m 6= 78
m 6= −6
91. 15 < −k + 8
k < −7
92. 11 ≤ −r + 5
r ≤ −6
a > 22
96. 17 − c > −5
c < 22
99. −16 > −8 − y
x < −17
r < −1
r≤3
64. −r ≥ −3
x≥8
101. 7 < −x − 10
c ≤ −3
x 6= 3
a<6
r>6
21
5
60. 96 6= 32x
63. −6 < −a
95. 14 − a < −8
98. −4 − x ≤ −12
m>
56. 10r < −10
x ≤ −1
w ≤ 20
r>5
c ≤ 36
48. 5m > 21
94. −13 ≤ −w + 7
97. −3 − r < −8
p<6
13
3
x≥
75. 30 > −8m
a ≥ − 11
2
x 6= −7
y ≥ 34
44. 6 ≥ 16 c
71. −4r < −24
x≤6
85. −12x =
6 84
93. −26 ≥ −y + 8
x > 25
>5
59. 25y =
6 100
x 6= −8
78. −12a > −20
82. −3 ≥ 18r
1
5x
55. −8 ≥ 8x
70. −6x ≥ −36
y<7
40. 9p < 54
51. 5y > −15
k ≤ −4
p > −5
66. −2k > 22
y ≤ −8
c<9
47. 26 ≤ 6x
11
2
a≥4
58. 9x 6= −72
62. 5 > −p
m≥2
65. −3y ≥ 24
r>
39. 45 > 5c
43.
a < 24
50. 9k ≤ −36
p>1
61. −m ≤ 2
<8
46. 22 < 4r
15
4
a < −6
57. −42 6= 14c
1
3a
x≥7
103. −y − 24 ≤ −21
107. 31 > 7 + 8a
111. −13 + 7y < 1
ALG catalog ver. 2.6 – page 304 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
y>8
y ≥ −3
a<3
y<2
100. −17 ≥ −9 − m
m≥8
104. −p − 15 ≥ −11
p ≤ −4
108. 26 ≤ 6x + 8
x≥3
112. 4w − 17 ≥ 3
w≥5
OA
113. 11 + 10m > 6
m > − 12
114. 8a + 15 ≥ 6
a ≥ − 98
117. −24 6= 15k − 4
k 6= − 43
118. 28 6= −7 + 4x
x 6=
121. −3 + 7c < −17
c < −2
122. 5p − 2 ≤ −27
p ≤ −5
35
4
115. −2 > 6y + 7
y < − 32
119. 14 + 5y 6= 23
y 6=
123. 19 ≥ 8x − 3
9
5
x ≥ − 14
120. 24a − 19 6= 33
a 6=
13
6
124. −28 < −10 + 6x
11
4
x≤
116. −1 ≤ 2 + 12x
x > −3
125. −9 ≤ 2a − 25
126. −2 < −17 + 3r
a≥8
129. −11 + 6a > −11
a>0
130. 9x − 8 ≤ −8
127. −25 + 4m > −5
r>5
m>5
131. −13 ≥ −7w − 13
x≤0
128. 2c − 15 ≥ −7
c≥4
132. −12 < −4k − 12
k<0
w≥0
133. −20 ≤ −25 − 5x
134. −15 > −23 − 8w
x ≤ −1
135. 3p − 13 < −10
p<1
136. 12c − 35 ≥ −23
c≥1
140. 39 ≥ −11x + 6
x ≥ −3
w > −1
137. −15a + 6 < 36
a > −2
138. 14 − 12m > 50
139. 31 ≤ 15 − 8x
x ≤ −2
k>4
m < −3
141. 21 < −4 − 5a
a < −5
142. 20 ≥ −9r − 7
r ≥ −3
143. −3k + 14 < 2
x≤6
146. 75 − 6w ≤ 27
w≥8
147. −25 < −4c − 13
145. −9x + 71 ≥ 17
149. −19 ≥ 17 − 12a
a≥3
150. −38 < −14p + 18
x>
157. 17 − 9w 6= −64
5
2
w 6= 9
x ≥ − 12
154. −27 + 12k ≤ 15
k≤
7
2
158. −25c − 58 6= 92
165. −17 ≤ −25 − 4n
162. 24 ≥ 20 − 6r
r≥2
148. −52 > −9m − 7
m>5
152. −9 − 11r < −64
r>5
x≥2
155. 22 > −5r + 17
r > −1
159. −22 6= −7 + 18m
156. 34 < −8a + 26
a < −1
160. 21 6= 45 + 30a
a 6= − 45
m 6= − 56
c 6= −6
161. −4x + 28 ≤ 30
c<3
151. −12 − 15x ≤ −42
p<4
153. 20x − 21 ≥ 29
144. 19 − 4r ≤ 11
r > − 23
166. −13 − 2a ≤ −7
a ≥ −3
163. 21 > −9x + 27
167. 15 − 2k < 7
x>
k>4
2
3
164. 30 − 8p > 26
p<
168. 8 < −3c + 17
c<3
1
2
n ≤ −2
169. −8x + 17 ≤ −17
x≥
17
4
m>7
173. −40 < −12 − 7w
w<4
177. −3p − 5 ≥ −5
170. −21 > −8m + 35
174. −14 − 6a < −74
178. −9 ≤ −7x − 9
172. 36 − 11x ≥ −63
x≤9
r<5
175. −39 − 17r ≥ −90
x≤0
179. 8 > −6k + 8
ALG catalog ver. 2.6 – page 305 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
176. −10 ≤ −26 − 12c
c≤
r≤3
a > 10
p≤0
171. −57 < −16r + 23
k>0
4
3
180. −5a + 11 > 11
a<0
OB
Topic:
Advanced inequalities. See also category OA.
Directions:
15—Solve. 31—Solve by graphing. 34—Solve each inequality and graph.
35—Solve and graph. 88—Graph the solution.
1.
3y + 5 − 4y > 10
2a + 6 + 3a < 16
2.
y < −5
5.
3.
4 ≥ 3x − 2 + 5x
7 ≤ 6m + 8 − 3m
6.
3
4
x≤
m ≥ − 13
9.
15 ≥ 5n + 4 − 3n − 3
13. 2w − 5 ≥ 3w
17. 25h > 3h − 33
h ≤ −8
h>
18. 5k + 32 > 13k
3
2
7.
c>1
5y − 6 − 3y ≥ −16
6r + 8 + r < −34
8.
r < −6
11. 3p + 13 − 4p − 6 ≥ 8
12. 10a − 12 + 4a − 11 > 19
p ≤ −1
14. −10x ≥ 4x + 7
w ≤ −5
1 > −4c + 7 − 2c
4.
y ≥ −5
10. 29 ≤ 4h − 6 − 9h − 5
n≤7
6 < 5w − 3 − 8w
w < −3
a<2
x ≤ − 12
k<4
15. 10y < 13y − 21
a>3
16. 12c + 18 < 14c
y>7
19. 25 − 4a ≥ −34a
20. −15m ≥ 35 − 10m
a ≥ − 56
21. 2n − 11 > 3n + 5
22. 12 − 5u < 3u + 13
u > − 18
n < −16
25. 15 − 18y < 12 − 14y
y>
26. 18 + 6c ≥ 4c − 5
c ≥ − 23
2
3
4
29. 3w − 5 + 4w < 5 − w + 11
w<
31. 3y + 6 − 8y ≤ 7 + 5y − 12
y≥
33. 8 − 2x + 13 < 5x − 3 − 2x
x>
35. 24 + 35a + 18 ≤ 3a − 8 + 2a
41. 12w − 35 6= 5w + 7
45. 3(2x − 5) > 5x − 3
r 6= 3
k≤
a<
y > −8
49. 3 − 2(4t − 7) > 5t − 2
t<
51. 23 < 10z − 3(5 − 3z)
z>2
53. 5(4x − 3) < 2(10x − 1)
IR
55. 4(5 − 4k) ≤ −2(8k + 1)
Ø
52. 95 < 10(14 − c) + 5c
56. 7(3c + 4) < 3(7c + 4)
a≥5
s≤
p>
a 6= − 72
1
4
IR
Ø
58. 7y − (6 − y) > 10(y − 1)
y<2
61. 5(2n − 13) − 1 6= 6(3n + 1)
n 6= −9
62. 3 − 4(5c + 9) 6= 2(6c + 11) + c
h > 13
64. 16 − (p − 4) 6= 5(2p − 3) + 2
w<3
c=
6 − 53
p 6= 3
66. 4x − [8 − (3 − x)] > 2 − (5 − x)
ALG catalog ver. 2.6 – page 306 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
m > −3
c<9
60. 5(2w + 3) < 4(15 − w) − 3
65. 8 + h < 3h − [7 − (2 − h)]
1
3
11
34
x≥7
h 6= 2
c≥
n>1
59. 11(2 − x) ≤ 3x − 4(2x + 5)
63. 25 + 7(h − 3) 6= 3(8 − h)
2
5
f ≥ − 16
3
44. 14a − 13 − 10a 6= 3a + 15 + 9a
54. −6(3h − 7) ≥ 9(1 − 2h)
57. 5 − 3(10 − 7a) ≥ 4(2a + 10)
k>
1
3
50. 13 − 5(2 − 7p) > 15 − 13p
19
13
3
4
p≥
40. 7m + 21 − 15m > −4m − 6 − 13m
y 6=
1
3
h < −2
36. −13 − 4f ≥ −5f − 8 − 2f − 21
48. 4s + 13 ≤ 6(4 − 5s)
15
11
28. 33a − 11 < 11 − 33a
x ≤ −1
46. 7(3 − 4n) < 12n − 19
x > 12
47. 7k − 12 ≥ 9(2k − 3)
27. 16 − 20x ≤ 15 − 21x
42. 6 − 17y 6= 5 − 14y
43. 3r − 25 + 4r 6= 5 + 2r − 15
x≥3
38. 13c + 11 + 12c ≥ 3c − 8c + 21
h≥1
w 6= 6
24. 7x + 18 ≥ 45 − 2x
r ≥ − 11
2
34. k + 10 + 6k − 12 > 4k − 2k
24
5
39. 18y − 25 − 13y < 35y + 15 − 25y
23. 8r − 6 ≤ 10r + 5
32. 18p + 4 + 11p ≥ 7 + 9p + 12
11
10
a ≤ − 53
37. 3h − 5h + 4 ≤ 4h − 5h + 3
m ≤ −7
30. 14h − 9 − 24h > 15 − 3h − 10
21
8
c>9
x>1
OB
67. 7 + 3 [2 − 5(4 − a)] > 2a − 1
69. 2(5 − 6m) ≥ 12 + 3 [2 − 4(2m − 5)]
71. 3n − 5 [4 + 2(n − 7)] ≤ 4(5 − 2n)
73.
2
3c − 4
c > 18
> 11 − 16 c
77.
2
3p − 6
< 12 p − 5
p<6
17
3
n ≤ −30
72. 8w + 2 [3(4 − w) + 7] ≥ 6(3w − 1)
75.
78.
1
2y − 7
y>3
79. 5 + 16 h ≥ 23 h − 4
≥ 23 (3x + 9)
x≥
83.
1
3 (12 − 3c)
≥ 45 (5c − 20)
c≤4
85.
5
4 (8m − 24)
87.
3
20 (60a + 120)
6=
89.
3
4 (2h − 5) + 4
> 8 − 25 (10h − 15)
6= 43 (9m − 27)
82.
15
4
≥ 12 u
91. 7 − 23 (9x − 2) ≤ 10 − 32 (3x + 1)
h>
5
2
5
2 (6t + 14)
a≥
> 29 (27t + 36)
t > −3
a<0
99. 5w(4w − 3) + 21 ≥ 2w(10w + 1) − 13
101. 5m(6m + 1) − 11 6= 3m(10m − 2) − 15
103. 3c(6c − 2) + 5 6= 2c(9c − 1) + 7
c 6= − 12
n ≤ − 23
86.
2
9 (81 − 54s)
6= 43 (12s − 16)
s=
6
88.
4
5 (30x − 20)
6= 47 (35x − 28)
x 6= 0
90.
4
3 (3 − 4w) + 2
10
7
< 35 (5w + 10) − 5
w≤2
4
m 6= − 11
3
5
w>
n ≥ − 13
c > −6
96. x2 − 7x + 20 ≤ 5 + x(x + 5)
17
14
x≥
5
4
98. 4h(3h + 5) − 2 ≤ 6h(2h + 3) + 1
h≤
100. 3y(3y + 5) − 20 > 9y(y + 2) + 4
y < −8
102. 2 − 4a(a + 6) 6= 3 + 2a(3 − 2a)
104. 6 − 4r(5 − 4r) 6= 8r(2r + 5) − 12
ALG catalog ver. 2.6 – page 307 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
34
5
11
4
+ 2 ≤ 34 r
a < 10
94. c(c + 3) < c2 + 4c + 6
2
9
a>
r≥8
80. 8 + 52 a > 12 a + 7
92. 3 + 41 (7 − 3n) ≥ 13 (2 − 3n) + 4
x ≥ − 19
11
14
w≤
1
2r
76.
u≥1
84. − 32 (6n − 12) ≤ 43 (12 − 9n)
a 6= −2
95. a2 + 11a − 15 ≥ a(a − 3) + 2
3
1
4u − 4
h ≤ 18
m 6= 3
20
3 (3a + 6)
y<
> −5 − 16 y
y≥
70. −3(2a − 9) < 4 [2(3a − 7) − 5a] + 15
3
1
8x + 1 ≤ 4x − 1
x ≤ −16
3
4 (8x − 12)
97. 2a(3a + 5) < a(6a − 3)
m≥
74.
81.
93. y(y − 4) > y 2 + 5y − 2
68. 4 − 2 [3y + 2(y − 5)] ≤ 4y + 13
46
13
a>
3
2
1
a 6= − 30
r=
6
3
10
OC
Topic:
Compound inequalities (and, or).
Directions:
15—Solve. 31—Solve by graphing.
35—Solve and graph. 85—Graph.
1.
x < 3 and x > −2
y ≥ 6 and y < 11
2.
−2 < x < 3
5.
m > 5 and m < 3
k ≤ −2 and k ≥ 1
6.
g≥3
13. d > 5 or d < −3
IR
18. b ≤ −7 or b > −10
IR
21. e > 3 or e ≥ 5
e>3
29. 7 < b − 3 < 11
22. t < −5 or t ≤ −2
26. −5 < y ≤ 0
same
2<x<8
35. −3 < z − 4 and 3 − z ≥ −4
1<z≤7
x < 9 or x > 13
39. 12 ≤ w + 3 or −5 < 1 − w
47. 52 < 4 − 3d < 13
same
19. z ≥ −11 or 2 > z
Ø
51. 19 + 2p < 5 or 4p − 6 ≥ 6
20. 6 ≤ f or 12 ≥ f
23. 6 ≤ x or x > 10
27. 1 ≤ k < 7
m≤5
28. −2 < a < 4
same
4 < k ≤ 13
34. y − 5 ≤ 1 and y + 2 ≥ 0
−2 ≤ y ≤ 6
36. 5 ≥ d + 2 and 3 > 1 − d
−2 < d ≤ 3
38. y + 6 ≤ 2 or 3 − y ≤ −3
y ≤ −4 or y ≥ 6
k ≤ 2 or k > 7
46. 29 ≥ 4x + 1 ≥ 13
7≥x≥3
48. 14 < 5 − 3f ≤ 53
−3 > f ≥ −16
52. 5n − 4 > 6 or 5 − 4n ≥ 1
−1 ≤ j ≤ 3
−2 ≤ t < 2
Ø
54. 9 − 3x ≤ 6 and 5 − 4x ≥ 13
55. 3 + 5h > 18 and 6 − 2h ≤ 18
h>3
56. 7 − 3k ≥ −5 and −2 ≤ 5 − 7k
59. 3t + 7 ≥ 13 or 8 − 5t ≥ −2
58. 6n − 11 < 1 or 3 − 2n < −1
60. 3m − 13 < −4 or 7 − 2m ≤ 5
IR
61. 4c − 6 + 2c < 36 and 5 − 3c − 10 ≥ −8
c≤1
w ≤ 2 or w ≥ 3
n ≤ 1 or n > 2
53. 4r − 19 ≥ −3 and 3 − 2r > 5
d≥3
same
1 ≥ r > −4
50. 5w + 3 ≤ 13 or 6 − 7w ≤ −15
u < 1 or u ≥ 3
p < −7 or p ≥ 3
57. 7d + 5 ≥ 26 or 45 ≤ 5 + 10d
IR
24. −4 ≥ m or m ≤ 5
44. 5 − 6t ≤ 17 and 3t + 8 < 14
−7 < k ≤ 1
−1 < y ≤ 13
49. 9 + 2u ≥ 15 or 3 − 4u > −1
16. −3 ≥ x or x > 5
42. 7j − 12 ≤ 9 and 4j + 5 ≥ 1
2<n<5
43. 8 − 5k ≥ 3 and 2k + 3 > −11
12. p < 8 and 8 ≥ p
40. 4 ≤ 6 − k or k + 3 > 10
w ≥ 9 or w < 6
41. 2n + 5 > 9 and 3n − 2 < 13
45. −5 < 2y − 3 ≤ 23
15. −2 > r or 1 ≤ r
32. 15 < k + 11 ≤ 24
33. x + 3 > 5 and x − 2 < 6
Ø
p<8
30. 4 ≥ r + 3 > −1
8 ≥ n ≥ −20
37. x − 4 < 5 or x + 2 > 15
11. 4 ≤ 6h and h > 10
x≥6
10 < b < 14
31. 31 ≥ n + 23 ≥ 3
3 > n and n ≥ 9
8.
IR
t < −2
same
−4 ≤ d and −6 ≥ d
same
same
17. c < 6 or c ≥ 4
7.
−20 ≤ r ≤ −10
h > 10
14. w ≤ 14 or w ≥ 20
same
−10 ≥ r and −20 ≤ r
4.
Ø
10. g ≥ 3 and 2 ≤ g
t < −5
4 > z and −3 ≤ z
−3 ≤ z < 4
Ø
t < −5 and −3 > t
25. −9 ≤ x ≤ −3
3.
6 ≤ y < 11
Ø
9.
34—Solve each inequality and graph.
88—Graph the solution.
Ø
k≤1
n 6= 2
IR
62. 18 + 4t − 12 ≤ 22 and 7 − t − 5 < 1
1<t≤4
63. 15 ≥ 3r − 9 − 7r and −12 + 5r + 32 ≥ 30
−6 ≤ r ≤ 2
64. 8f + 5 − 2f + 11 > 4 and −13 < 9 + 3f − 1
65. 3x + 23 − 8x ≥ 3 or 5 − 2x − 3 + 7x ≤ 12
x≤4
66. 2y − 3 + 5y ≥ 18 or 6 − 4y − 11 > −1
y < −1 or y ≥ 3
ALG catalog ver. 2.6 – page 308 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
f > −2
OC
67. 23 > 5w + 18 − 10w or −14 ≤ 13 − 3w − 6
IR
68. 45 < 12 + 5v − 2 + 2v or 9v + 6 − 13v > −10
v < 4 or v > 5
69. 4h − 5 − 6h < 1 and 5 > 2 + 5h − 8
−3 < h <
11
5
71. 15 < 5k − 12 + 4k and 8k − 11 + 4k − 12 ≥ 25
70. 9 + 10u + 6 ≥ 5 and 4u − 11 − 7u ≤ 12
u ≥ −1
72. −31 > 5 − 2b + 18 − 5b and 4b + 5 + 6b + 7 ≤ 8
Ø
k≥4
73. 7d − 5 − 8d > 5 or 4 + 3d − 9 > 15
d < −10 or d >
74. 12 ≤ 17q + 6 − 8q or 5q − 7 + 3q ≤ 15
75. −16 ≥ 6p − 7 − 10p or 13 > 5 + 3p + 9
p < − 13 or p ≥
76. −3a − 18 + 7a − 12 > 6 or 5 ≤ 8 − 5a − 3 − 6a
9
4
a ≤ 0 or a > 9
77. 15g − 9 < 9 − 6g and 3g + 8 < 5g − 12
79. 3x − 5 > 2x − 1 and 4 − 9x ≤ 3 + 5x
Ø
x>4
81. 14w − 6 + 3w < 5 + 2w and 16 − w ≥ 6 + 3w − 4
w<
IR
20
3
11
15
78. 8j + 11 ≥ 14 + 3j and 5j − 12 < 2j + 9
80. 2 − 7k ≤ 5 + 13k and 4 + 11k ≥ 8 − k
k≥
≤j<7
1
3
82. 5 − 7t + 12 ≤ 5t + 23 and −10 < 2t − 12 − 5t − 13
Ø
83. 3f − 8 + 7f > 2 − 9f and 65 + 4f < 3f + 25 − 5f
84. 21 − 8h ≥ 3h − 13 + 2 + 5h and 7h + 8 − 3h ≥ 4 − 4h
− 12 ≤ h ≤
Ø
85. x + 2 < 2x − 5 < x + 4
87. 7 − v < 5 ≤ −v + 15
3
5
7<x<9
2<v
89. 6y − 8 ≤ 8 − 5y or 3 + 7y ≥ 5 − 9y
91. 3k + 5 < 2k − 12 or 5 − 4k ≥ 6 + 9k
IR
1
k ≤ − 13
1
2
86. 5 + 3c ≥ 4c − 9 > 3c − 1
14 ≥ c > 8
88. 3 + 5p > 6p − 1 ≥ 5p + 7
Ø
90. 3x + 4 − 5x > 2x − 11 or 16x ≥ 3x − 9 − 5x
IR
92. 2n + 13 > 5 − 8n or 12n + 25 > 13n + 26
n < −1 or n > − 45
93. 8k + 10 < 3k − 4k − 5k or 5k − 9 ≥ 15 − 3k + 8
k < − 57 or k ≥ 4
95. 5u + 8 + 4u ≥ 3u − 8 and 6 − 7u ≥ 4u − 13 − 11u
u ≥ − 83
97. 10d + 8 + 2d > 3d − 7 + 4d and 2d + 8 ≥ 5 + 4d − 11
−3 < d ≤ 7
99. 2f + 5 − 8f ≤ 3f − 1 or 4f + 7 < 9f + 2 − 5f
f≥
2
3
94. 4w − 7 + 2w > 6w − 5 and 3w + 4 − 19w ≤ 5w + 8
Ø
96. 8x − 5 + 3x ≤ 5 − 3x + 8 and 4x − 7 − x < 7 + 3x
x≤
9
7
98. 6 − 5r + 21 < r + 5 − 7r or 4r + 7 ≤ 3r − 11 + 7r
IR
100. 10z − 7 − 3z > 4z + 6 + 3z or 5z − 1 − 2z > 6 + 3z
Ø
ALG catalog ver. 2.6 – page 309 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
OD
Topic:
Equations with absolute value.
Directions:
15—Solve.
1.
|−25| = r
5.
x + 5 = |3|
9.
k + |4| = −2
16—Solve and check.
25
−2
2.
t = |−19|
6.
|12| = 4 − d
13. |w | = 12
12, −12
14. |x| = 7
17. |s| = −2
Ø
18. −18 = |c|
25. |3c| = 24
−8
10. |−14| − m = 3
−6
21. −13 = − |d|
19
7.
|−9| = a + 21
15. |y | = 18
35, −35
6, −6
8.
f − 4 = |−2|
−7
12. −8 = |−8| + r
16. |m| = 23
20. 7 = − |w |
27. |−8r | = 16
2, −2
28. |6p| = 0
Ø
34. −64 = 16 |n|
Ø
35. 8 = −32 |u|
38. |p| + 12 = 12
0
39. 6 − |z | = 4
Ø
43. −11 = |p| − 8
Ø
2, −2
41. |y | + 6 = 1
Ø
42. 17 − |k | = 20
45. 9 = |x + 9|
0, −18
46. |h + 23| = 1
−22, −24
49. |t − 4| = 5
9, −1
50. 32 = |m − 8|
40, −24
51. |a − 12| = 6
18, 6
54. − |x + 4| = 5
Ø
55. −6 = |k − 9|
Ø
58. |u + 12| = 0
−12
57. 0 = |k − 6|
Ø
6
61. |15 − b| = 1
62. |23 − y | = 23
0, 46
63. |18 − c| = −6
2, 6
66. 11 = |5 − 2k |
8, −3
67. |5h + 8| = 3
69. −6 = |2n − 10|
70. |5g + 32| = −52
Ø
73. − |4x + 1| = −5
1, − 32
74. −11 = − |2r − 5|
Ø
40. 8 = 23 − |m|
15, −15
44. −5 − |r | = 9
Ø
48. |x + 5| = 21
16, −26
15, 35
56. − |h − 12| = 1
Ø
−7
64. −14 = |4 − x|
Ø
−1, − 11
5
−2, − 32
3
75. |34 − 8p| = −22
5, −5
36. 15 |x| = −15
60. |7 + s| = 0
71. |3a + 19| = 13
Ø
0
52. |y − 25| = 10
5
16, 14
65. |3z − 12| = 6
Ø
−10, −6
59. |5 − w | = 0
24, −24
32. −40 = −8 |k |
0
47. |y + 8| = 2
−16
Ø
24. − |y | = −24
31. 0 = 14 |t|
6
23, −23
5, −5
33. |7v | = −21
53. |m + 10| = −3
15
23. −5 = − |n|
30. 28 = 7 |a|
17, −17
−12
Ø
1, −1
37. |a| + 5 = 22
|15| = z
18, −18
29. 12 |j | = 12
4, −4
4.
31
19. − |a| = 10
Ø
26. |−5b| = 30
8, −8
g = |31|
11. x + |−11| = 4
11
7, −7
22. − |h| = −35
13, −13
3.
Ø
68. |5 − 4d| = 9
Ø
7
2
−1,
72. |7 − 2r | = 21
14, −7
76. |6y + 8| = −3
Ø
8, −3
77. |7h + 3| = 21
24
18
7 ,− 7
78. |3x + 1| = 19
6, − 20
3
79. |5t − 8| = 28
36
5 , −4
80. −31 = − |6p − 5|
6, − 13
3
81. |−10u − 9| = −11
Ø
82. − |−8k + 2| = 18
Ø
83. 4 = |−3s + 17|
7,
13
3
84. |−10w − 1| = 9
−1, − 45
85. |5s − 13| = 0
y
89. 12 = 5
86. 0 = |18 + 3d|
13
5
c
90. = 1
2
60, −60
w
93. − 2 = 3
3
15, −3
97. 3 = |u − 8| − 4
15, 1
87. |−21y − 9| = 0
−6
8a 91. = 6
3
2, −2
m 94. 4 − = 10
5
98. |y + 6| + 2 = 9
70, −30
1, −13
− 37
2u 92. = 8
5
9
9
4,−4
k
95. 3 + = 1
4
−8, −16
99. |d + 1| + 4 = 10
88. |34 − 2x| = 0
5, −7
17
20, −20
c
96. 18 = + 6 3
36, −72
100. |h + 11| − 3 = 7
−21, −1
101. |c − 6| + 12 = 2
Ø
102. |z + 10| − 4 = −9
Ø
103. 5 − |y + 6| = 12
ALG catalog ver. 2.6 – page 310 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
Ø
104. 4 = 7 + |x + 1|
Ø
OD
105. 11 − |4k − 4| = −13
109. 4 + |8y − 2| = 12
− 34 ,
106. 8 = 6 + |2x + 4|
107. |3n − 9| + 4 = 10
− 12 ,
113. − |5x + 8| + 2 = −11
1, − 21
5
111. −2 = |3x + 2| − 4
7
10
2, −4
121. 4 |c − 9| + 5 = 17
114. 7 − |4y + 1| = −2
−1,
118. 5 |k + 1| = 15
2,
129. 2 |3m − 1| − 12 = 4
5, 13
127. 8 = 6 − 2 |4p + 9|
Ø
10
3
137. |2z − 6| = 3z − 9
3
141. 6 − n = |1 + 3n|
Ø
1,
131. 3 |5 − k | − 11 = 4
135. |1 − 3a| = a
2
3
138. |3x + 12| = x + 4
−4
142. |5a + 2| = 14 + a
5
7
4,−2
128. 7 − 3 |2m − 3| = 1
132. 27 = 6 + 7 |5 − 4n|
2,
136. r = |2r + 5|
1 1
2, 4
139. 5u + 6 = |2 + 3u|
1
2
Ø
−1
140. |6k − 5| = 8k − 9
2
143. |1 − 2x| = x + 4
5, −1
144. |8 − 7p| = 3p + 4
3,
147. |4y + 10| = y + 1
Ø
148. |3x + 14| = x + 2
Ø
2
5
3, − 83
145. |5r + 6| = r + 1
Ø
149. |4c + 2| = |c + 3|
146. d + 2 = |2d + 18|
Ø
150. |u − 6| = |2 − 3u|
−2, 2
1
3
151. |3 − g | = |1 − 2g |
−2,
152. |5 − 3w | = |2w + 10|
−1, 15
4
3
153. y 2 + 3 + 1 = 13
154. t2 + 1 + 3 = 8
157. x2 + 5x = 6
−6, −3, −2, 1
158. 6 = w2 − 5w 159. 24 = z 2 − 10z 160. 10c + c2 = 24
161. 30 = b2 + 13b 162. 30 = 13c − c2 163. 54 = y 2 + 15y 164. −15m + m2 = 54
3, −3
2, −2
−1, 2, 3, 6
−10, −3, 2, −15
165. |d − 2| = d − 2
Ø
1 5
2, 2
130. 4 − 2 |5a + 3| = 6
134. |5y − 4| = y
− 32 , − 52
124. 4 − 6 |x + 1| = 13
0, 10
Ø
120. 12 |h + 2| = 6
18, −4
3, − 73
−1,
1 7
4, 4
123. 8 + 3 |y − 7| = 41
126. 7 |3w − 8| − 2 = 12
1, −6
116. 7 = − |4 − 4a| + 10
1
3
119. −7 |9 − j | = −28
2, −4
−8, 0
125. 3 |2g + 5| − 5 = 16
11 5
2 , 2
115. 10 − |6r + 2| = 6
122. 3 = 23 − 5 |a + 4|
12, 6
112. −6 + |8 − 2z | = −3
0, − 43
2, − 52
117. 24 = 8 |d + 1|
108. − |2y − 5| + 8 = 1
6, −1
110. |10p − 1| + 3 = 9
5
4
133. |3 + 2x| = x
1, 5
−1, −3
7, −5
166. |5 − h| = h − 5
5, −5
−9, −6, 3, −18
h≥5
167. |3 − a| = 3 − a
ALG catalog ver. 2.6 – page 311 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
156. 20 = 2 u2 − 6 4, −4
−12, −6, −4, 2
−2, 4, 6, 12
10, 3, 2, −15
d≥2
155. 3 x2 − 14 = 33
9, 6, −3, 18
a≤3
168. 6 − x = |x − 6|
x≤6
OE
Topic:
Inequalities and absolute value.
Directions:
15—Solve. 31—Solve by graphing.
35—Solve and graph. 85—Graph.
1.
|−a| < 0
5.
|p| 6= 8
9.
|x| ≤ 4
2.
|x| > 0
p 6= 8, −8
6.
|d| 6= −5
−4 ≤ x ≤ 4
10. |−h| ≤ 12
Ø
13. |u| > 5
17. |v | > −3
|t| ≤ 0
IR
7.
|−x| =
6 12
18. −8 ≤ |w |
−12 ≤ h ≤ 12
− 52 < y <
11. − |y | > −6
c>
1
7
37. |−4r | ≥ −12
|k |
>3
3
42.
Ø
|m|
≥ −1
4
20. − |d| ≤ 1
IR
24. − |x| > 5
Ø
7
2
35. |5y | ≤ 45
32. 27 |a| ≥ −36
− 74 < c <
7
4
36. |−7h| > 39
−9 ≤ y ≤ 9
39. |−3n| ≤ −18
43.
IR
1 46. a ≥ 3
6
50. |p| − 8 ≤ −11
IR
h > 7 or h < −7
a ≥ 18 or a ≤ −18
49. |n| − 9 < −1
−14 ≤ n ≤ 14
2
3
|a|
1
>−
−4
6
40. |45c| ≤ 36
Ø
− 23 < m <
2
3
44.
Ø
−8 < n < 8
−k > −2
47. 3 3
IR
51. |y | + 3 > 9
− 45 ≤ c ≤
4
5
− |u|
3
<−
2
4
u>
− 72 ≤ y ≤
p 6= 3, −3
−9 < z < 9
28. 7 |n| ≤ 98
k > 9 or k < −9
1 7
45. y ≥
5
10
12. |z | < 9
or t < − 13
4
38. |−8c| < 14
IR
IR
31. −18 |m| < −12
Ø
34. |12t| > 39
13
4
− |m| =
6 −3
or c < − 17
− 23 < m <
t>
8.
IR
c ≥ 10 or c ≤ −10
27. 35 |−c| > 5
30. 10 |−k | ≤ −24
or x ≤ − 10
3
|y | ≥ 0
16. |c| ≥ 10
23. |−m| ≤ −10
Ø
5
2
33. |−9x| ≥ 30
−6 < y < 6
19. −5 < |−a|
−12 < w < 12
29. −6 |y | > 15
p 6= 12, −12
4.
f > 4 or f < −4
26. −2 |w | > −24
h ≥ 8 or h ≤ −8
t=0
15. |−f | > 4
IR
22. −2 ≥ |−z |
Ø
25. 4 |h| ≥ 32
41.
3.
r ≥ 19 or r ≤ −19
IR
21. −14 > |n|
10
3
x 6= 0
14. − |r | ≤ −19
u > 5 or u < −5
x≥
34—Solve each inequality and graph.
88—Graph the solution.
3
2
or u < − 32
r 1
48. <
10
2
−5 < r < 5
52. |x| + 15 > 0
IR
y > 6 or y < −6
53. − |d| − 4 < 0
IR
54. 5 − |w | ≤ 2
−3 ≤ w ≤ 3
55. 4 − |z | ≥ 10
Ø
56. − |v | + 5 ≤ −12
v ≥ 17 or v ≤ −17
57. |x + 5| > 3
58. |h − 3| ≥ −2
IR
x > −2 or x < −8
61. |9 − z | ≤ 5
4 ≤ z ≤ 14
59. |−8 + m| ≤ 6
60. |c + 7| > 12
2 ≤ m ≤ 14
62. |7 − u| < 6
1 < u < 13
c > 5 or c < −19
63. |6 − n| ≥ 5
64. |5 − b| < 14
n ≤ 1 or n ≥ 11
65. |32 − 8y | < 32
1
69. c − 4 ≤ 10
3
−18 ≤ c ≤ 42
73. |6x − 3| 6= 9
x 6= −1, 2
y≤
or y ≥
68. |3t + 9| ≤ −12
Ø
−3 < w < 5
1
71. u − 8 ≥ 5
2
3 72. 9 − b ≥ 21
2
74. |10n + 35| 6= 5
75. |8 − 2r | 6= 12
76. |−3w + 15| 6= −6
n 6= −3, −4
r 6= −2, 10
78. |5k − 4| ≤ 6
11
6
67. |4 − 4w | < 16
2 70. 4 − y > 6
5
y > 25 or y < −5
77. |4 − 6y | ≥ 7
− 12
66. |5t − 10| ≥ 15
t ≥ 5 or t ≤ −1
0<y<8
−9 < b < 19
− 25
≤k≤2
u ≥ 26 or u ≤ 6
b ≥ 20 or b ≤ −8
79. |12 − 6g | > 14
g<
ALG catalog ver. 2.6 – page 312 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
− 13
or g >
13
3
80. |5r + 21| ≥ 34
r≥
13
5
or r ≤ −11
IR
OE
81. |3c − 5| ≤ −8
82. |5m + 6| ≥ −2
Ø
IR
83. |2p − 7| ≤ 13
84. |3 − 5u| ≥ 33
−3 ≤ p ≤ 10
u≥
5
1
85. k − ≥ 2
6
2
a
7
86. + 3 >
2
2
3 1
87. 2 − p < −
4
2
89. |5.9 + 2x| < 4.3
90. |7n − 1.4| ≤ 3.5
91. |5y − 1.5| ≥ 2
a > 1 or a < −13
k ≥ 3 or k ≤ − 95
−5.1 < x < −0.8
−0.3 ≤ n ≤ 0.7
93. 3 |2y − 1| < 27
94. −2 |4 − 5y | > 8
Ø
1
10
<y<
1
6
92. |3.6 + 3k | ≥ 2.1
k ≥ −0.5 or k ≤ −1.9
95. 9 |6j + 3| < 27
96. 2 |5n + 9| ≤ −6
Ø
−1 < j < 0
97. 5 |5 − 3k | ≥ −25
IR
98. −8 |6 − 7m| ≥ −24
3
7
101. 6 |2 − 3u| > 30
≤m≤
9
7
g ≤ − 95 or g ≥
105. |3w + 5| − 2 ≤ 4
0<r<
109. 1 − |2p + 5| ≥ 13
Ø
m≤
117. 13 − 5 |9 − 2j | ≤ 8
or m ≥
− 13
IR
121. 10 |8x − 1| + 2 ≥ −2
IR
125. |11 − (5 − x)| − 6 > −1
x > −1 or x < −11
126. 9 + |2 − (y + 8)| > 12
130. −4 > |z | ≥ −2
Ø
141. |x| > x + 6
−10 ≤ e ≤ −7 or 4 ≤ e ≤ 73
5 < c < 7 or 0 < c < 2
142. |a| ≥ a + 10
x < −3
g<
− 72
116. 3 |5m + 1| − 6 ≥ 12
m ≤ − 75 or m ≥ 1
or g > 0
119. 4 |3n + 5| − 2 ≤ 18
120. 5 − 7 |4 − g | > 12
Ø
≤n≤0
123. 7 − 6 |3c + 7| > 25
Ø
127. |−3 − (n − 7)| − 4 ≤ 0
124. 5 − 3 |4k + 3| > 2
131. 1 ≤ |w | ≤ 7
132. 5 > |r | > 0
a ≤ −5
135. −2 < |8y + 5| ≤ 3
−1 ≤ y <
− 14
≤x≤
18
5
0 < r < 5 or −5 < r < 0
136. −5 < |2 − 3w | < 7
− 53 < w < 3
139. 24 ≤ 3 |5x − 8| ≤ 30
16
5
128. 8 − |8 − (3 − r)| ≥ 3
−10 ≤ r ≤ 0
1 ≤ w ≤ 17 or −17 ≤ w ≤ −1
137. 13 ≤ 1 − 3 |4r − 9| ≤ 19 138. 15 < 5 |2c − 7| < 35
Ø
115. 7 − 3 |4g + 7| < −14
0≤n≤8
134. 11 ≤ |2e + 3| ≤ 17
2 < h < 4 or −3 < h < −1
IR
− 12 < k < −1
4 < d ≤ 8 or −8 ≤ d < −4
133. 3 < |2h − 1| < 7
3
2
112. 7 − |5w + 2| ≤ 14
− 52
y > −3 or y < −9
129. 4 < |d| ≤ 8
111. |4x − 9| + 20 > 35
− 10
3
122. 13 − 9 |4k + 11| > 4
−3 < k <
1<d<
x < − 32 or x > 6
118. 9 − 5 |2v + 13| < 24
k ≤ 4 or k ≥ 5
17
7
108. 6 + |5 − 4d| < 7
−1 ≤ c ≤ 2
110. 4 + |3k + 6| ≥ 10
− 43
2
3
104. −3 |4 − 7y | ≤ −39
IR
107. 5 − |2c − 1| ≥ 2
9
2
114. 2 |6m + 5| + 5 ≥ 11
x < −3 or x > 2
103. −8 |7y − 1| ≤ 4
− 16
3 ≤e≤
y ≤ − 97 or y ≥
k ≤ −4 or k ≥ 0
113. 4 |2x + 1| − 11 > 9
100. 5 |3e + 7| ≤ 45
1
2
7
5
106. |4r − 9| + 5 < 14
1
3
99. 6 |10d − 1| ≥ 24
3
d ≤ − 10
or d ≥
102. −4 |5g + 1| ≤ −32
7
3
u < −1 or u >
− 11
3 ≤w ≤
or u ≤ −6
2 1
88. 5y − <
3
6
Ø
y ≥ 0.7 or y ≤ −0.1
−5 < y < 6
36
5
or
143. |y | ≤ 3y − 2
− 25
≤x≤0
y≥1
140. 25 ≤ 4 |3 − 2f | − 3 ≤ 33
−3 ≤ f ≤ −2 or 5 ≤ f ≤ 6
144. |m| ≥ 5m + 12
m ≤ −2
145. |2 − 3x| > 3x
x<
149. |y | < |y + 4|
y > −2
1
3
146. |k + 5| < 3 − k
150. |x| ≥ |x − 2|
k < −1
x≥1
147. |p − 8| ≥ p + 4
151. |a + 2| > |3a|
− 12 < a < 1
ALG catalog ver. 2.6 – page 313 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
p≤2
148. |c − 3| ≤ 3 − c
152. |n + 3| ≤ |2n|
n ≥ 3 or n ≤ −1
c≤3
PA
Topic:
Slope between two points. See also categories PG (mixed practice and review) and PH (perpendicular
and parallel lines).
Directions:
80—Find the slope.
1.
(0, 0)(−6, 2)
− 13
2.
(3, 9)(0, 0)
5.
(−7, 0)(0, 7)
1
6.
(0, −10)(−4, 0)
9.
(9, 8)(3, 0)
4
3
13. (4, 3)(5, −2)
−5
3
− 52
13
5
15. (7, −5)(8, 4)
0
22. (1, 11)(5, 11)
37. (−3, −7)(−1, −7)
41. (2, 14)(2, 16)
− 47
53. ( 73 , 5)( 43 , 2)
3
57. (1, 13 )(3, − 23 )
61. ( 65 , 0)(3, 6)
65. (2, − 14 )(1 12 , 14 )
54. (−4, 3 25 )(2, 1 25 )
− 13
78. (1, 6)(−2, 0.9)
− 13
20
81. (−3, 0.2)(1.5, 0.7)
1
9
24. (8, −1)(−8, −1)
0
3
2
35. (9, 9)(−11, 7)
1
10
0
undef.
−1
−5
− 85
7
32. (3, 12)(9, 10)
− 13
36. (6, 0)(−6, −9)
3
4
40. (7, −5)(2, −5)
0
44. (8, 5)(8, 2)
undef.
48. (−4, −4)(4, −9)
1
6
56. (2 23 , 0)(− 13 , 3)
−1
2
3
60. (− 72 , 0)(− 12 , 2)
63. ( 13 , 2)(− 13 , 3)
− 32
64. (3, 23 )(1, 1)
− 16
68. ( 12 , 73 )(4, 13 )
2
− 47
72. (6, − 12 )(6 12 , 0)
3
7
75. (1.7, 2)(−0.3, −4)
3
− 10
9
83. (2.8, 3.1)(2, −1.9)
25
4
− 58
52. (−10, 1)(2, 3)
1
79. (−1.3, 0)(−4, 3)
17
10
1
2
3
4
71. (5, 27 )(3, − 47 )
− 54
82. (3.5, 0.2)(0.5, 5)
−1
0
28. (8, 3)(7, −4)
59. (−1, 12 )(0, 32 )
67. (1, 35 )( 45 , 15 )
4
15
74. (7.4, −3)(1.4, 3)
1
2
undef.
55. ( 21 , −3)(4 12 , 0)
1
8
70. (1, −2)( 35 , − 32 )
5
3
20. (−3, 6)(−3, 0)
51. (−6, 9)(−3, −6)
2
66. (− 34 , 35 )( 34 , 1)
−1
73. (0, −2.8)(6, 0.2)
77. (6, 2.4)(2, 5)
1
62. (2, 14 )(−2, − 14 )
10
3
undef.
47. (−4, −5)(−8, −7)
− 11
6
1
6
− 74
43. (−2, 4)(−2, 8)
undef.
(0, −2)(12, 0)
16. (6, −8)(−2, 6)
39. (8, 0)(−6, 0)
0
50. (−4, −8)(1, −3)
58. ( 25 , 1)( 75 , 3)
− 12
69. (4, 56 )(3, − 56 )
38. (1, 12)(−3, 12)
46. (5, 6)(11, −5)
9
4
− 53
8.
− 35
9
3
5
31. (−2, −11)(−15, 2)
34. (−3, −3)(−9, 7)
(5, −3)(0, 0)
−1
27. (−4, 2)(4, 14)
7
2
42. (9, −5)(9, 5)
undef.
45. (0, 14)(−8, −4)
49. (4, 9)(11, 5)
0
19. (5, 2)(5, 8)
4.
12. (−4, 0)(−9, 5)
23. (0, 6)(−4, 6)
−5
30. (2, −5)(6, 9)
− 15
undef.
0
26. (1, −6)(−1, 4)
33. (−2, −6)(8, −8)
−3
14. (7, 6)(2, −7)
21. (2, −8)(3, −8)
8
(3, 0)(0, 9)
11. (−1, 0)(−11, −6)
18. (−4, −3)(−4, 5)
29. (1, 12)(−1, −4)
7.
1
7
−4
undef.
− 29
(0, 0)(−7, −1)
10. (2, −5)(0, 3)
17. (10, −1)(10, 1)
25. (−9, 5)(9, 1)
3.
1
76. (2.5, 1)(−2.5, 3)
80. (2, 3)(1.4, −1)
− 25
20
3
84. (−1.6, 1.4)(0.6, −0.6)
− 10
11
85. (4, −1.7)(3.3, −0.3)
86. (4.1, 4.2)(3.8, 3.6)
2
−2
89. (a, 3)(3, a)
88. (−0.5, 3)(−3, 2.5)
1
5
−10
−1
93. (−4, −b)(4, b)
97. (c, d)(c, −d)
87. (−1, 1.2)(−0.9, 0.2)
b
4
undef.
101. (a, −a)(a − 2, a)
105. (2x, y)(7x, 2y)
a
y
5x
90. (−1, n)(−n, 1)
1
91. (c, 0)(0, −c)
94. (x, y)(−x, −y)
y
x
95. (−2, e)(2, −e)
98. (s, 5)(t, 5)
99. (r, 3r)(r, 4r)
0
92. (r, s)(s, r)
1
−1
− 2e
96. (−p, 1)(p, 0)
1
2p
undef.
100. (−2, y)(2, y)
0
102. (k + 3, 2k)(k, k)
k
3
103. (2p, 3)(2p − 2, 1)
106. (k, n)(k − n, 3n)
2
107. (3c, 5d)(c, −d)
−1
3d
c
104. (c + 3, −c)(c, 2c)
−c
108. (5a, a + 2)(−a, a − 1)
1
2a
109. (r − p, 3s)(p − r, s)
s
r−p
110. (x − 2y, 0)(x − 2y, −w)
undef.
111. (n, −4k)(2k + n, −4k)
0
ALG catalog ver. 2.6 – page 314 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
112. (y, x − 1)(3y, 1 − x)
1−x
y
PB
Topic:
Midpoint. See also category PG (mixed practice and review).
Directions:
81—Find the midpoint.
1.
(−1, 9)(−1, 1)
5.
(−4, 0)(5, 0)
(−1, 5)
( 12 , 0)
2.
(−7, 0)(3, 0)
6.
(1, −2)(10, −2)
(−2, 0)
3.
(6, 5)(−6, 5)
7.
(0, −11)(0, 2)
(0, 5)
(0, −4 21 )
4.
(0, −12)(0, −4)
8.
(4, −3)(4, 8)
(0, −8)
(4, 2 12 )
(5 21 , −2)
9.
(2, 3)(10, 7)
10. (12, 5)(8, 11)
(6, 5)
13. (−7, −8)(−3, 8)
(−5, 0)
(10, 8)
14. (−6, 4)(0, −4)
(−3, 0)
11. (9, 16)(3, 4)
(6, 10)
15. (−2, −1)(10, −9)
12. (11, 0)(7, 14)
(9, 7)
16. (−5, 2)(3, −8)
(−1, −3)
20. (7, −2)(9, −9)
(8, −5 12 )
(4, −5)
17. (0, 2)(−8, 5)
(−4, 3 12 )
18. (2, −9)(−9, 1)
19. (6, 6)(4, 13)
(5, 9 12 )
(−3 12 , −4)
21. (6, −13)(1, 2)
(3 21 , −5 12 )
22. (−3, 7)(2, −6)
(− 12 , 12 )
23. (−8, −3)(5, 0)
24. (5, 10)(4, −7)
(4 21 , 1 12 )
(−1 12 , −1 12 )
25. (−4, −2)(3, 9)
(− 12 , 3 12 )
26. (−8, −5)(−1, 6)
27. (8, 0)(−3, −7)
(−4 12 , 12 )
29. (− 12 , 1)(2 12 , 3)
(1, 2)
30. (0, 23 )(−2, 1 31 )
28. (−7, 5)(−6, −5)
(2 12 , −3 12 )
(−1, 1)
(−6 12 , 0)
31. (−1, −3 12 )(1, − 12 )
32. (2 54 , −3)(− 45 , 1)
(1, −1)
(0, −2)
33. (5, −4 13 )(−2, 5)
1
37. ( 34 , 12 )(− 34 , 10
)
(1 21 , 13 )
(0,
3
10 )
34. ( 12 , −3)(−3, 2)
35. (−1, 2)( 51 , −5)
(−1 14 , − 12 )
(− 25 , −1 12 )
38. (4, −2)(−2 25 , 1 12 )
39. ( 43 , 13 )(− 14 , 16 )
36. (7, 0)(2, 2 23 )
( 41 , 14 )
(1.6, −5.2)
45. (3.3, −2.8)(7.1, 0)
(5.2, −1.4)
49. (2p, p − 3)(4p, p + 1)
(3p, p − 1)
53. (x, y − 8)(x + 6, 2 − y)
(x + 3, −3)
42. (3, 1)(−4.8, 0.3)
43. (−4, −4)(1.5, 1)
(−0.9, 0.65)
(−1.25, −1.5)
46. (−2, −2.5)(4.6, 5.5)
(1.3, 1.5)
50. (a − 6, a)(3a, 8 − a)
(2a − 3, 4)
54. (−12, 3y)(4x, y + 2)
(2x − 6, 2y + 1)
40. (− 72 , −1)( 12 , 52 )
(−1 12 , 34 )
( 54 , − 14 )
41. (−2, 0.6)(5.2, −11)
(4 12 , 43 )
47. (−0.7, 4)(−2.3, −3.9)
(−1.5, 0.05)
51. (−7k, 2k − 3)(−k, −9)
(−4k, k − 6)
55. (a + b, b)(a − b, −b)
(a, 0)
ALG catalog ver. 2.6 – page 315 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
44. (7.5, 9)(−3, −2)
(2.25, 3.5)
48. (1.5, −0.6)(−2, 2.8)
(−0.25, 1.1)
52. (n − 5, 10)(5 − n, 4n)
(0, 2n + 5)
56. (3b, −5a)(2a − b, a + 2b)
(a + b, −2a + b)
PC
Topic:
Distance between two points. See also category PG (mixed practice and review).
Directions:
82—Find the distance.
1.
(0, 0)(5, 0)
5.
(−3, −11)(−3, −2)
9.
(4 21 , 0)(−5 12 , 0)
5
9
(−8, 0)(3, 0)
11
3.
(0, 10)(0, 2)
6.
(8, 1)(8, −6)
7
7.
(−9, 5)(5, 5)
10. (−0.8, 1)(−4.8, 1)
10
13. (7, 6.2)(7, 3.4)
2.
14. (−4, −3)(−4, 32 )
2.8
4
8
14
11. (−2, 53 )(−2, 23 )
1
(0, −6)(0, 0)
8.
(12, −1)(10, −1)
2
12. (0, 5.6)(0, −1.4)
7
6
16. (− 74 , 2)(−3, 2)
15. (1.5, −6)(−2.8, −6)
9
2
4.
5
4
4.3
1
17. ( 10
, 3)(− 45 , 3)
21. (3, −4)(0, 0)
18. (3 13 , −5)(4 16 , −5)
9
10
22. (0, 0)(−8, −15)
5
25. (4, −6)(−1, 6)
13
26. (7, 0)(10, −4)
29. (0, −13)(7, 11)
25
30. (−3, 6)(6, −6)
√
37. (−7, 4)(−2, 5)
26
38. (8, −7)(5, −9)
41. (5, 5)(11, 3)
45. (10, −8)(8, 0)
√
2 10
√
2 17
49. (4, −3)(−9, −4)
53. (1, 2)(− 12 , 0)
42. (−2, 2)(0, −2)
√
170
50. (1, 12)(4, 1)
61. (−2n, n)(2n, −2n)
√
130
58. (1.7, −3)(2, −2.6)
5n
√
3 10
62. (5k, −5k)(0, 7k)
0.5
13k
13
32. (−8, −7)(12, 8)
√
5 2
40. (−2, −2)(2, −1)
√
4 5
√
4 10
48. (1, 9)(10, −4)
51. (−4, 12)(3, 9)
√
58
52. (−5, −3)(2, −8)
56. ( 53 , 4)(3, 5)
59. (−0.2, 5)(0.6, 3.5)
1.7
63. (7a, a + 3)(a, −7a + 3)
10a
ALG catalog ver. 2.6 – page 316 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
√
17
√
5 2
44. (−6, 3)(−5, 10)
47. (3, −10)(7, 2)
15
4
10
25
36. (−7, 5)(−2, 0)
√
29
55. ( 41 , −2)(−2, 1)
5
3
20
√
4 2
43. (−6, −4)(−2, 4)
5 16
28. (−10, −3)(−4, 5)
17
35. (1, 1)(−3, −3)
39. (0, 1)(2, 6)
√
2 5
54. (−4, 13 )(−3, −1)
1
2
13
46. (−5, −3)(−2, 6)
5
2
57. (0, 1)(−0.6, 0.2)
√
√
20. (0, −1 32 )(0, −6 56 )
24. (12, 5)(0, 0)
31. (−15, −1)(1, −13)
15
7 34
10
27. (−2, −7)(6, 8)
5
34. (−4, 10)(−3, 11)
19. (−6, 2 41 )(−6, −5 21 )
23. (0, 0)(−8, 6)
17
√
3 2
33. (−2, −7)(1, −4)
5
6
√
5 10
√
74
13
5
60. (3, 0)(2.5, −1.2)
1.3
64. (p − 1, −8p)(9p − 1, 7p)
17p
PD
Topic:
Points on a line.
Directions:
0—(No explicit directions.)
1.
Find four points on the line: y = −7x
3.
Find four points on the line: y = 61 x
5.
Find four points on the line: 2y = x
7.
Find four points on the line: −8y = 6x
9.
Find four points on the line: y = −3
11. Find four points on the line: y = 2
2.
Find four points on the line: y = 5x
[points]
4.
Find four points on the line: y = − 15 x
[points]
6.
Find four points on the line: −9y = −x
8.
Find four points on the line: 6y = 10x
[points]
[points]
10. Find four points on the line: x = 8
[points]
13. Find four points on the line: 9x − 12y = 0
[points]
[points]
17. Find three points on the line: y = 25 x + 6
[points]
19. Find three points on the line: x + 6y = −6
[points]
21. Find three points on the line: 3x + 4y + 8 = 0
[points]
[points]
[points]
[points]
14. Find four points on the line: −x + 4y = 0
[points]
16. Find four points on the line: y = −3x + 9
[points]
18. Find three points on the line: y = − 23 x − 8
20. Find three points on the line: −x − 4y = 12
[points]
[points]
22. Find three points on the line: 7x − 2y − 6 = 0
24. Find three points on the line: −2x + 10y = 25
[points]
25. Which points are on the line: y = −3x ?
A(−3, 1)B(1, −3)C(−2, 6)D(3, 0)
A(4, 2)B(−3, 1 21 )C(0, 12 )D(−6, −3)
A, D
31. Which points are on the line: x = −3 ?
A
37. Which points are on the line: y = −4x + 7 ?
D
39. Which points are on the line: y = 6x + 1 ?
A, C, D
41. Which points are on the line: 2x + 5y = −3 ?
A(0, 0.6)B(1, −1)C(−1.5, 0)D(−6, 3)
32. Which points are on the line: y = 8 ?
A, B, D
34. Which points are on the line: 2x + 8y = 0 ?
B, C
all
36. Which points are on the line: −4x + y − 1 = 0 ?
A(−27)B( 21 , −3)C(2, 9)D(0, 1)
B, D
A(7, 0)B(−2, 1)C(−1, −11)D(3, −5)
C, D
A(0, 0)B(8, −2)C(−1, 14 )D(1, − 14 )
C
35. Which points are on the line: x − 8y + 5 = 0 ?
A(−1, −5)B(−5, 0)C(0, 1)D( 12 , 4)
A(1, 23 )B(2, 3)C( 31 , − 12 )D(−3, 2)
A(0, 8)B(−8, 8)C(8, −8)D(4, 8)
none
33. Which points are on the line: −6x − 3y = 0 ?
3
4 )B(3, 1)C(10, −2)D(−5, 0)
28. Which points are on the line: −3y = −2x ?
A(0, 1)B(−1, 0)C(0, −1)D(−2, −1)
all
A( 21 , −2)B(1, 2)C(−3, 6)D(6, −3)
B, C, D
30. Which points are on the line: y = −1 ?
29. Which points are on the line: x = 7 ?
A(0, −3)B(6, −3)C(6, 3)D(3, −3)
26. Which points are on the line: −2y = 5x ?
A(5, 2)B(−2, 5)C(4, −10)D(0, 0)
B, C
27. Which points are on the line: y = 21 x ?
A(2,
[points]
[points]
23. Find three points on the line: 10x + 15y = −9
A(7, −7)B(7, 3)C(7, 7)D(7, 0)
[points]
[points]
12. Find four points on the line: x = −5
[points]
15. Find four points on the line: y = 5x − 1
[points]
C, D
38. Which points are on the line: y = 23 x − 9 ?
A(6, −5)B(0, 9)C(−3, 8)D(9, −15)
A
40. Which points are on the line: y = − 12 x − 5 ?
A(0, 5)B(−4, 3)C(2, −4)D(−5, 0)
none
42. Which points are on the line: −x + 3y = 8 ?
A(7, 5)B(−5, 1)C(1, 3)D(−8, 0)
ALG catalog ver. 2.6 – page 317 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
all
PD
43. Which points are on the line: −5x − y = −3 ?
A(1, 2)B(−2, 7)C(0, −3)D(4, −15)
none
45. Find the missing coordinates, so that each point is
on the line: y = 5x.
(0, )( , 15)(−2, )
0, 3, −10
47. Find the missing coordinates, so that each point is
on the line: −y = 4x.
( , 8)( , −4)(0, )
−2, 1, 0
49. Find the missing coordinates, so that each point is
on the line: −15y − 6x = 0.
( , 25 )(5, )( , −10)
−1, −2, 25
51. Find the missing coordinates, so that each point is
on the line: x − 8y = 0.
(−8, )(4, )( , 2)
−1, 12 , 16
53. Find the missing coordinates, so that each point is
on the line: x + 6y = 6.
(12, )( , −3)(0, )
−1, 24, 1
55. Find the missing coordinates, so that each point is
on the line: 6x + 5y = 15.
(0, )(5, )( , 9)
3, −3, −5
57. Find the missing coordinates, so that each point is
on the line: y = 10
3 x − 3.
(3, )( , 0)(−6, )
7,
9
10 , −23
59. Find the missing coordinates, so that each point is
on the line: y = 5x − 8.
( , 0)(−2, )( , 4)
8
12
5 , −18, 5
44. Which points are on the line: 4x + 6y = 15 ?
A(3, 12 )B(2, 1)C(5, −1)D(0, 2 12 )
A, D
46. Find the missing coordinates, so that each point is
on the line: y = − 13 x.
( , 0)(−12, )( , 2)
0, 4, −6
48. Find the missing coordinates, so that each point is
on the line: 21y = 3x.
(14, )( , 0)(−7, )
2, 0, −1
50. Find the missing coordinates, so that each point is
on the line: −12x + 2y = 0.
( , −18)( 12 , )( , 4)
−3, 3,
2
3
52. Find the missing coordinates, so that each point is
on the line: 2y + 8x = 0.
( , 12)(2 12 , )( , −6)
−3, −10,
3
2
54. Find the missing coordinates, so that each point is
on the line: −3x + y = −1.
( , 5)(0, )(4, )
2, −1, 11
56. Find the missing coordinates, so that each point is
on the line: 2x − 9y = 12.
(−3, )( , −4)( , 0)
−2, −12, 6
58. Find the missing coordinates, so that each point is
on the line: y = 12 x + 12 .
(−1, )(5, )( , 3 12 )
0, 3, 6
60. Find the missing coordinates, so that each point is
on the line: y = 4x − 43 .
(0, )( 13 , )(1, )
ALG catalog ver. 2.6 – page 318 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
− 43 , 0,
8
3
PE
Topic:
Graphing lines.
Directions:
85—Graph. 36—Solve for y and graph. 83—Find the slope and y-intercept for each line. 84—Find
the x- and y-intercepts for each line. 86—Graph on the same coordinate system. 87—Graph on
separate coordinate systems.
1.
x=5
5.
y = −4
9.
y=x
[graph]
[graph]
[graph]
13. y = − 14 x
[graph]
17. y = −x + 2
[graph]
21. y = 2x + 3
[graph]
25. y = − 47 x − 7
29. y = 21 x −
1
2
33. y = − 83 x +
[graph]
[graph]
5
6
[graph]
37. y = 52 x − 2 45
[graph]
41. y = 0.3x − 1.5
[graph]
2.
y = −7
6.
x=1
[graph]
[graph]
3.
x = −4
7.
y=6
[graph]
[graph]
10. y = −x
[graph]
11. y = −3x
14. y = 21 x
[graph]
15. y = 38 x
18. y = x − 5
[graph]
22. y = −6x − 1
[graph]
26. y = 25 x + 4
30. y = − 16 x +
34. y = 43 x −
[graph]
5
6
1
8
[graph]
[graph]
38. y = − 12 x − 3 12
[graph]
42. y = −0.5x + 0.5
8.
x = −2
[graph]
[graph]
[graph]
16. y = − 35 x
[graph]
[graph]
23. y = −4x + 2
y=3
12. y = 4x
[graph]
19. y = x + 6
4.
[graph]
[graph]
20. y = −x − 1
[graph]
24. y = 5x − 7
[graph]
27. y = 51 x − 5
[graph]
28. y = − 13 x + 1
31. y = 49 x +
[graph]
3
32. y = − 10
x−
3
4
35. y = − 13 x − 2 31
39. y = 21 x +
9
2
[graph]
[graph]
43. y = −2.2x + 4
[graph]
[graph]
1
10
36. y = 41 x + 3 34
40. y = − 76 x +
[graph]
[graph]
11
6
44. y = 1.6x − 4
[graph]
[graph]
[graph]
45. y = −1.5x + 3.5
46. y = 1.8x + 0.2
[graph]
47. y = 0.25x + 1
[graph]
48. y = −0.1x − 2
[graph]
[graph]
49. −x + y = 0
53. 2x − y = 0
[graph]
[graph]
50. x − y = 0
[graph]
54. −5x − y = 0
[graph]
57. −x − 3y = 0
[graph]
58. x + 8y = 0
61. 6x + 20y = 0
[graph]
62. −12x + 9y = 0
65. −5y = −7x
[graph]
69. 3x − y = −6
[graph]
66. 3y = 2x
[graph]
[graph]
70. −4x + y = 2
[graph]
73. x − y − 4 = 0
[graph]
74. x + y − 8 = 0
77. 4x + 4y = 12
[graph]
78. x − 3y = 3
81. −x + 5y + 15 = 0
[graph]
[graph]
[graph]
[graph]
86. −4x + 9y − 18 = 0
90. 5x − 4y = −16
[graph]
[graph]
56. 3x + y = 0
59. 2x − 14y = 0
[graph]
60. 5x − 10y = 0
63. 2x + 5y = 0
[graph]
[graph]
68. 9y = −6x
71. −x − y = 3
[graph]
72. x − y = −5
75. 5x + y − 5 = 0
79. x + 4y = −8
[graph]
[graph]
[graph]
[graph]
[graph]
[graph]
76. 2x + y + 8 = 0
[graph]
80. 10x − 5y = 30
[graph]
84. x − 3y − 12 = 0
[graph]
87. 3x + 2y − 8 = 0
91. 10x − 2y = 4
[graph]
64. −11x + 4y = 0
67. −4y = 10x
[graph]
[graph]
52. −x − y = 0
55. −9x + y = 0
[graph]
[graph]
[graph]
[graph]
83. −6x + 2y + 2 = 0
82. 8x + 2y + 6 = 0
[graph]
85. 6x − 2y + 14 = 0
89. 2x + 7y = 21
[graph]
51. x + y = 0
88. 15x + 5y + 10 = 0
[graph]
[graph]
92. −21x + 7y = −7
[graph]
93. 6x + 5y = 30
[graph]
94. −3x + 8y − 24 = 0
[graph]
97. 18x − 21y + 28 = 0
[graph]
98. 20x + 35y = −14
[graph]
95. 5x − 7y + 35 = 0
[graph]
99. −10x − 4y = 30
[graph]
ALG catalog ver. 2.6 – page 319 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
96. 10x + 2y = −20
[graph]
100. 8x + 18y − 12 = 0
[graph]
PE
101. −15x + 10y − 25 = 0
102. 12x + 9y − 15 = 0
[graph]
105.
x y
− =1
3
2
109. x − 16 y =
103. 36x + 15y + 24 = 0
[graph]
2
3
[graph]
[graph]
113. 2x + 43 y = −4
[graph]
117. 4y − 3x = 2y + x − 8
106. −
x y
+ =1
5
4
[graph]
110. 3x + 12 y = −4
114. 6x − 92 y =
3
2
[graph]
[graph]
118. y + 5x = 4y − x + 9
[graph]
104. −30x − 45y + 18 = 0
[graph]
[graph]
x y
+ =1
6
3
x y
− =1
2
8
[graph]
108.
111. −x − 15 y = 1
[graph]
112. 2x + 14 y = − 12
[graph]
116. x − 54 y = −10
[graph]
107. −
115. −4x + 85 y =
1
8
[graph]
119. −6x − y = x − 5y − 6
[graph]
120. 10x + 7y = 2x − y + 4
[graph]
[graph]
121. 2x − 7y + 6 = 3(y − x) − 4
[graph]
122. 2(x + 3y) + 3 = −7(x − 1)
[graph]
123. 4(x + y) − 5 = 3x − 2y − 1
[graph]
124. −5(2x − y) = 3(y + 1) − 8
[graph]
125.
2y
= −6
x−2
[graph]
126.
2y + 1
=4
x−1
129.
y−1
2
=
x + 10
5
[graph]
130.
y+4
1
=−
x−6
2
[graph]
[graph]
[graph]
127.
y−4
= −1
x+2
[graph]
128.
3y
=9
x+1
131.
2y − 1
1
=
3x
6
[graph]
132.
y+5
5
=−
x+8
4
ALG catalog ver. 2.6 – page 320 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
[graph]
[graph]
PF
Topic:
Writing equations of lines.
Directions:
91—Write
92—Write
93—Write
94—Write
95—Write
96—Write
97—Write
the
the
the
the
the
the
the
equation
equation
equation
equation
equation
equation
equation
1.
slope = −2, y-intercept = 7
3.
slope = − 23 , y-intercept = −8
5.
slope = 16 , y-intercept = − 56
7.
slope = 5, y-intercept = −2 12
9.
slope = 3, y-intercept = 0
11. slope = −2, y-intercept = 0
of
of
of
of
of
of
of
the
the
the
the
the
the
the
line.
line in standard form.
line in general form.
line in point/slope form.
line in point/point form.
line in slope/intercept form.
line and graph.
y = −2x + 7
y = − 23 x − 8
y = 61 x −
5
6
y = 5x −
5
2
y = 3x − 5
y = −6x − 6
17. slope = −5, contains point (−3, −5)
19. slope = 3, contains point (5, 10)
y = −5x − 20
passes through the origin
25. slope = − 14 , contains point (4, −2)
29. slope = 23 , contains point (−9, 0)
5
12 ,
y=
slope = − 92 , y-intercept =
8.
slope = −1, y-intercept = − 34
2
3
37. slope = 15 , contains point (−6, 2 45 )
y = − 58 x +
3
4
y = − 35 x
y = 73 x
14. slope = 1, contains point (−4, −6)
y =x−2
16. slope = −4, contains point (2, −5)
y = −4x + 3
18. slope = −4, contains point (1, −3)
y = −4x + 1
y = 7x + 14
28. slope = 13 , contains point (6, 4)
y = 13 x + 2
38. slope = − 38 , contains point (4, 3 21 )
7
3
y = − 85 x − 9
y = 74 x +
y = 3x + 4
42. slope = 3, contains point (2, 10)
43. slope = −1, contains point (−4, 4)
y = −x
44. slope = 2, contains point (−3, −6)
45. slope = − 13 , contains point (9, −3)
y = − 13 x
46. slope = 52 , contains point (10, 4)
47. slope = − 74 , contains point (−4, 7)
y = − 74 x
48. slope = 65 , contains point (12, 10)
y = −0.5x + 2
y = −0.4x + 1
y = 2x
y = 25 x
y = 56 x
50. slope = 0.25, contains point (−2, 3.5)
52. slope = 0.7, contains point (5, 2.5)
ALG catalog ver. 2.6 – page 321 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
3
5
y = − 38 x + 5
y = 74 x − 2
y = 8x + 6
5
2
3
y = − 10
x−
40. slope = 74 , contains point (2, 32 )
41. slope = 8, contains point (−1, −2)
51. slope = −0.4, contains point (1.5, 0.4)
y = − 56 x − 4
3
36. slope = − 10
, contains point (8, −3)
5
25
12 x + 4
y = −8x
y = 12 x + 4
34. slope = 47 , contains point (−2, −1)
7
4
y = − 13 x −
49. slope = −0.5, contains point (−2.8, 3.4)
y = 4x
32. slope = − 85 , contains point (−5, −1)
y = 15 x + 4
39. slope = − 13 , contains point (−2, − 53 )
y = −x −
30. slope = − 56 , contains point (6, −9)
y = 92 x − 15
y=
2
3
26. slope = 21 , contains point (−8, 0)
y = − 17 x − 3
contains point (−15, 0)
y = − 92 x +
24. slope = −8, passes through the origin
10
3 x
y = 23 x + 6
33. slope = − 58 , contains point (−2, 3)
35. slope =
6.
y = 15 x + 2
22. slope = 4, passes through the origin
y = − 12 x
y = − 14 x − 1
27. slope = − 17 , contains point (−7, −2)
31. slope = 92 , contains point (4, 3)
slope = 51 , y-intercept = 2
y = 4x − 1
20. slope = 7, contains point (−2, 0)
y = 3x − 5
21. slope = − 12 , passes through the origin
10
3 ,
4.
12. slope = 37 , y-intercept = 0
y = −2x
15. slope = −6, contains point (−1, 0)
23. slope =
slope = 4, y-intercept = −1
10. slope = − 35 , y-intercept = 0
y = 3x
13. slope = 3, contains point (2, 1)
2.
y = 0.25x + 4
y = 0.7x − 1
PF
53. vertical, passes through (−3, −6)
x = −3
54. vertical, passes through (2, −1)
55. vertical, contains the point (4, 7)
x=4
56. vertical, contains the point (−5, 0)
x=2
x = −5
57. horizontal, contains the point (8, 2)
y=2
58. horizontal, contains the point (−4, 4)
59. horizontal, passes through (−1, −5)
y = −5
60. horizontal, passes through (3, −9)
61. zero slope, passes through (1, −4)
62. zero slope, passes through (3, 5)
y = −4
63. zero slope, contains the point (0, −6)
65. undefined slope, contains (−5, 0)
y=4
y = −9
y=5
64. zero slope, contains the point (−7, −2)
y = −6
66. undefined slope, contains (−7, −8)
x = −5
y = −2
x = −7
67. undefined slope, passes through (9, −3)
x=9
68. undefined slope, passes through (4, 1)
69. y-intercept = 4, contains point (10, −8)
y = − 65 x + 4
70. y-intercept = 6, contains point (−6, 2)
y = 23 x + 6
72. y-intercept = −2, contains point (9, 1)
y = 13 x − 2
71. y-intercept = −3, contains point (−4, −5)
73. x-intercept = 2, y-intercept = 8
y = 12 x − 3
74. x-intercept = −3, y-intercept = 9
y = −4x + 8
y = 3x + 9
75. x-intercept = 6, y-intercept = −6
y =x−6
76. x-intercept = −1, y-intercept = −5
77. x-intercept = 6, y-intercept = −3
y = 12 x − 3
78. x-intercept = −8, y-intercept = −10
79. x-intercept = 7, y-intercept = 2
80. x-intercept = −2, y-intercept = 5
y = − 27 x + 2
x=4
y = −5x − 5
y = − 54 x − 10
y = 25 x + 5
81. x-intercept = −5, contains point (1, 6)
y =x+5
82. x-intercept = −1, contains point (4, −5)
y = −x − 1
83. x-intercept = 8, contains point (2, −3)
y = 12 x − 4
84. x-intercept = 4, contains point (−4, −6)
y = 34 x − 3
85. contains (−2, 5) and (−2, −1)
x = −2
86. contains (4, 0) and (4, −6)
87. passes through (−1, 7) (−1, 3)
x = −1
88. passes through (5, −3) and (5, −9)
89. passes through (−8, 6) and (0, 6)
91. contains (5, 2) and (−5, 2)
x=4
90. passes through (−1, −3) and (1, −3)
y=6
y=2
y = −4
y = 5x
y = 3x
94. contains (3, 15) and (−2, −10)
95. contains (−7, 7) and (5, −5)
y = −x
96. contains (−3, 6) and (3, −6)
97. contains (10, −6) and (−5, 3)
y = − 35 x
98. contains (4, 14) and (2, 7)
99. contains (3, 8) and (−6, −16)
y = 83 x
100. contains (8, −2) and (−4, 1)
y = −x + 6
103. passes through (1, 6) and (−2, −9)
y = 5x + 1
y = −2x
y = 27 x
y = − 14 x
102. passes through (−2, 2) and (2, −10)
y = −3x − 4
104. passes through (−3, −4) and (5, 12)
y = 2x + 2
105. contains (−1, 7) and (3, −9)
y = −4x + 3
106. contains (3, −4) and (−5, −12)
107. contains (−5, 8) and (−3, 0)
y = −4x − 12
108. contains (−1, 16) and (2, −5)
109. contains (−5, −3) and (10, 0)
111. contains (3, −5) and (−6, −2)
110. contains (8, −3) and (−2, 2)
y = 15 x − 2
112. contains (−12, 0) and (4, 4)
y = − 13 x − 4
113. passes through (6, −5) and (−2, 7)
y = − 32 x + 4
y =x−7
y = −7x + 9
y = − 12 x + 1
y = 14 x + 3
114. passes through (12, 7) and (−6, −8)
115. passes through (−4, −4) and (8, 11)
y = 54 x + 1
116. passes through (6, −9) and (−9, 1)
117. contains (−4, −5) and (−8, −16)
11
4 x+6
118. contains (−4, 1) and (8, −8)
119. contains (−10, 3) and (5, 12)
y=
121. passes through (12, 2) and (3, 1)
y = 19 x +
123. passes through (−3, 1) and (9, −1)
y = − 16 x +
125. contains (2, −2) and (−6, 1)
y = − 38 x −
127. contains (−1, −5) and (3, 9)
y = 72 x −
3
2
5
4
y = − 23 x − 5
y = − 76 x + 6
122. passes through (−5, 3) and (11, −5)
2
3
1
2
124. passes through (8, 6) and (−4, 2)
126. contains (8, 1) and (−7, −5)
y=
128. contains (12, −10) and (−4, 0)
ALG catalog ver. 2.6 – page 322 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
y = 56 x − 3
y = − 34 x − 2
120. contains (−6, 13) and (6, −1)
y = 35 x + 9
y = −3
92. contains (4, −4) and (−7, −4)
93. contains (−1, −3) and (2, 6)
101. passes through (2, 4) and (−1, 7)
x=5
y = − 12 x +
y = 13 x +
6
33
15 x − 15
y = − 58 x −
5
2
10
3
1
2
PG
Topic:
Mixed practice and review (slope, midpoint, distance, equations of lines).
Directions:
0—(No explicit directions.)
1.
What is the slope of the y-axis?
3.
What is the slope of all horizontal lines?
5.
Find the slope between (−2, 7) and (4, −3).
7.
Find the slope between (−5, −4) and (7, −1).
9.
What is the slope between (−4, −7) and (10, 0) ?
undef.
0
− 53
1
4
1
2
2.
What is the slope of the x-axis?
4.
What is the slope of all vertical lines?
6.
Find the slope between (1, 6) and (−8, 12).
− 23
8.
Find the slope between (6, 5) and (−6, 14).
− 34
0
undef.
10. What is the slope between (11, −5) and (−5, 11) ?
−1
11. What is the slope between (0, 8) and (8, −2) ?
− 54
12. What is the slope between (−3, −4) and (−1, 4) ?
4
13. Find the slope of the line which passes through (2, 9)
and (7, 4). −1
14. Find the slope of the line which passes through
(−3, −8) and (0, −2). 2
15. What is the slope of the line which contains (14, −6)
and (4, −4) ? − 1
16. What is the slope of the line which contains (−15, 5)
and (12, 2) ? − 1
17. A line contains the points (−12, 5) and (−8, 5).
What is the slope of the line? 0
18. A line contains the points (6, −1) and (6, 4). What
is the slope of the line? undef.
19. A line passes through (9, −2) and (6, −2). What is
the slope of the line? 0
20. A line passes through (−7, 0) and (−7, 10). What is
the slope of the line? undef.
21. Find the slope of the line which contains ( 32 , −4) and
passes through the origin. −6
22. Find the slope of the line which contains (−3, 1 12 )
and passes through the origin. − 1
23. Find the slope of the line which contains (−7, −2.1)
and passes through the origin. 3
24. Find the slope of the line which contains (2.4, 4) and
passes through the origin. 5
5
9
2
10
3
25. Find the slope of the line which contains (5, −2) and
whose y-intercept is 8. −2
26. Find the slope of the line which contains (8, 3) and
whose y-intercept is −7. 5
27. Find the slope of the line which contains (1, −3) and
whose x-intercept is 4. 1
28. Find the slope of the line which contains (−6, 6) and
whose x-intercept is −2. − 3
29. Find the slope of the line whose y-intercept is −5
and x-intercept is −7. − 5
30. Find the slope of the line whose y-intercept is 8 and
x-intercept is −6. 4
31. Find the slope of the line whose y-intercept is 4 and
x-intercept is 10. − 2
32. Find the slope of the line whose y-intercept is −9
and x-intercept is 3. 3
33. Find the slope of the line whose y-intercept is 7.5
and x-intercept is −1.5. 5
34. Find the slope of the line whose y-intercept is 8 and
x-intercept is 3.2. − 5
4
2
7
5
35. Find the slope of the line whose y-intercept is
x-intercept is − 32 . 4
2
3
9
37. Find the slope of the line: x − y = 8.
2
36. Find the slope of the line whose y-intercept is
x-intercept is 18 . −2
38. Find the slope of the line: x + y = 0.
1
39. Find the slope of the line: −x + y = −3.
and
3
40. Find the slope of the line: −x − y = 0.
1
−1
−1
41. What is the the slope of the line y = − 25 x ?
− 25
42. What is the the slope of the line y = 3x ?
3
43. What is the the slope of the line y = −5x ?
−5
44. What is the the slope of the line y = 21 x ?
1
2
45. Find the slope of the line: −5x + y = 1.
47. Find the slope of the line: −x − 3y = −6.
5
− 13
46. Find the slope of the line: 2x − y = 0.
2
48. Find the slope of the line: x + 7y = 5.
− 17
ALG catalog ver. 2.6 – page 323 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
1
4
and
PG
49. What is the the slope of the line x = −2 ?
51. What is the the slope of the line x = 5 ?
53. Find the slope of the line: 9x + 6y = 0.
undef.
undef.
− 32
55. Find the slope of the line: −8x + 2y = 14.
4
50. What is the the slope of the line y = 8 ?
0
52. What is the the slope of the line y = −1 ?
0
54. Find the slope of the line: −5x − 15y = 3.
− 13
56. Find the slope of the line: 6x − 10y = −1.
3
5
57. Given M (3x, 8) and N (−6, −x). Find the value of x,
←−→
if the slope of MN is 23 . 4
58. Given B(x, 2x) and C(6, 0). Find the value of x, if
←→
the slope of BC is −6. 9
59. Given H(−1, 0) and J(x, x). What is the value of x,
←
→
if the slope of HJ is − 23 ? − 2
60. Given A(−5, x) and D(2x, 5x). What is the value
←→
of x, if the slope of AD is 3? 2
61. A line contains the points (x − 4, x) and (2, x + 5),
and has a slope of −10. What is the value of x ?
62. A line contains the points (x − 4, −4) and (2x, 2),
and has a slope of 32 . What is the value of x ? 0
5
2
13
2
64. Line PQ contains the points (x, x + 3) and (1, x), and
63. Line MN contains the points (0, 2x + 1) and
(3x, x + 5), and has a slope of 5. Find the value of x.
has a slope of − 13 . Find the value of x. −8
1
4
65. Find the midpoint between (3, 12) and (−7, 2)
66. Find the midpoint between (5, −8) and (−9, −8)
(−2, −8)
(−2, 7)
67. Find the midpoint between (9, 0) and (11, −10)
(10, −5)
68. Find the midpoint between (−2, −3) and (16, 7)
(7, 2)
69. Given E(−3, 9) and F (−8, 0). Find the midpoint
−−−
of EF . (− 11 , 9 )
70. Given K(5, 6) and N (10, −6). Find the midpoint
−−−
of KN . ( 15 , 0)
−−−
71. Find the midpoint of GH , if G = (15, −3) and
H = (4, 4). ( 19 , 1 )
−−−
72. Find the midpoint of AB, if A = (8, 6) and
B = (−5, 11). ( 3 , 17 )
−−−
73. What is the midpoint of RS , if R = (−4, −1) and
S = (4, 6). (0, 5 )
−−−
74. What is the midpoint of PQ, if P = (2, 2) and
Q = (−3, 14). (− 1 , 8)
75. Given Q(−5, −2) and R(7, −9). What is the
−−−
midpoint of QR ? (1, − 11 )
76. Given E(9, −7) and H(0, −7). What is the midpoint
−−−
of EH ? ( 9 , −7)
77. What point is halfway between the origin and
(−5, 2.2) ? (−2.5, 1.1)
78. What point is halfway between the origin and
(4, − 23 ) ? (2, − 1 )
79. What point is halfway between the origin and ( 51 , 1) ?
80. What point is halfway between the origin and
(−1.6, −0.4) ? (−0.8, −0.2)
2
2
2
2
2
2
1 1
( 10
, 2)
81. Find the midpoint between (3 12 , 13 ) and (−3 12 , 23 ).
(0,
1
2)
83. Find the midpoint between (2, −2) and (−0.4, −0.6).
(0.8, −1.3)
2
2
2
2
2
3
82. Find the midpoint between (−0.5, 3.9) and
(1.7, −0.1). (0.6, 1.9)
84. Find the midpoint between (− 15 , 6) and (2 15 , −5).
(1, 12 )
85. A segment has endpoints at (a, a − 1) and (3a, a + 1).
What is the midpoint? (2a, a)
86. A segment has endpoints at (n − 6, 5p) and
(6 − n, −p). What is the midpoint? (0, 2p)
87. A segment has endpoints at (k, k + 4) and
(8 − k, k + 2). What is the midpoint? (4, k + 3)
88. A segment has endpoints at (x − 2, −7y) and
(x + 2, 3y). What is the midpoint? (x, −2y)
89. Given X(4, −2) and Q(7, −6). Find a point Y such
−−−
that Q is the midpoint of XY . (10, −10)
90. Given P (9, 3) and X(−5, −4). Find a point Q such
−−−
that X is the midpoint of PQ. (−19, −11)
ALG catalog ver. 2.6 – page 324 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
PG
91. Given A(8, −1) and M (0, 6). Find a point B such
−−−
that M is the midpoint of AB. (−8, 13)
92. Given Y (−10, 2) and Z(−3, −7). Find a point X
−−−
such that Y is the midpoint of XZ . (4, −16)
93. A segment has endpoints at (2x, 0) and (7, y + 5).
Solve for x and y if the midpoint is (−3, −1).
94. A segment has endpoints at (x − 8, 3y) and (−6, y).
Solve for x and y if the midpoint is (−4, 4).
x = 6, y = 2
x = − 13
2 , y = −7
95. A segment has endpoints at (x + 1, y + 3) and
(x, −3). Solve for x and y if the midpoint is (5, −2).
96. A segment has endpoints at (−x, 5) and (5x, 2y − 1).
Solve for x and y if the midpoint is (8, 0).
x = 4, y = −2
x = 29 , y = −4
97. Find the distance between (9, −1) and (−3, 4).
13
99. Find the distance between (−8, −5) and (0, 10).
17
101. Given A(0, 3) and B(−10, −7). Find the length
−−−
of AB. 10√2
−−−
103. Given P (5, −3) and Q(3, 1). Find the length of PQ.
√
2 5
98. Find the distance between (−2, 5) and (−5, 9).
5
100. Find the distance between (−1, 2) and (7, −4).
10
102. Given C(−3, −1) and D(−10, 0). Find the length
−−−
of CD. 5√2
104. Given M (−2, 5) and N (−10, 9). Find the length
−−−−
of MN . 4√5
−−−
105. What is the length of RT , if R = (2, −2) and
T = (8, 2) ? 2√13
−−−
106. What is the length of AD, if A = (−4, −6) and
D = (6, −4) ? 2√26
−−−
107. What is the length of NK , if N = (7, 4) and
√
K = (4, −5) ? 3 10
−−−
108. What is the length of EH , if E = (9, −6) and
√
H = (−3, 0) ? 6 5
109. Given A(−3, 6), B(2, 1) and C(4, −5). Find the
−−−
distance between A and the midpoint of BC . 10
110. Given A(1, −9), B(−6, 0) and C(−2, 6). Find the
−−−
distance between A and the midpoint of BC . 13
111. Given D(−1, −3), E(−2, −7) and F (8, 7). Find the
−−−
distance between D and the midpoint of EF . 5
112. Given D(2, −11), E(−3, 9) and F (−9, −1). Find the
−−−
distance between D and the midpoint of EF . 17
113. Given X(0, 5), Y (−4, −3) and Z(10, −1). Find the
−−−
−−−
distance between the midpoints of XY and XZ .
114. Given X(−2, −5), Y (−8, 5) and Z(2, −9). Find the
−−−
−−−
distance between the midpoints of XZ and YZ . √34
√
5 2
115. Given P (−7, 8), Q(3, −2) and R(−1, 6). Find the
−−−
−−−
distance between the midpoints of PQ and RQ.
√
10
116. Given P (5, 4), Q(−3, 0) and R(−11, −6). Find the
−−−
−−−
distance between the midpoints of PR and QR.
√
2 5
117. What is the slope and y-intercept for the line:
y = 2x + 25 ? 2 and 2
118. What is the slope and y-intercept for the line:
y = 21 x − 32 ? 1 and − 3
119. What is the slope and y-intercept for the line:
y = −5x − 7 ? −5 and −7
120. What is the slope and y-intercept for the line:
y = −4x + 12 ? −4 and 1
121. What is the slope and y-intercept for the line:
9y + 6x − 9 = 0 ? − 2 and 1
122. What is the slope and y-intercept for the line:
−10x − 5y + 20 = 0 ? −2 and 4
123. What is the slope and y-intercept for the line:
−2x − 8y = −4 ? − 1 and 1
124. What is the slope and y-intercept for the line:
−6x + 15y = 25 ? 2 and 5
125. What are the x- and y-intercepts of the line:
10x − 4y = −20 ? −2 and 5
126. What are the x- and y-intercepts of the line:
−12x + 6y = 12 ? −1 and 2
127. What are the x- and y-intercepts of the line:
−9x − 12y = 36 ? −4 and −3
128. What are the x- and y-intercepts of the line:
15x − 10y = 30 ? 2 and −3
129. What are the x- and y-intercepts of the line:
4y = 21x − 14 ? 2 and − 7
130. What are the x- and y-intercepts of the line:
10x − 9y − 24 = 0 ? 12 and − 8
5
3
4
3
2
2
2
2
2
5
ALG catalog ver. 2.6 – page 325 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
3
5
3
PG
131. What are the x- and y-intercepts of the line:
12x + 8y = −6 ? − 1 and − 3
2
133. Graph: x = −5.
132. What are the x- and y-intercepts of the line:
−5x + 16y − 6 = 0 ? − 6 and 3
4
134. Graph: y = 2.
5
[graph]
135. Graph: x = 4.
8
136. Graph: y = −7.
[graph]
[graph]
[graph]
138. Graph: y = − 73 x.
137. Graph: y = 6x.
[graph]
[graph]
[graph]
141. Graph: y = − 25 x − 1.
142. Graph: y = 4x − 12 .
[graph]
[graph]
[graph]
[graph]
143. Graph: y = −3x + 23 .
144. Graph: y = 21 x + 5.
[graph]
148. Graph:
−12x − 4y = 16.
147. Graph:
−10x + 8y + 8 = 0.
[graph]
149. Graph:
(y + 4) = −6(x − 2)
150. Graph:
(y − 2) = − 32 (x + 6)
[graph]
153. Graph:
[graph]
[graph]
146. Graph: x + 3y = 12.
145. Graph:
9x − 3y + 6 = 0.
140. Graph: y = 41 x.
139. Graph: y = −3x.
154. Graph:
[graph]
151. Graph:
(y + 1) = 5(x − 1)
152. Graph:
(y + 6) = 14 (x + 8)
[graph]
[graph]
y−1
2
=
x
3
[graph]
y+2
=4
x−4
155. Graph:
[graph]
y
1
=−
x+4
2
[graph]
[graph]
157. Solve for y and graph: −4x + 6y = −18.
y = 23 x − 3
159. Solve for y and graph: −10x − 2y = 10.
y = −5x − 5
y−3
= −3
x+5
156. Graph:
[graph]
158. Solve for y and graph: 3x − 5y = −20.
160. Solve for y and graph: 7x + 7y = 14.
y = 35 x + 4
y = −x + 2
161. Write in slope-intercept form and graph: x + 5y = 10. 162. Write in slope-intercept form and graph:
−4x − 2y = 1. y = −2x − 1
y = −1x + 2
5
2
163. Write in slope-intercept form and graph:
7x − 3y = −3. y = 7 x + 1
164. Write in slope-intercept form and graph:
−2x + 6y = −4. y = 1 x − 2
165. Write the equation of the vertical line through
(−5, 9). x = −5
166. Write the equation of the vertical line through
(7, −1). x = 7
167. Write the equation of the horizontal line through
(−8, −3). y = −3
168. Write the equation of the horizontal line through
(4, 6). y = 6
169. Write the equation of the line which passes through
the origin and contains (−3, −5). y = 5 x
170. Write the equation of the line which passes through
the origin and contains (4, −6). y = − 3 x
171. Write the equation of the line which passes through
the origin and contains (10, 2). y = 1 x
172. Write the equation of the line which passes through
the origin and contains (−7, 7). y = −x
173. Write the equation of the line which has a y-intercept
of −4 and slope of 53 . y = 5 x − 4
174. Write the equation of the line which has a y-intercept
of 21 and slope of − 32 . y = − 3 x + 1
175. Write the equation of the line which has a y-intercept
of 1 and slope of −2. y = −2x + 1
176. Write the equation of the line which has a y-intercept
of − 13 and slope of 3. y = 3x − 1
177. Write the equation of the line which contains (4, −10)
and has a slope of −3. y = −3x + 2
178. Write the equation of the line which contains
(−2, −12) and has a slope of 6. y = 6x
179. Write the equation of the line which contains (7, 1)
and has a slope of 14 . y = 1 x − 3
180. Write the equation of the line which contains (10, −6)
and has a slope of − 25 . y = − 2 x − 2
181. Write the equation of the line which contains (3, 11)
and has a slope of 37 . y = 7 x + 4
182. Write the equation of the line which contains
(−1, −5) and has a slope of 4. y = 4x − 1
3
3
5
3
4
3
4
3
3
2
2
2
3
5
ALG catalog ver. 2.6 – page 326 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
PG
183. Write the equation of the line which contains (−8, 4)
and has a slope of − 12 . y = − 1 x
184. Write the equation of the line which contains (6, 0)
and has a slope of − 53 . y = − 5 x + 10
185. Write the equation of the line which contains (−3, 6)
and (4, −8). y = −2x
186. Write the equation of the line which contains (3, −1)
and (−5, 5). y = − 3 x + 5
187. Write the equation of the line which contains (12, −8)
and (6, −9). y = 1 x − 10
188. Write the equation of the line which contains
(−2, −7) and (2, 13). y = 5x + 3
189. Write the equation of the line which contains
(−1, −1) and (−5, 6). y = − 7 x − 11
190. Write the equation of the line which contains (8, 0)
and (−8, 2). y = − 1 x + 1
191. Write the equation of the line which contains (1, 10)
and (−2, −2). y = 4x + 6
192. Write the equation of the line which contains (7, 3)
and (14, 6). y = 3 x
193. Write the equation of the line which contains (−3, 1)
and whose y-intercept is −8. y = −3x − 8
194. Write the equation of the line which contains
(−6, −2) and whose y-intercept is 1. y = 1 x + 1
195. Write the equation of the line which contains (8, 4)
and whose y-intercept is −4. y = x − 4
196. Write the equation of the line which contains (14, −1)
and whose y-intercept is 3. y = − 2 x + 3
197. Write the equation of the line which contains (10, 11)
and whose x-intercept is 5. y = 11 x − 11
198. Write the equation of the line which contains (2, −15)
and whose x-intercept is −1. y = −5x − 5
199. Write the equation of the line which contains (−10, 3)
and whose x-intercept is 20. y = − 1 x + 2
200. Write the equation of the line which contains
(−9, −2) and whose x-intercept is −7. y = x + 7
201. Write the equation of the line whose y-intercept is 2
and x-intercept is −6. y = 1 x + 2
202. Write the equation of the line whose y-intercept is 5
and x-intercept is −5. y = x + 5
203. Write the equation of the line whose y-intercept is 8
and x-intercept is 4. y = −2x + 8
204. Write the equation of the line whose y-intercept
is −7 and x-intercept is −3. y = − 7 x − 7
205. Write the equation of the line which has an
x-intercept of 6 and slope of 32 . y = 3 x − 9
206. Write the equation of the line which has an
x-intercept of −2 and slope of −6. y = −6x − 12
207. Write the equation of the line which has an
x-intercept of 15 and slope of − 45 . y = − 4 x + 12
208. Write the equation of the line which has an
x-intercept of −9 and slope of 31 . y = 1 x + 3
209. Given D(1, 7), E(−3, 3) and F (0, 4). Write the
equation of the line which passes through F and the
−−−
midpoint of DE . y = 2x + 4
210. Given A(−4, 5), B(4, −9) and C(3, −2). Write the
equation of the line which passes through C and the
−−−
midpoint of AB. y = −2
211. Given P (−2, 3), Q(3, −6) and R(6, −5). Write the
equation of the line which passes through Q and the
−−−
midpoint of PR. y = −5x + 9
212. Given K(0, 5), M (−1, −2) and N (7, 10). Write the
equation of the line which passes through K and the
−−−−
midpoint of MN . y = − 1 x + 5
2
6
4
4
5
10
3
2
5
3
4
4
8
7
2
7
3
3
ALG catalog ver. 2.6 – page 327 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
3
PH
Topic:
Parallel and perpendicular lines.
Directions:
0—(No explicit directions.)
1.
What is the slope of all lines parallel to the x-axis?
2.
0
What is the slope of all lines parallel to the y-axis?
undef.
3.
What is the slope of all lines perpendicular to the
x-axis? undef.
4.
What is the slope of all lines perpendicular to the
y-axis? 0
5.
What is the slope of all lines parallel to the line
x = 5? undef.
6.
What is the slope of all lines parallel to the line
y = −2? 0
7.
What is the slope of all lines perpendicular to the
line x = −3? 0
8.
What is the slope of all lines perpendicular to the
line y = 7? undef.
9.
What is the slope of all lines parallel to y = 5x − 2 ?
10. What is the slope of all lines parallel to
y = − 43 x + 13 ? − 4
5
3
11. What is the slope of all lines parallel to y = x +
3
2
?
1
12. What is the slope of all lines parallel to y = − 12 x − 4 ?
− 12
13. What is the slope of all lines perpendicular to
y = − 52 x ? 2
14. What is the slope of all lines perpendicular to
y = x + 3 ? −1
15. What is the slope of all lines perpendicular to
y = 41 x − 74 ? −4
16. What is the slope of all lines perpendicular to
y = −4x − 1 ? 1
17. What is the slope of all lines parallel to 8x − 2y = 5 ?
18. What is the slope of all lines parallel to
−5x + 6y = 0 ? 5
5
4
4
6
19. What is the slope of all lines parallel to
−10x − 8y = 2 ? − 5
20. What is the slope of all lines parallel to x + 4y = −6 ?
21. What is the slope of all lines perpendicular to
−3x − y = 9 ? 1
22. What is the slope of all lines perpendicular to
x − 5y = −10 ? −5
4
3
23. What is the slope of all lines perpendicular to
4x + 14y = 0 ? 7
2
− 14
24. What is the slope of all lines perpendicular to
−15x + 9y = 3 ? − 3
5
25. A line contains the points P (−3, 7) and Q(2, −3).
←→
What is the slope of all lines parallel to PQ ? −2
26. A line contains the points N (0, −2) and K(6, 12).
←→
What is the slope of all lines parallel to NK ? 7
27. Line AB contains (4, −10) and (−5, −1). What is
the slope of all lines parallel to line AB ? −1
28. Line CD contains (−1, 8) and (3, 6). What is the
slope of all lines parallel to line CD ? − 1
29. Line EF contains (0, 4) and (6, 0). What is the slope
of all lines perpendicular to line EF ? 3
30. Line QR contains (2, −4) and (−3, 1). What is the
slope of all lines perpendicular to line QR ? 1
2
31. A line contains the points M (9, 6) and N (3, −2).
←−→
What is the slope of all lines perpendicular to MN ?
− 34
3
2
32. A line contains the points B(−4, −2) and C(5, −5).
←→
What is the slope of all lines perpendicular to BC ?
3
33. Write the equation of the line that contains (2, 4)
and is parallel to the x-axis. y = 4
34. Write the equation of the line that contains (−5, 8)
and is parallel to the x-axis. y = 8
35. Write the equation of the line that contains (1, −10)
and is parallel to the y-axis. x = 1
36. Write the equation of the line that contains (−12, 5)
and is parallel to the y-axis. x = −12
ALG catalog ver. 2.6 – page 328 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
PH
37. Write the equation of the line that contains (−1, 7)
and is parallel to the line y = −2x − 5. y = −2x + 5
38. Write the equation of the line that contains (2, 9)
and is parallel to the line y = 3x. y = 3x + 3
39. Write the equation of the line that contains (8, 0)
and is parallel to the line y = 43 x + 1. y = 3 x − 6
40. Write the equation of the line that contains (−4, −3)
and is parallel to the line y = − 12 x. y = − 1 x − 5
41. Write the equation of the line that contains (−5, 1)
and is perpendicular to the line y = 52 x + 2.
42. Write the equation of the line that contains (−12, 5)
and is perpendicular to the line y = 3x + 6.
4
2
y = − 13 x + 1
y = − 25 x − 1
43. Write the equation of the line that contains (4, −3)
and is perpendicular to the line y = −4x. y = 1 x − 4
4
44. Write the equation of the line that contains (−2, 4)
and is perpendicular to the line y = − 13 x + 23 .
y = 3x + 10
45. Given A(5, 2), B(−1, 4) and C(6, −5). Write the
equation of the line which passes through C and is
←→
parallel to AB. y = − 1 x − 3
46. Given A(0, −3), B(3, 8) and C(−2, 5). Write the
equation of the line which passes through B and is
←→
parallel to AC . y = −4x + 20
47. Given D(8, 0), E(6, −3) and F (−2, 3). Write the
equation of the line which passes through D and is
←
→
parallel to EF . y = − 3 x + 6
48. Given D(−4, 2), E(−2, 4) and F (−3, −3). Write the
equation of the line which passes through F and is
←→
parallel to DE . y = x
49. Given A(−6, 0), B(−2, 2) and C(5, 9). Write the
equation of the line which passes through A and is
←→
perpendicular to BC . y = −x + 6
50. Given P (−3, −4), Q(−8, −3) and R(−1, 4). Write
the equation of the line which passes through Q and
←
→
is perpendicular to PR. y = − 1 x − 5
51. Given A(7, −1), B(2, 2) and C(−8, 4). Write the
equation of the line which passes through B and is
←→
perpendicular to AC . y = 3x
52. Given P (9, −9), Q(−5, 1) and R(−3, 4). Write the
equation of the line which passes through P and is
←→
perpendicular to QR. y = − 2 x − 3
53. Given A(9, −1) and B(3, −4). Write the equation of
←→
the line which is perpendicular to AB and contains
−−−
the midpoint of AB. y = −3x + 15
54. Given R(−3, −4) and S(5, 4). Write the equation of
←
→
the line which is perpendicular to RS and contains
−−−
the midpoint of RS . y = −x + 1
55. Given Q(1, −11) and R(7, 1). Write the equation of
←→
the line which is perpendicular to QR and contains
−−−
the midpoint of QR. y = − 1 x − 3
56. Given E(−7, 4) and F (1, −8). Write the equation of
←
→
the line which is perpendicular to EF and contains
−−−
the midpoint of EF . y = 2 x
3
4
2
4
3
ALG catalog ver. 2.6 – page 329 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
3
PI
Topic:
Graphing inequalities in two variables.
Directions:
85—Graph. 31—Solve by graphing. 86—Graph on the same coordinate system.
87—Graph on separate coordinate systems. 88—Graph the solution.
1.
y≥x
[graph]
2.
y > −x
[graph]
3.
x+y ≤0
5.
y<3
[graph]
6.
y ≤ −2
[graph]
7.
y > −5
9.
x ≤ −4
[graph]
10. x < 7
13. y < 13 x
[graph]
14. y ≤ − 52 x
17. y > −x + 6
21. y < 3x + 2
[graph]
[graph]
18. y ≥ x + 1
[graph]
22. y ≤ 6x − 1
[graph]
26. y > − 14 x + 3
29. y ≥ − 23 x −
[graph]
30. y > 74 x −
33. y < −0.3x − 1
[graph]
[graph]
1
4
[graph]
[graph]
34. y ≥ 0.5x + 3
[graph]
[graph]
15. y > − 12 x
[graph]
25. y ≤ − 47 x − 4
7
6
11. x ≥ 1
[graph]
[graph]
[graph]
[graph]
19. y < x − 2
27. y ≥ 15 x + 1
31. y < − 16 x +
[graph]
[graph]
2
3
[graph]
16. y ≥ 37 x
[graph]
[graph]
[graph]
24. y ≥ −2x + 2
[graph]
28. y < 65 x − 5
[graph]
32. y ≤ 14 x +
[graph]
1
2
[graph]
40. y − x + 4 < 0
[graph]
44. −8x + 2y > 0
[graph]
39. y + x − 1 ≤ 0
41. −10y − 4x ≤ 0
[graph]
42. 15y + 5x > 0
[graph]
43. 9x − 6y ≥ 0
[graph]
47. 6y − 12x < −6
[graph]
[graph]
[graph]
12. x > −6
[graph]
[graph]
49. x − 2y < 4
[graph]
36. y > 0.75x − 4
38. x − y − 5 < 0
46. −10x + 2y ≤ −8
y≥0
[graph]
[graph]
[graph]
8.
[graph]
35. y ≤ 0.25x + 2
37. −x − y + 2 ≤ 0
45. 9x − 3y > 12
x−y<0
20. y ≤ −x − 3
[graph]
23. y > −4x − 3
4.
48. −5y − 20x ≥ 20
[graph]
50. −8y + 6x ≥ −16
51. 5y − 2x > −5
[graph]
52. 6x + 4y ≤ 16
[graph]
[graph]
53. 4y + x > −24
[graph]
54. −6x − 9y ≤ 18
[graph]
55. −4y + 15x < 28
56. 15x − 10y ≥ −30
[graph]
57. 6x + 5y − 30 ≤ 0
[graph]
61. 5x − 5y + 20 ≥ 0
[graph]
65. −6y + 10x − 6 > 0
[graph]
69. 12y + 6x + 15 > 0
[graph]
58. 4y + 9x + 36 > 0
59. x − 3y − 9 ≥ 0
[graph]
[graph]
[graph]
62. 2y + 12x + 12 > 0
[graph]
66. 4x − 14y − 28 ≥ 0
[graph]
70. x − 4y + 10 ≤ 0
[graph]
60. −3y − 2x + 6 > 0
[graph]
63. 4y − 16x − 16 < 0
[graph]
67. x + 6y + 24 ≤ 0
[graph]
71. 4x − 20y − 16 < 0
[graph]
ALG catalog ver. 2.6 – page 330 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
64. −9x − 3y − 18 ≤ 0
[graph]
68. 12y + 8x − 12 < 0
[graph]
72. −4y − 10x + 1 ≥ 0
[graph]
PJ
Topic:
Graphing the intersection of inequalities.
Directions:
85—Graph.
1.
−3 < x < 5
5.
y ≤ −4 and y ≤ 0
31—Solve by graphing.
[graph]
2.
0≤y≤4
6.
x > 2 and x ≥ −3
[graph]
9.
Ø
13. 0 ≤ y < 4 and x ≤ 3
−6 ≤ x < −2
7.
x < 5 and y ≥ 2
[graph]
11. y ≥ 0 and y ≤ −4
18. y ≤ 4 and y < −x
[graph]
y < −1 and x ≥ 6
[graph]
24. x > 4 and y ≤ x − 4
[graph]
[graph]
[graph]
[graph]
26. x > 1, y ≤ 2 and x − y ≥ 4
[graph]
[graph]
28. x ≤ −2, y > 1 and x + 2y ≥ 2
[graph]
30. x − 3y < 0 and y ≥ −x + 4
[graph]
31. y ≤ −x and 2x − 5y > −15
[graph]
32. 5y < 2x and 2x + y ≥ 10
33. 2x + 3y < 12 and y ≥ x − 6
[graph]
34. 4y − 7x ≤ 8 and 2x + y < 2
35. 3x + y > −4 and x − 3y > 4
41. 6x + 5y ≤ 35 and 3x − y > 14
45. y < −2x and y > −2x + 3
47. 3x + y > 4 and y ≤ −3x
[graph]
[graph]
[graph]
44. 5x − 3y < 21 and 2x + y ≤ 7
[graph]
46. y ≥ x and y < x − 5
[graph]
51. y ≥ 0, x + 6y − 5 < 0 and x − 4y + 5 > 0
[graph]
[graph]
[graph]
[graph]
[graph]
Ø
48. 4x − 2y ≤ 0 and 2x − y > 2
Ø
49. x < 5, 10y − x ≤ 25 and 3x + 2y ≥ 5
[graph]
42. 2x + y + 5 > 0 and x − 2y + 10 > 0
Ø
55. x < 2, 3 ≤ y < 7 and y ≤ 2x + 9
[graph]
40. 2x + y + 1 ≤ 0 and 2x + y − 2 < 0
[graph]
53. −3 ≥ x < 3, y > −2 and x + y ≤ 4
[graph]
38. 3y ≤ −2x and 2x + 3y > −12
[graph]
[graph]
43. 4x − y ≤ −6 and 3x + 2y ≥ −10
[graph]
36. 2y − x ≤ 12 and x + 4y < −12
[graph]
37. x − 2y − 10 < 0 and x − 2y + 4 ≥ 0
39. y − x ≥ 0 and x − y + 3 < 0
[graph]
[graph]
22. x ≤ 0 and y > 23 x + 1
[graph]
29. y ≤ 2x and 2y − x > −12
8.
[graph]
12. x > 5 and x ≤ 0
20. y ≤ 0 and y ≥ −3x
[graph]
27. y ≥ −1, x ≥ 0 and x + y < 3
1<y≤6
16. −4 < x ≤ 0 and −1 ≥ y < 2
[graph]
25. y < 3, x ≥ −2 and y > 4x − 1
4.
[graph]
14. y ≥ 0 and −5 < x ≤ 1
15. 1 < x < 5 and −2 ≤ y ≤ 0
23. y < −2 and y ≤ −4x + 2
3.
Ø
[graph]
21. y ≤ 2 and y < −x − 1
89—Graph the intersection.
[graph]
10. y > 5 and y ≤ −1
Ø
19. x ≤ −3 and y > x
[graph]
[graph]
x ≥ 4 and x < −2
17. x > 0 and y ≤ − 12 x
88—Graph the solution.
Ø
50. x ≥ 0, x + y ≤ 7 and 4x − 5y < 10
[graph]
52. y < 2, 3x − 2y < 4 and 3x + 2y > −4
[graph]
54. −4 < x ≤ 0, 2y ≥ x and x + 2y < 4
[graph]
56. x > −3, y > 2x − 4 and −4 < y ≤ 2
[graph]
ALG catalog ver. 2.6 – page 331 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
Ø
PK
Topic:
Other equations and graphs.
Directions:
85—Graph. 31—Solve by graphing. 86—Graph on the same coordinate system.
87—Graph on separate coordinate systems. 88—Graph the solution.
1.
y = |x|
5.
y = |x| + 5
9.
x = |y + 2|
2.
x = |y |
[graph]
6.
y = − |x| − 2
[graph]
10. y = |x − 4|
[graph]
13. x = |y − 1| − 4
17. x = − |y + 6|
21. x = −5 |y |
[graph]
[graph]
25. y = 4 |x + 2|
x
29. y = − 6
33. y =
[graph]
[graph]
[graph]
|5 − x|
10
[graph]
37. x = 4 |y + 2| + 3
[graph]
[graph]
[graph]
[graph]
14. y = |x + 8| + 3
18. y = − |1 − x|
22. y = 6 |x|
[graph]
[graph]
y
30. x = 3
34. x =
[graph]
[graph]
|y + 3|
−2
[graph]
38. y = −2 |x − 5| − 7
[graph]
41. y = −x2
[graph]
45. x = y 2 − 4
[graph]
49. x = (y + 2)2
53. y = 3x2
[graph]
[graph]
1
57. x = − (y − 6)2
3
7.
x = |y | − 3
11. x = |5 − y |
4.
x = − |y |
[graph]
8.
x = − |y | + 6
[graph]
12. y = |x + 3|
[graph]
15. y = |x + 7| − 1
19. y = − |x + 8|
[graph]
[graph]
27. y = |5x + 2|
31. y =
|x|
−2
35. y =
|5 − x|
4
[graph]
[graph]
[graph]
[graph]
39. y = 3 |x − 1| + 6
46. x = y 2 + 6
[graph]
50. y = (x − 1)2
−x2
4
[graph]
[graph]
58. x = 5(y + 2)2
[graph]
[graph]
16. x = |y − 4| + 2
20. x = − |5 − y |
24. y = |2x|
[graph]
[graph]
[graph]
28. x = |3 − 4y |
32. x =
|y |
10
36. x =
|y + 6|
−3
[graph]
[graph]
44. x = y 2
[graph]
47. y = x2 + 1
[graph]
51. x = −(y − 5)2
55. x =
[graph]
[graph]
[graph]
43. y = x2
[graph]
[graph]
40. x = − |y + 2| − 4
[graph]
42. x = −y 2
54. y =
y = − |x|
23. x = |−3y |
[graph]
26. x = −2 |1 − y |
3.
y2
2
[graph]
[graph]
48. y = x2 − 10
52. y = −(x + 3)2
56. x = −5y 2
[graph]
59. y = −3(x + 1)2
[graph]
[graph]
60. y =
[graph]
[graph]
1
(x − 4)2
2
[graph]
[graph]
61. y = 4(x − 1)2 − 3
[graph]
65. y =
[graph]
1
x
66. x =
[graph]
[graph]
70. y = −
1
|x|
[graph]
74. x =
77. y = x3
81. x = −y 3 − 4
[graph]
67. y = −
[graph]
10
x
1
|y |
78. y = −x3
[graph]
63. x = 2(y + 6)2 + 1
[graph]
1
y
4
y
69. x = −
73. y =
62. x = −3(y − 5)2 + 8
[graph]
[graph]
82. x = y 3 + 2
[graph]
1
x
71. x =
8
y
75. y =
−1
|x|
79. x = y 3
[graph]
[graph]
64. y = −5(x + 2)2 − 4
68. x = −
[graph]
[graph]
[graph]
72. y =
5
x
76. x =
−1
|x|
[graph]
[graph]
80. x = −y 3
[graph]
83. y = −x3 + 5
1
y
[graph]
[graph]
[graph]
84. y = x3 − 1
[graph]
85. x = (y + 2)3 [graph]
√
89. y = x [graph]
86. y = (x − 1)3 [graph]
√
90. y = − x [graph]
87. y = (x + 1)3 [graph]
√
91. x = y [graph]
88. x = (y − 3)3 [graph]
√
92. x = − y [graph]
93. x2 + y 2 = 25
94. x2 + y 2 = 100
95. x2 + y 2 = 16
96. x2 + y 2 = 1
[graph]
97. 4x2 + y 2 = 4 [graph]
√
101. y = x [graph]
√
105. y = 9 − x2 [graph]
[graph]
98. x2 + 9y 2 = 9 [graph]
√
102. y = − x [graph]
p
106. x = 16 − y 2 [graph]
[graph]
99. 25x2 + y 2 = 25
√
103. x = y [graph]
√
107. y = 100 − x2
ALG catalog ver. 2.6 – page 332 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
[graph]
[graph]
[graph]
100. x2 + 16y 2 = 16 [graph]
√
104. x = − y [graph]
p
108. x = 25 − y 2 [graph]
PL
Topic:
Ordered pairs and equations of lines. See also categories PD (points on a line) and
PF (writing equations of lines).
Directions:
175—Tell whether the points listed in the table are collinear.
If so, write the equation of the line that passes through them.
176—Tell whether the points listed in the table are collinear.
If so, determine the slope of the line that passes through them.
177—Graph the points whose coordinates are shown in the table.
178—Write a rule of correspondence for the set of ordered pairs.
179—State whether the ordered pairs belong to linear function.
1.
x −3 −3 −3 −3 −3
y −5
0
1
5
2.
10
x = −3
4.
x −6 −3
0
3
6
5.
10.
x −4 −2
0
y
0 −3
2
1
8.
6
x −11
−3
1
y
11
0 −11
3
3
11.
x −2
4
6
y −8 −4 −2
2
0
14.
x −9 −5
0
2
y −1
8
10
3
17.
x −8 −4
0
12
y
4
7
2
3
20.
2 −2
2 −2
y
4
8
6
23.
10
x −2 −1
4
1
non-colinear, y =
28.
x −1
1
3
5
y −4
4
12
20
6.
1
2
0
1
4
x2
26.
−2
−1
y
10
5
0
y −3 −1
0 −1 −3
1
8
8
8
y −2 −1
0
1
2
x
0
3
6
12
0
1
2
4
1
2
9.
x −4 −1
0
2
−5 −10
y
1
4
x
1
3
6
8
y
2
4
5
7
x −5 −3
6
12.
8
2
2
x −1
0
1
2
y −2
2
3
4
non-colinear, no simple rule of
correspondence
15.
4
4 −1 −3
x −4 −2
y
2
3
5
0 −5 −7
y = −x − 2
x −4 −1
0
y
1 −5
13
1
3x
non-colinear, no simple rule of
correspondence
4
18.
2
x −1
1
2
3
y −7
3
8
13
0 −5
5
0
0
5
y = 5x − 2
x −4 −2
2
4
y −9 −6
0
3
21.
x
y −5
0
not a function
1 −3 −3
x
0
1
y
0
1 −1
24.
9 −9
x −1
3
5 −1
y
5
7 −3
1
not a function
x −2 −1
0
y −4 −1
0 −1 −4
non-colinear, y =
x −3 −1
8
y=
x
8
y
not a function
0
x
x=8
y = 32 x − 3
x
y
5
y = −3x + 1
not a function
25.
5
y = −x + 1
y = 41 x + 4
22.
5
y
y =x+8
19.
5
5
non-colinear, no simple rule of
correspondence
y =x−6
16.
y
3.
y = −5x
− 12 x
non-colinear, no simple rule of
correspondence
13.
4
y = 4x
y = −2 (not a function)
y=
2
y = 5 (not a function)
y −2 −2 −2 −2 −2
7.
x −8 −4 −2
1
−x2
3
non-colinear, y = − |x|
ALG catalog ver. 2.6 – page 333 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
2
27.
x −4 −2
0
2
4
y
0
2
4
4
2
non-colinear, y = |x|
PM
Topic:
Writing systems of inequalities. See also category PI (graphing inequalities in two variables).
Directions:
174—Write a set of equations that describes the shaded region.
1.
2.
SHDREG02.PCX
SHDREG01.PCX
x ≥ −5 and y < 4
x ≥ 3 and y ≥ 2
5.
6.
SHDREG05.PCX
SHDREG03.PCX
SHDREG04.PCX
x > 6 and y ≤ −3
x ≤ 2 and y < 1
7.
8.
SHDREG06.PCX
x ≤ −10 and y ≥ 35 x
y ≤ 8 and y ≥ 43 x
10.
9.
SHDREG07.PCX
SHDREG08.PCX
y > − 32 x and y ≥ x + 5
y ≤ − 34 x and y < 34 x − 6
12.
11.
SHDREG10.PCX
SHDREG09.PCX
y≥
y ≥ 32 x + 4 and y < − 12 x + 4
13.
− 12 x − 4
SHDREG13.PCX
17.
15.
16.
SHDREG15.PCX
SHDREG16.PCX
x ≤ 3, y > −6 and y ≤ 23 x − 4
x ≤ −3, y ≥ 0 and y < x + 7
18.
y ≤ x + 2 and y ≤ − 12 x − 1
y > x − 3 and y > −x + 3
SHDREG14.PCX
− 43 x + 16
SHDREG12.PCX
SHDREG11.PCX
and y < x − 4
14.
x > 6, y ≥ 0 and y ≤
4.
3.
19.
x > −9, y ≤ 5 and y ≥ 23 x + 7
20.
SHDREG17.PCX
y < −2, y ≥
y ≥ −x − 6
4
3x
− 6 and
SHDREG18.PCX
y ≥ 3, y < −x + 7 and
y <x+7
SHDREG19.PCX
y > −7, y ≤ 2x − 5 and
y ≤ −2x − 1
ALG catalog ver. 2.6 – page 334 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
SHDREG20.PCX
x ≤ 5, y ≥ −x + 3 and
y < 12 x + 32
PN
Topic:
Word problems involving graphing. See also category QE (word problems involving two variables).
Directions:
0—(No explicit directions.)
1.
Mr. La Fleur pays $130 to rent a car for one week.
He also pays $5 for every 100 miles that he drives.
Write an equation and draw a graph which shows
the relationship between the total cost of renting the
car and the number of miles that are driven.
2.
y = 5x + 130, (y is cost, x is hundreds of miles)
An artist spends $80 on brushes and paint. For
canvas and frames, she has to spend an additional
$20 per painting. Write an equation and draw a
graph which shows the relationship between the
artist’s total expenses and the number of paintings
that she makes.
y = 20x + 80 (y is cost, x is number of paintings)
3.
An author sells his latest book to a publisher for
$5000, plus $300 for every thousand books that are
sold. Write an equation and draw a graph which
shows the relationship between the author’s income
and the number of books that are sold.
4.
y = 300x + 5000 (y is income, x is thousands of books)
5.
A car from U-Save Rentals costs $22 per day and
5 cents per mile. The same car from Xpress Rentals
costs $28 per day and 3.5 cents per mile. For what
number of miles is the cost of the two cars the same?
y = 50x + 3000 (y is income, x is number of items)
6.
A moving van from Loc-n-Stor costs $40 per day
and 12 12 cents per mile. The same vehicle from
TravelTime costs $32 per day and 15 cents per mile.
For what number of miles is the cost of the two
vehicles the same? 320
8.
Photo World charges $4.00 for the first 5 × 7
enlargement and $2.50 for each additional. Camera
Center charges $3.30 for the first enlargement
and $2.60 for each additional. For how many
enlargements will the cost at each place be the same?
400
7.
Speedy Print Shop charges $1.50 for the first color
copy and $0.60 for each additional. Quality Print
Shop charges $0.90 for the first copy and $0.75 for
each additional. For how many copies will the cost
at each place be the same? 5
At an electronics store, a salesperson gets a monthly
salary of $3000, plus $50 for every stereo system that
she sells. Write an equation and draw a graph which
shows the relationship between the saleperson’s
income and the number of stereos that she sells.
8
9.
A checking account at Northern Bank costs $5 per
month and 20/c per check. At County Bank, the cost
is $2 per month and 50/c per check. Write equations
and draw a graph which shows the cost of each
account in relation to the number of checks that are
written. For what number of checks would the cost
of the two accounts be the same? 10
11. Ms. Swanson drove 300 miles one week in a rental
car. The total cost of renting the car was $120,
which included the weekly rate plus a mileage charge.
Another week she drove 560 miles in the same rental
car, and the total cost was $133.
10. StarCom advertises a cellular phone at a basic rate
of $12 a month, plus 14/
c per minute of usage. The
same phone from ProTel costs $15 a month plus
8/
c per minute. Write equations and draw a graph
which shows the cost of each account in relation to
the number of minutes of usage. After how many
minutes would the cost of the two phones be the
same? 50
12. In January, Edward spent 320 minutes on the
telephone. His phone bill for the month was $75,
which included a monthly charge plus a “per minute”
charge. The next month he spent 250 minutes on
the phone and was billed $61.
a) Determine the charge per mile and the weekly
rate for the car.
a) Determine the “per minute” charge and the
monthly charge for the telephone.
b) Write an equation which relates the total cost
of the rental to the number of miles that are
driven.
b) Write an equation which relates the total cost
of the telephone to the number of minutes of
usage.
c) Graph the equation, then determine the cost of
renting a car if it is driven 800 miles in a week.
c) Graph the equation, then determine the cost of
the telephone if it is used for 90 minutes in a
month.
y = 0.05x + 105 (y is total cost, $0.05 is charge per mile, x is number of
miles, $105 is weekly rate); $145
y = 0.2x + 11 (y is total cost, $0.20 is charge per minute, x is number of
minutes, $11 is monthly rate); $29
ALG catalog ver. 2.6 – page 335 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
PN
13. Assume there is a linear relationship between the
cost of building a house and its size (the number of
square feet of living space). If a 1700 sq ft house
costs $60,000 to build, and a 2300 sq ft house costs
$75,000, write an equation that relates cost to
the square footage. Then graph the equation and
determine the cost of building a 2500 sq ft house.
y = 25x + 17500 (y is cost, x is square feet); $80,000
14. Assume there is a linear relationship between the
cost of running a widget factory and the number
of widgets that are produced each year. If it costs
$30,000 to produce 700 widgets and $40,000 to
produce 1500 widgets, write an equation that relates
cost to the number of widgets. Then graph the
equation and determine the cost of producing 2000
widgets.
y = 12.5x + 21250 (y is cost, x is number of widgets); $46,250
15.
In an experiment, various
objects are suspended from
the ceiling by a spring (see
figure). There turns out to be
a linear relationship between
the length s of the spring and
the weight w of the object.
SPRING01.PCX
The relationship is given by the
equation s = 0.05w + 14, where
s is measured in centimeters and
w in grams. Graph the equation and find the length
of the spring when it holds a 150 gram object. What
is the length of the spring when it is not suspending
any objects? 21.5 cm; 14 cm
17. Tina’s TV & Appliance Store charges a flat rate of
$70 for up to 2 hours of repair work. After that, the
charge is $28 per hour. Fred’s Fixit charges $40 for
the first hour of repair work, and then $23 for each
additional hour. (At both stores, fractions of an
hour count as a whole hour.) For what number of
hours would the repair cost at the two stores be the
same? Draw a graph which illustrates the problem
and explain your answer.
16.
In an experiment, various
objects are held up by a
spring (see figure). There
turns out to be a linear
relationship between the
length s of the spring
SPRING02.PCX
and the weight w of the
object. The relationship
is given by the equation
s = −0.02w + 8, where s is measured in centimeters
and w in kilograms. Graph the equation and find
the length of the spring when it holds a 40 kg object.
What is the length of the spring when it is not
suspending any objects? 7.2 cm; 8 cm
18. Shiny Auto Body charges a flat rate of $240 for up
to 6 hours of repair work. After that, the charge is
$35 per hour. NuBody’s Business charges $180 for
up to 4 hours of repair work and then $40 for each
additional hour. (At both stores, fractions of an
hour count as a whole hour.) For what number of
hours would the repair cost at the two stores be the
same? Draw a graph which illustrates the problem
and explain your answer.
Never the same. Fred’s Fixit always costs less, because of the minimum
charges.
Never the same. For 5 hours or less work, NuBody’s offers a better
deal; otherwise, Shiny is better.
20. The relationship between degrees Celsius and degrees
Fahrenheit is given by the formula: C = 59 (F − 32).
19. The relationship between degrees Fahrenheit
and degrees Celsius is given by the formula:
F = 95 C + 32.
a) Draw a graph which illustrates the relationship.
a) Draw a graph which illustrates the relationship.
◦
b) How many degrees Celsius is 32 ◦ F? 68 ◦ F?
320 ◦ F?
◦
b) How many degrees Fahrenheit is −5 C? 20 C?
100 ◦ C?
c) 100 ◦ C is equivalent to how many degrees
Fahrenheit? 0, 20, 160; 212
◦
c) 32 F is equivalent to how many degrees Celsius?
23, 68, 212; 0
17.
18.
ALG-P001.PCX
19.
20.
ALG-P002.PCX
ALG-P003.PCX
ALG catalog ver. 2.6 – page 336 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
ALG-P004.PCX
QA
Topic:
Solving systems by substitution, graphing. See also categories QB and QC.
Directions:
15—Solve. 16—Solve and check.
26—Solve by any method.
28—Solve by transitivity. 31—Solve by graphing.
77—Find the intersection. 90—Graph each system.
1.
2x = 10
y = −1
(5, −1)
2.
3y − 6 = 0
x+2=0
5.
y = −x
x = −5
(−5, 5)
6.
4y = 4x
y=6
9.
5x + 5y = 0
x−y =0
10.
x − 4y = 0
2y = −2x
13.
x+5=0
2x − 6 = 0
14.
2y − 12 = 0
y−4=0
17.
y = 4x
4x = 8
18.
3y = −6x
y−6=0
21.
3y − 12 = 0
2x + y = 10
22.
− 3y = −15
x − 2y = −1
25.
2x = 6
6x = −2y
26.
− 3x − 12 = 0
2x = −2y
(0, 0)
Ø
(2, 8)
(3, 4)
(3, −9)
4.
− 2y = −8
x=0
8.
x+y =0
3x = 9
12.
y = −2x
2y = −x
(0, 0)
Ø
16.
x+4=1
x+1=4
Ø
(−1, −3)
20.
5x + y = 0
2y = 20
2x = 8
x + 3y = 25
(4, 7)
24.
− 2x + 10 = 0
2x − 2y = 6
27.
5y + 30 = 0
− 4x = 2y
(3, −6)
28.
2y = 12
− 4x = −6y
31.
y = 2x + 2
y+3=0
(− 52 , −3)
32.
y = 6x − 3
x − 12 = 0
35.
y = −4x − 3
y=5
(−2, 5)
36.
y = −2x − 9
x = −7
(−7, 5)
39.
x = −3
3x − 2y = −17
40.
x−4=0
2y + 3x = 16
(4, 2)
44.
2y = 0.8
− 2x + y = −1
3.
x+6=0
4y + 12 = 0
7.
y−x=0
y = −1
(0, 0)
11.
y = −x
2y − 6x = 0
Ø
15.
y−1=7
y + 3 = −4
(−3, 6)
19.
3x − y = 0
5x + 5 = 0
23.
(−2, 2)
(6, 6)
(9, 5)
(−6, −3)
(−1, −1)
(0, 0)
(0, 4)
(3, −3)
(−2, 10)
(5, 2)
(9, 6)
(−4, 4)
29.
y = −x + 2
x = 23
33.
y =x+3
x−7=0
37.
y+3=0
4x − 5y = −9
( 23 , 43 )
(7, 10)
30.
y = −x − 2
y = − 12
34.
y =x−5
y+3=0
38.
y=1
3x + 5y = 20
(− 32 , − 12 )
(2, −3)
(5, 1)
(−6, −3)
41.
(−3, 4)
− 2x = 1.2
x + 6y = 1.8
42.
(−0.6, 0.4)
3x = 0.3
2x − 5y = −1.3
43.
y − 3x = 0
2y − 6x = 0
49.
y = 13 x
y = 3x
53.
y = −5x
2x − 3y + 17 = 0
coincide
(0, 0)
(−10, 8)
(0.7, 0.4)
46.
2y = x
− 4y = −2x
coincide
47.
2x + 2y = 0
x+y =0
50.
x + 4y = 0
−x+y =0
(0, 0)
51.
y = − 25 x
y = −5x
54.
y = 4x
x + 2y = 18
(2, 8)
55.
y = 3x
− 9x − y − 4 = 0
coincide
(0, 0)
48.
3x − 3y = 0
5x − 5y = 0
52.
2x + y = 0
x − 2y = 0
56.
y = −4x
6x + 2y = −3
60.
y = 27 x
− 9x + 2y + 12 = 0
coincide
(0, 0)
( 32 , −6)
(− 13 , −1)
(−1, 5)
y = − 45 x
7x + 10y = 10
3y = −0.9
3x + 4y = 0
(0.4, −0.3)
(0.1, 0.3)
45.
57.
( 12 , 0)
58.
y = 14 x
3x − 8y = −16
(−16, −4)
59.
y = − 13 x
x + 6y + 9 = 0
(9, −3)
ALG catalog ver. 2.6 – page 337 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(6, 21)
QA
61.
y = − 32 x + 14
2x + 4y − 11 = 0
62.
(− 52 , 4)
65.
y = 6x − 32
7x − 2y − 3 = 0
63.
y = 94 x + 9
9x + 4y = 36
67.
y = − 12 x − 92
3x − 5y = 17
(0, 9)
64.
y = − 23 x + 13
4x + 3y = 11
68.
y = 57 x − 9
2x − 3y − 16 = 0
(5, −3)
(0, − 32 )
y = 13 x + 17
8x − y − 6 = 0
(3, 18)
66.
y = 32 x − 9
x − 5y − 6 = 0
(6, 0)
(−1, −4)
(5, −2)
Slope-intercept form
69.
y = −x + 5
y=x
73.
y = −3x + 6
y = −3x
77.
y =x+3
y = −x + 3
81.
y = 5x
y = 2x − 1
85.
y =x+2
y = −x + 4
89.
y = −6x + 1
y = −6x − 3
93.
y = −x + 6
y = 5x − 2
( 52 , 52 )
70.
y = −x
y =x−6
Ø
74.
y = 4x
y = 4x − 1
(0, 3)
78.
y =x−4
y = −2x − 4
82.
y = 2x + 6
y = −4x
(1, 3)
86.
Ø
(− 13 , − 53 )
( 43 ,
14
3 )
71.
y =x+4
y = 5x
(1, 5)
72.
y = 2x
y = −x − 2
(− 23 , − 43 )
75.
y =x−1
y=x
Ø
76.
y = −x
y = −x + 3
Ø
79.
y = −x + 2
y = 5x + 2
(0, 2)
80.
y = 3x − 5
y =x−5
(−1, 4)
83.
y = −x
y = 3x + 8
(−2, 2)
84.
y = −5x + 3
y=x
y =x−3
y = 2x − 1
(−2, −5)
87.
y = −x − 2
y = −3x + 4
88.
y = 2x + 6
y =x+4
90.
y = 5x − 5
y = 5x − 1
Ø
91.
y = 2x − 2
y = 2x + 2
92.
y = −3x + 1
y = −3x + 7
94.
y = −4x + 5
y = 2x − 4
95.
y =x−4
y = −3x + 10
96.
y = 8x − 7
y = 4x − 2
(3, −3)
Ø
(0, −4)
( 32 , −1)
(3, −5)
Ø
(0, −5)
( 21 , 12 )
(−2, 2)
Ø
( 45 , 3)
( 27 , − 12 )
97.
y = 2x − 1
y = 3x + 4
(−5, −11)
101. y = −3x
y = − 12 x + 5
105. y = 43 x + 5
y = 2x + 7
(−3, 1)
109. y = x − 13
y = 3x + 3
113. y = 12 x −
(−2, 6)
(− 53 , −2)
(4, − 13 )
2
3
117. y = 0.2x − 0.5
y = −0.3x
(1, −0.3)
y = 2x − 4
y = −3x + 6
102. y = x − 3
y = 23 x
(2, 0)
(9, 6)
106. y = 12 x + 10
y = −4x + 1
110. y = 15 x − 6
(−2, 9)
(10, −4)
y = − 65 x + 8
7
3
y = − 14 x +
98.
114. y = − 23 x +
1
6
y = − 12 x + 1
118. y = −x + 1.3
y = 2x − 0.2
(0.5, 0.8)
y = 4x + 1
y = −5x − 8
(−1, −3)
100. y = −3x + 2
y = −6x + 8
103. y = − 52 x
y = −4x + 6
(4, −10)
104. y = 43 x − 5
y = 2x
99.
107. y = 15 x − 2
y =x−6
111. y = 2x −
1
2
(5, −1)
(1, 32 )
115. y = 2x − 6
y = 25 x + 65
(−4, −8)
(−3, 2)
112. y = − 23 x − 7
(−6, −3)
y = 31 x − 1
y = − 12 x + 2
(−5, 72 )
108. y = 32 x + 4
y =x+5
(2, −4)
( 29 , 3)
116. y = x +
1
4
y = 43 x −
(−2, − 74 )
1
4
119. y = 0.25x − 1
y = 0.75x
120. y = x + 1
y = −2x − 0.2
(−2, −1.5)
(−0.4, 0.6)
ALG catalog ver. 2.6 – page 338 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
QA
General form
121. x + y = 0
x−y =4
122. x − y = −12
4x + 4y = 0
(2, −2)
(−6, 6)
123. − 2x + 2y = 0
x+y+8=0
124. x + y − 10 = 0
x−y =0
(5, 5)
128. − x + y = −6
2x + 2y = 0
(3, −3)
(−4, −4)
125. − 5x − 5y = 0
2x + y + 8 = 0
126. − x + y − 2 = 0
3x − y = 0
(−8, 8)
127. x + 3y = 0
x+y =4
(1, 3)
129. 6x + 2y = 0
4x − y = −7
(−1, 3)
130. − x − y = 9
14x − 4y = 0
133. 5x − 4y = 0
x − 4y − 16 = 0
(−5, 5)
134. − 2x + 3y + 22 = 0
x + 4y = 0
(−4, −5)
132. x + 7y − 12 = 0
x + 3y = 0
131. x + y = 0
4x + 3y + 5 = 0
(−2, −7)
137. 2x − 6y = 0
3y − x = 0
(6, −2)
(−9, 3)
135. − 4x + 2y = 0
5x − 2y = 3
(3, 6)
(8, −2)
coincide
(5, −1)
138. 3x − 9y = 12
− x + 3y + 4 = 0
145. − x + y = −6
3x + y = 2
coincide
Ø
142. 12x + 2y − 2 = 0
− 6x − y = 0
(2, −4)
146. − 5x − y + 7 = 0
2x + 2y = 22
Ø
(−1, 12)
149. x + y − 9 = 0
− 10x + 6y − 6 = 0
154. − 2x + 2y = 6
3x − y = 3
Ø
147. x − y + 4 = 0
− 2x + y − 8 = 0
(3, 6)
158. 2x − 2y + 1 = 0
x+y+1=0
( 31 , 23 )
148. 2x − 8y + 8 = 0
3x − 8y + 4 = 0
(−3, 28)
155. 5x − 7y − 4 = 0
x − 3y − 4 = 0
159. 6x + 2y = 5
x − 2y = −5
Ø
152. − 10x − y = 2
9x + y = 1
(−2, −2)
157. x + y − 1 = 0
9x − 3y − 1 = 0
144. 3x + 3y = −15
x + y = −4
(4, 2)
(2, −7)
(−8, 0)
161. x − y = 3
− 3x + 3y = −21
143. − 5x + y = −6
5x − y = −6
151. − 6x − 2y = 2
4x + y = 1
(−2, −5)
153. − x + 4y = 8
2x − 3y = −16
coincide
(−4, 0)
150. x + 2y + 12 = 0
4x − y + 3 = 0
(3, 6)
140. − 10x + 2y = 0
5x − y = 0
139. x + y + 1 = 0
− 4x − 4y = 4
coincide
141. x − 5y − 1 = 0
− 2x + 10y = 0
136. − 2x + y = −11
2x + 10y = 0
156. 3x + 2y − 6 = 0
x + 2y + 6 = 0
(6, −6)
(0, 52 )
160. x + 6y = −9
x − 3y = 6
(1, − 53 )
(− 34 , − 14 )
Ø
162. − 3x + y = 12
6x − 2y = 8
Ø
163. − 2x + 10y − 2 = 0
x − 5y + 2 = 0
164. x + y + 4 = 0
4x + 4y + 8 = 0
Ø
165. 10x + 2y + 2 = 0
5x + y + 1 = 0
coincide
166. x + y + 1 = 0
− 2x − 2y − 2 = 0
coincide
167. 6x − 3y = −6
− 2x + y = 2
coincide
ALG catalog ver. 2.6 – page 339 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
168. − 2x − 12y = −6
x + 6y = 3
coincide
Ø
QB
Topic:
Solving systems by elimination, determinants. See also categories QA and QC.
Directions:
15—Solve. 16—Solve and check.
26—Solve by any method.
27—Solve by substitution. 30—Solve by elimination. 32—Solve using determinants.
1.
x+y =0
x − y = −14
5.
x − 3y = 7
x + 3y = 7
(−7, 7)
(7, 0)
2.
y − x = −3
y + x = −3
6.
x − y = −2
− 2x + y = 7
(0, −3)
3.
y − x = −4
y+x=0
7.
− x − 6y = −3
x − 2y = 11
(−5, −3)
9.
− 3x − 9y + 6 − 0
3x + 5y − 14 = 0
10.
(8, −2)
13.
(5, 1)
14.
− x + 2y − 17 = 0
x − 3y + 28 = 0
11.
10x − y + 24 = 0
− x + 3y + 15 = 0
18.
(−3, −6)
4x + 6y = 0
− x + 2y = −14
15.
19.
Ø
22.
− x − 3y = 2
x + 3y = −4
25.
x+y =4
3x + 3y = 12
coincide
26.
2x − 4y = 6
x − 2y = 3
4x − 5y = −28
− 4x + y = −4
5x − 3y − 2 = 0
− 4x + 3y − 2 = 0
12.
4x + 5y − 7 = 0
− 3x − 5y + 14 = 0
− 3x + y = −3
5x − 2y = 10
16.
2x + 3y = 20
6x − y = 20
− 8x + y = 0
x + 2y + 17 = 0
20.
− x − 7y − 11 = 0
4x + 11y − 7 = 0
2x − 7y = 6
2x − 3y = 14
(10, 2)
30.
Ø
coincide
23.
x − 5y − 1 = 0
− x + 5y − 1 = 0
27.
x + 3y − 5 = 0
2x + 6y − 10 = 0
− 11x + 2y = −8
− 11x − 3y = 12
31.
(0, −4)
33.
− 5x + 2y − 9 = 0
12x − 3y = 0
34.
(3, 12)
37.
Ø
24.
− 3x + y + 5 = 0
6x − 2y = 0
28.
−x+y−1=0
2x − 2y + 2 = 0
38.
(14, −5)
41.
5x − 3y = 0
− 4x + 5y = 26
35.
42.
3x + 4y + 1 = 0
− 5x − 9y + 17 = 0
39.
− 9x − 10y = −124
4x + 3y = 19
46.
6x + 5y = 12
9x + 11y = 39
(−3, 6)
43.
3x + 11y + 77 = 0
10x + 13y − 51 = 0
47.
(22, −13)
2x − 3y − 1 = 0
− 8x + 12y − 4 = 0
8x − 3y − 12 = 0
5x − 3y − 21 = 0
(−3, −12)
2x − 3y − 35 = 0
9x + 4y = 0
36.
3x + 2y = 0
8x + 7y = 5
− 4x + 5y + 15 = 0
7x − 6y − 7 = 0
40.
− 11x − 2y = −23
8x − 3y = −10
(−2, 3)
(1, 6)
24x + 13y = −4
5x + 7y = −18
44.
(2, −4)
(−14, 25)
49.
32.
(−5, −7)
(0, 7)
45.
3x + 9y − 24 = 0
− 2x + 9y + 16 = 0
(4, −9)
(−11, 8)
7x + 15y − 105 = 0
− 15x − 4y + 28 = 0
(−8, −5)
6x − 35y = 0
9x − 5y − 285 = 0
48.
− 6x + 3y = −24
4x − 2y = 12
54.
− 14x + 6y = −8
7x − 3y = 4
Ø
51.
2x + y + 5 = 0
4x + 2y + 2 = 0
55.
4x − 6y − 20 = 0
2x − 3y − 10 = 0
− 25x + 2y = 70
15x + 7y = −360
(−6, −40)
(35, 6)
50.
5x − 11y − 15 = 0
− 12x + 5y − 71 = 0
Ø
52.
12x − 10y = 0
− 6x + 5y = 2
56.
− 5x + 8y − 1 = 0
5x − 8y + 1 = 0
Ø
Ø
53.
10x + 5y = 10
6x + 3y = 6
coincide
57.
x+y =5
x−y =2
coincide
( 27 , 32 )
58.
2x + 3y = 6
x − 3y = 0
Ø
coincide
(8, 0)
(6, 10)
− 2x − 7y = 7
3x + 10y = −8
(4, 4)
(10, −3)
coincide
29.
(3, 8)
(−7, 7)
(−1, −8)
(4, 7)
2x − y = 5
− 2x + y = 7
8.
(4, 0)
(−4, −15)
2x + y − 15 = 0
5x − 6y + 22 = 0
21.
x+y =4
x−y =4
(4, 6)
(6, −4)
17.
4.
(9, −1)
(5, 11)
2x + 4y = 14
3x − y = 14
(2, −2)
coincide
(2,
2
3)
59.
− 4x − y = 0
4x − 3y − 8 = 0
( 12 , −2)
ALG catalog ver. 2.6 – page 340 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
coincide
60.
3x − 2y = −5
− 3x − y = −1
(− 13 , 2)
QB
61.
− 4x − 3y + 12 = 0
3x + y − 7 = 0
62.
2x + y = 9
6x + y = 14
( 54 ,
13
2 )
63.
( 59 , 85 )
65.
6x − 2y = 2
x + 8y = 2
( 25 , 15 )
66.
3x − 4y = −16
3x − 12y = −18
70.
(−5, 14 )
73.
x + 2y − 5 = 0
2x − 3y + 7 = 0
67.
17
7 )
71.
− 5x + 8y + 22 = 0
5x + 4y + 5 = 0
74.
3x − 2y = −9
− 3x + 5y = 21
75.
(− 13 , 4)
6x − 5y = −4
9x + 8y = −6
(− 23 , 0)
− 4x + 7y = 3
7x + 4y = 6
6
( 13
,
85.
78.
82.
9
13 )
86.
21
22 )
− 3x + 6y − 2 = 0
7x − 4y − 7 = 0
4x − 5y = 4
8x − 5y = 11
( 74 , 35 )
72.
6x + 6y + 1 = 0
6x − 6y − 7 = 0
76.
31
7 )
10x + 3y − 3 = 0
2x + 3y + 1 = 0
79.
83.
2x − 3y + 5 = 0
− 3x + 2y − 6 = 0
80.
2x − 7y + 8 = 0
3x + y − 8 = 0
87.
40
23 )
1
10 )
4x − 6y = −1
7x − 9y = −2
(− 12 , − 16 )
5x + 8y + 2 = 0
3x + 2y + 8 = 0
(− 30
7 ,
2x + 10y = −4
− 4x − 10y = 9
(− 52 ,
(− 85 , 35 )
12x + 5y = 8
4x + 9y = 0
48
( 23
,
2x − y + 7 = 0
3x + 2y − 5 = 0
(− 97 ,
( 12 , − 23 )
9
4
( 11
, − 11
)
3x − 5y + 6 = 0
2x + 4y − 3 = 0
9
(− 22
,
68.
( 21 , − 23 )
( 35 , 76 )
81.
− 4x + 3y = −7
2x + 9y = 14
( 52 , 1)
5x + 7y + 38 = 0
5x − 3y + 23 = 0
2x − 15y − 7 = 0
− x + 6y + 4 = 0
(6, 13 )
3
(− 11
2 ,−2)
( 54 , − 94 )
77.
64.
( 12 , 5)
( 71 ,
69.
8x − y = −1
− 10x + 2y = 5
84.
17
7 )
7x + 9y − 4 = 0
5x + 12y − 8 = 0
8
( 13
,
11x − 2y = 8
8x + 3y = 0
88.
64
( 24
49 , − 49 )
12
13 )
− 3x + 5y = −7
2x + 3y = −5
4
(− 19
, − 29
19 )
Fraction and decimal coefficients
89.
1
2 x + 5y
=0
6x + 4y = −14
90.
(− 52 , 14 )
93.
1
3
2x − 2y
+4=0
x + 7y − 12 = 0
94.
3
4x −
1
2x +
92.
x − 4y = −2
3
2 x − 8y = 0
(−8, − 32 )
96.
3x + y = −1
9
3
2x − 4y = 3
( 31 , −2)
100.
2
3x +
1
3x +
5
6y
4
3y
=1
3
5x −
1
2x +
2
3y
1
3y
=4
( 32 , 34 )
− 17 x + 74 y = 1
x + 2y = 1
1
3
2x − 4y
95.
= −1
5x + 12y = 29
1
3
2 x − 8 y = −2
− 32 x + 43 y = 9
(−10, −8)
99.
− 13 x + 56 y = 14
1
1
5x + 4y = 0
− 12 x − 54 y =
(10, −8)
103. − 35 x + 12 y = −5
(1, 2)
(− 53 , 43 )
5
8y
3
4y
=1
(8, 8)
1
1
3x − 2y
(12, 6)
=1
2
3x −
1
2x +
= − 65
4
5y
1
4y
98.
= 10
101. − 12 x + 13 y = −4
105.
− 3x + 38 y = 0
− 3x + 8y − 4 = 0
91.
(− 12 , 6)
(−2, 2)
97.
8x + y − 2 = 0
3x + 14 y = 0
=
102.
(3, 4)
5
2
109. 0.5x + 2y = 13
4x − 3y = 9
(6, 5)
106.
1
3x +
1
2x −
1
6y
1
4y
=
2
1
3x − 6y
(18, 24)
3
1
2x + 4y
5
5
3
= 10
104.
= −5
(15,
15
2 )
= 10
( 25
2 , 5)
( 72 , 3)
1
3x −
5
3x +
107.
=1
110. 6x + 5y = 7
1.5x − 5y = 8
=8
(9, −6)
(2, −1)
5
3y
7
2y
=8
=
(9, −3)
9
2
111. 3x − 2.5y = −14
4x − y = 0
108. − 23 x − 12 y = 1
1
1
2x − 8y
(−3, 2)
= − 74
112. 7x + 4y = 24
0.5x − 3y = 5
(4, −1)
(2, 8)
113. 0.4x + 5y = 12
0.8x − 3y = −2
(5, 2)
114. 5x − 0.5y = 31
2.5x + 3y = 9
(6, −2)
117. 0.2x − 0.6y = −5
0.1x + 0.2y = 0
(−10, 5)
118. 0.5x − 0.2y = 19
0.4x − 0.3y = 11
(50, 30)
115. 0.3x + 2y = −3
1.2x − 7y = 33
(10, −3)
119. 0.3x + 0.1y = 3
0.2x − 0.3y = 13
(20, −30)
ALG catalog ver. 2.6 – page 341 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
116. 0.4x + y = 6
1.2x − 5y = 18
(15, 0)
120. 0.1x + 0.3y = 7
− 0.7x + 0.1y = −5
(10, 20)
QB
121. 0.13x + 0.08y = 50
0.09x + 0.34y = 120
122. 0.12x + 0.09y = −21
0.01x + 0.05y = −40
(500, −900)
(200, 300)
125. 1.5x − 2.3y = 4.6
1.5x − 3.4y = 6.8
128. 0.3x + 1.1y = −6.2
0.6x − 0.7y = −0.8
(−6, −4)
(3, 2)
130. 1.2x − 0.6y = 4
0.4x + 0.2y = 2.4
20
11 )
(−50, 2000)
127. 0.8x + 0.3y = 3
1.3x + 0.5y = 4.9
(8, 4)
129. 0.3x − 0.5y = 1
0.2x + 0.4y = 2
124. x + 0.03y = 10
10.4x + 0.75y = −980
(−1200, 750)
126. 0.2x + 0.6y = 4
0.3x − 0.2y = 1.6
(0, −2)
70
( 11
,
123. − 0.25x + 0.2y = 450
0.05x + 0.08y = 0
131. 0.4x + 1.5y = 1.5
0.8x + 0.5y = −3
8
( 14
3 , 3)
(− 21
4 ,
132. 0.8x + 1.1y = 0
− 0.4x + 0.8y = 1.5
12
5 )
(− 55
36 ,
10
9 )
Mixed order
133. 2x = 4
4y = 6x
134. y − 5x = 0
3x + 3 = 0
(2, 3)
(−1, 5)
138. 5x = 70 − 2y
x−y =0
137. x = y
− 3y + 2x = 7
(10, 10)
135. − 3y = x
y−2=0
(−6, 2)
139. 2y − 2x = 0
3y = 2x − 3
(−3, −3)
136. y = −4
2y + 4x = 0
140. 4y + 3x − 5 = 0
− 4x = 4y
(−7, −7)
(−5, 5)
141. 5x + 9 = 11y
2y = x
(18, 9)
142. − x = 11y
15y − 21 = −2x
143. 17y − 3x + 4 = 0
5y − x = 0
(33, −3)
145. y + 4x = 0
x=9−y
(−3, 12)
144. x = −6y
y + 4x = 23
(2, 12)
146. y − 5x + 12 = 0
3y = 6x
147. − 3x = y
8x − 11 = y
(1, −3)
148. x = 2y + 18
2y − 10x = 0
(−2, −10)
150. y − 2x + 3 = 0
4x = y − 11
151. − 5y − 6 = x
4y + 3 = −x
(9, −3)
152. 5y + x − 14 = 0
− x + 2 = 3y
(−7, −17)
(−2, 11)
158. 9x + 2y + 2 = 0
− 3y = x − 22
(1, −2)
(−4, −3)
156. 7x = 3y
x+8=y
159. 3x − 2y = 9
y − 7x = 12
(−3, −9)
160. y + 4x = 31
x + 17 = 2y
162. 3x + 7y = −23
− 3x + 2y − 5 = 0
163. 8y + 19 = 5x
3y = 5x − 29
165. − 3x + 13y = 12
7y − 5x − 20 = 0
(5, 11)
164. − 10x − 3y = 25
3y + 7x + 22 = 0
(−1, −5)
166. 2x = 5y − 15
5x − 6y + 5 = 0
(−4, 0)
167. 11x + 4y − 20 = 0
8x = 5y − 25
(5, 5)
169. 6x = 3y + 45
8y + 5x − 6 = 0
2y + 23 x − 4 = 0
(−3, −3)
171. 3x − 2y + 2 = 0
5y + 4x = 51
(3, −2)
= −x − 4
168. 2x − 3y − 3 = 0
11y − 8x = −9
(0, 5)
170. 3y + 5x = 9
− 2y = 13 − 3x
(6, −3)
172. 5x = 7y − 19
4x + 3y + 41 = 0
(−8, −3)
(4, 7)
174. 6x = 3y − 8
3
1
3y + 5x = 3
( 53 , 6)
175.
3
5x
= 21 y + 11
3x + 8y + 13 = 0
176. 2y − 4x = −19
3
1
2y = −4x + 2
(5, 12 )
(15, −4)
(−3, 3)
2
1
5x + 2y + 5 =
6
1
2 y + 5 x = −1
(5, −14)
(7, 2)
(−3, −2)
(−3, 9)
177.
(6, 14)
(−2, 8)
161. 7x + 2y + 3 = 0
7x − 2y = −39
1
3y
155. 4y − 3x = 0
x = 2y + 2
(−16, 10)
157. − 3x + 1 = y
5y + 2x + 8 = 0
173.
(−16, 6)
154. − 5x = 8y
2x + y = −22
153. 2y + 11x = 0
7x + y + 3 = 0
(6, −1)
(−10, −2)
(4, 8)
149. 8x = 4 + y
y = 7x − 2
(2, −4)
0
178.
3
4y
1
2x
= 4 − 23 x
=8−
7
8y
(−12, 16)
179.
1
2x =
3
4y −
(12, 8)
ALG catalog ver. 2.6 – page 342 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
7
4y − 8
2
3x + 2 =
180.
0
1
1
5y − 3x + 1 =
7
3
2y = 9 + 5x
(15, 20)
0
QC
Topic:
Solving systems by various methods. See also categories QA and QB.
Directions:
15—Solve. 16—Solve and check.
26—Solve by any method.
27—Solve by substitution. 30—Solve by elimination.
1.
x + 5 = 3y − 2
2x + 7 = y + 3
2.
− 11y + 4 = 4x + 5y
4y − (x − 9) = 2x
6.
( 25 , − 12 )
9.
4(x + y) + 5 = −15
3 − (x − y) = 2 − x
10.
14.
y + 1 = 12 (x + 3)
y − 8 = 73 (x − 1)
y−1=
18.
7
3 (x + 2)
x y
+ =6
2
3
3x − 2y = 12
(8, 6)
22.
4.
26 − 5y = 2(x − 3)
2x − (y + 1) = x + 1
11.
− 3x + 6y = −7 − 6x
11x + 4y = −5(6 + y)
8.
15.
5x − 6(y + 2) = 31
6(x + 7) = 4 − 4y
12.
19.
y + 4 = 12 (x − 4)
y − 4 = −7(x − 5)
16.
y − 4 = 43 (x − 4)
y + 5 = −(x + 1)
(−4, −2)
y − 3 = 15 (x + 1)
y−2=
− 3(y − x) = 14 − 4y
2x + 26 = 4(x + y)
(3, 5)
(6, −3)
y + 4 = −2(x − 9)
y + 4 = −2(x − 6)
x + 39 = 4(x + 2y)
y − 5x = 3y + 3
(−3, 6)
(−1, −8)
y − 2 = 5(x + 2)
y + 2 = 3(x + 4)
− 3x + 4 = 4 − 3y
− x + 3y = 12
(6, 6)
(−3, 13 )
Ø
coincide
21.
7.
(−1, 7)
(3, 2)
17.
2(x + y) = −2 − 4x
7x + 27 = 3x + y
(6, 4)
y − 6 = 2(x − 5)
y − 2x = 3y + 10
x+2=y−3
(−5, 0)
(−2, 5)
(−4, −1)
13.
3.
(3, −4)
(−1, 2)
5.
3x − 8 = 2y + 9
x − y + 3 = 10
20.
y − 7 = 4(x − 3)
y + 3 = 4(x − 1)
24.
5x + 4y = 0
x
y
+
=2
4
10
1
5 (x + 6)
Ø
coincide
−5x + 3y = −14
x y
− = −2
5
3
23.
x y
+ =0
3
2
x + 2y = 4
(−12, 8)
(16, −20)
(10, 12)
25.
x y
+ = −5
3
2
x y
+ =0
3
7
(6, −14)
26.
x y
+ =2
6
2
x y
− − =0
5
3
33.
37.
x y
+ =1
2
3
x y
− − =1
2
3
−
y−x
= −4
3
1
3
4x + 2y = 9
(−2, 0)
(12, 0)
x
2
=
y
3
x+8
3y − 2
=
3
4
30.
34.
38.
45.
y+3
=4
x+6
y + 11
= −2
x−1
y−1
=3
x+5
y+1
=3
x−5
Ø
(4, 23 )
1
3y
= 31 x + 6
x+y
=3
2
(−6, 12)
x−4
y+7
=
5
4
y
=3
x
(−5, 1)
42.
46.
y−2
= − 23
x
y
1
=
x − 10
2
y−3
1
=
x − 10
2
y
1
=
x−4
2
x y
+ =1
4
9
x y
− − =1
4
3
31.
x − 4y
= −1
4
1
2
3x + 4y = 9
35.
(8, −9)
32.
(12, 4)
36.
x−y
x+y
=
4
20
x
3
=
y
2
39.
(−1, −3)
(4, 6)
41.
x y
+ =1
6
2
x y
− =1
3
2
28.
x y
− = −1
2
3
x y
+ =7
4
3
(8, 15)
(10, −4)
(−15, 9)
29.
x y
+ = −4
5
2
x y
− =6
2
4
−
27.
40.
43.
coincide
47.
ALG catalog ver. 2.6 – page 343 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
y+1
= −1
x−1
y−6
=1
x+2
y+3
5
=−
x
2
y−3
5
=−
x−3
2
x = 20 + 54 y
4x + y
=7
8
(7, 0)
(15, −4)
y
=2
x
x + 2y
2x + y
=
4
5
(5, 10)
(−6, −4)
(6, −2)
x y
− =1
7
5
x y
+ =1
7
5
(−4, 4)
Ø
44.
48.
y+4
7
=
x−4
3
y−4
1
=−
x
7
y+1
= −4
x+2
y+5
= −4
x+1
(7, 3)
coincide
QC
49.
x+y
1
x−y
− =
2
2
3
x+2
y+4
=
+4
2
3
50.
(8, −1)
53.
57.
61.
1
1
1
+ =
x y
12
1
1
7
= +
x
y
12
(3, −4)
1
1
− =2
x y
3
2
+ = 21
x y
( 15 , 13 )
2x − y = 0
√
x + 3y = −7 3
54.
58.
62.
√
√
x+y 3=4 2
√
√
x 3+y =2 6
√ √
( 2, 6)
69.
70.
√ √
( 5, 2 10)
73.
y = x2 + 3
x = −2
77.
y = x2
y = 16
1
1
5
− =
x y
3
1
1
7
+ =
x y
3
( 12 , 3)
2
1
− =6
x y
3
4
− =4
x y
(− 12 , − 14 )
x = 3y
√
x−y =2 5
63.
(−2, 43 )
56.
( 12 , 15 )
60.
√
x − y = −3 2
√
x+y =5 2
64.
√ √
( 2, 4 2)
√
√
x 2−y+3 5=0
√
x+y 2=0
67.
71.
74.
y = −x2 − 1
x=4
78.
y = −x2
y=9
(4, −17)
Ø
√
√
−x+y 2+2 6=0
√
√
x 2+y = 3
68.
y = (x − 4)2
y = −1
√
√
x 5+y 7=0
√
√
√
x 7 + y 5 = −2 2
72.
√ √
(− 14, 10)
y = (x − 3)2
x=3
(3, 0)
76.
y = (x + 5)2
x = −6
79.
y = −x2 + 8
y=8
(0, 8)
80.
y = x2 − 11
y = −2
Ø
82.
y = (x + 1)2
y=4
83.
86.
y = x2
y = 6x − 9
y = (x + 3)2 − 5
y = −4
84.
(−4, −4) and (−2, −4)
(3, 9)
87.
y = 2x2 − 4
y = x2 + 5
(−6, 1)
y = x2
y = 2x − 3
(2, 3)
88.
Ø
y = (x − 2)2 + 3
y=3
y = −x2
y = x − 12
(−4, −16) and (3, −9)
(2, 1)
90.
y = x2 + 2
y = −3x + 2
91.
(0, 2) and (−3, 11)
93.
√
√
x 6 − y 15 = 9
√
√
√
x 2 + y 5 = −7 3
(3, −2) and (−3, −2)
y = −x2
y = 2x
y = (x − 3)2
y = −2x + 5
√
√
− x 7 + y = 2 14
√
√
x − y 7 = −8 2
√
√
(− 6, − 15)
(0, 0) and (−2, −4)
89.
( 31 , 12 )
√
3x − y = 6
√
x−y =3 6
75.
(1, 4) and (−3, 4)
85.
2
3
+ = 12
x y
1
1
− =1
x y
(6, 9)
√ √
(− 2, 14)
(4, 16) and (−4, 16)
81.
1
1
5
+ =
x y
18
1
1
1
− =
x y
18
√
√
(− 6, −4 6)
√
√
( 6, − 3)
√
√
√
x 7−y 2=5 3
√
√
x 2−y 7=0
x−y
3
5y
− =
2
2
4
2x y
− +4=0
3
2
(−3, 4)
1
1
+ =7
x y
2
3
+ = 19
x y
59.
√ √
( 21, 6)
(−2, 7)
1
1
1
+ =
x y
4
1
1 5
= −
x
y
4
55.
√ √
(− 10, 5)
√
√
−x+y 2=3 5
√
√
x 5 + y 10 = 25
52.
(1, 5)
√ √
(3 5, 5)
66.
x+2 y−1
−
=0
6
8
y−x y+1
+
=4
2
3
51.
(2, 3)
√
√
(− 3, −2 3)
65.
x−1 y−1
x
+
=
3
3
2
x+y
x
3y
+ =
4
2
4
94.
y = −3x2
y = x2 + 4
Ø
y = x2 + 3
y = 4x
92.
y = (x + 2)2
y = 4x − 5
96.
y = −x2
y = x2 − 8
Ø
(1, 4) and (3, 12)
95.
y = x2 + 3
y = −x2 + 3
(3, 14) and (−3, 14)
ALG catalog ver. 2.6 – page 344 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
(0, 3)
(−2, −4) and (2, −4)
QD
Topic:
Systems of 3 variables.
Directions:
15—Solve.
1.
x+3=0
− 3y + z = 1
2x + y + z = 7
26—Solve by any method.
2.
y = −5
x+y+z =1
x − y − 5z = 5
6.
(5, −5, 1)
9.
− 3y = z
3x + z = 3
2x − y + z = 0
2x − y = 0
3y − 2z = 0
x+y−z =0
z=2
x−y+z =2
x + y − 3z = 0
10.
2x = y
3x − y = −1
x + 2y + z = −5
7.
11.
(2, 4, 6)
14.
x + y + z = −2
2x − y + z = −1
x − 4y + z = 3
18.
2x + y + 3z = 4
− 4x − y + 6z = −2
4x − y − 3z = −1
22.
y + 4z = −1
2x − y = 1
x−y+z =4
15.
x + 3y + z = 3
2x + 5y − 2z = −4
x + 6y + 2z = 0
2x − 3y + 6z = 1
− 4x + y + 4z = −1
x+y+z =1
( 21 , 13 , 16 )
4x − y + z = 6
x + y − z = −1
− x − 2y + z = −2
x + 4y + z = 1
2x − 4y + z = −5
x + 8y − z = 8
26.
5x − y − 2z = 1
− 3x + 2y + 3z = 2
x − 2y − z = −10
19.
(2, 3, −3)
(3, 3, 3)
12.
− 2x + 6y + z = 8
8x − y + 2z = 1
6x + 3y − z = 7
( 83 , 32 , − 14 )
− 2x + z = −2
y+z =1
4x + y − z = 5
x+z =0
2x + z = −4
x + y − z = −8
16.
x − 3y = 4
x + z = 12
x−y−z =1
x+y−z =1
− x + y − 4z = 0
2x + y + z = 1
20.
2x − y + z = 5
x−y+z =1
x + y + 3z = 7
(2, −2, −1)
23.
(7, 1, 5)
− 2x + 5y − z = −4
4x − 5y + z = 9
2x + 10y + 3z = 12
(4, 3, 0)
24.
( 52 , 25 , 1)
27.
(0, 7, −4)
30.
y−3=0
x − 2y − z = −1
2x + y + z = 4
(0, 3, −2)
(0, 34 , −2)
(6, −2, 3)
29.
8.
(−4, 0, 4)
(1, 3, 5)
( 21 , 2, 13 )
25.
x+1=0
2x + y + z = 5
3x − y + z = 2
y−z =0
y + 4z = 15
x+y+z =9
z = −3
x + 3y = −7
−x−y+z =0
(−1, −2, −3)
(−1, −2, 0)
(−1, −1, 0)
21.
4.
(−1, 1, 6)
(−2, −5, 1)
17.
y=8
− 2x + y = 0
x − y + 2z = 0
(4, 8, 2)
(3, 3, 2)
(2, 1, −3)
13.
3.
(2, −6, 4)
(−3, 3, 10)
5.
y+6=0
2x − z = 0
x + y + 3z = 8
30—Solve by elimination.
− 3x − 2y + z = −3
2x + 3y + 2z = 7
x+y+z =0
(−3, 16 , 83 )
28.
(−3, 5, −2)
31.
x + y + 3z = 10
2x − 2y + z = 11
− 4x − 2y + 3z = 2
( 25 , − 32 , 3)
ALG catalog ver. 2.6 – page 345 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
x + 2y + z = 0
− x − 4y + z = 5
2x − 2y + 2z = −1
5x + y + 2z = 7
− 2x + 2y + 3z = −2
2x + 3y + 2 = −12
(1, −6, 4)
32.
x + 2y + 5z = −3
3x + 6y + z = −9
− x − 2y + 2z = 3
(− 23 , − 76 , 0)
QE
Topic:
Word problems involving two variables. See also categories HA–HJ (first degree equations)
and PN (word problems involving graphing).
Directions:
0—(No explicit directions.) 16—Solve and check.
21—Solve using two variables.
39—Translate and solve. 41—Write a system of equations and solve.
1.
One number is 8 more than 3 times another number.
Their sum is 68. What are the numbers? 15, 53
2.
One number is 10 less than 4 times another number.
Their sum is 20. What are the numbers? 6, 14
3.
A number is 1 more than half another number.
Their sum is 25. What are the numbers? 9, 16
4.
A number is 12 less than a third of another number.
Their sum is 56. What are the numbers? 5, 51
5.
The difference of two numbers is 19. Their sum
is 27. Find the numbers. 23, 4
6.
The difference of two numbers is 18. Their sum
is 42. Find the numbers. 30, 12
7.
Find two numbers whose sum is 66 and whose
difference is 40. 53, 13
8.
Find two numbers whose sum is 71 and whose
difference is 21. 46, 25
9.
The sum of the digits of a two-digit number is 10.
Five times the tens digit plus six times the units
digit is 57. Find the number. 37
10. The sum of the digits of a two-digit number is 4.
Two times the tens digit plus three times the units
digit is 10. Find the number. 22
11. The sum of the digits of a two-digit number is 11.
The ones digit is 5 more than twice the tens digit.
What is the number? 29
12. The sum of the digits of a two-digit number is 13.
The tens digit is 2 less than twice the ones digit.
What is the number? 85
13. The sum of the digits of a two-digit number is 14.
If the digits are reversed, the number is increased
by 18. What is the number? 68
14. The sum of the digits of a two-digit number is 9.
If the digits are reversed, the number is increased
by 63. What is the number? 18
15. The sum of the digits of a two-digit number is 12.
If the digits are reversed, the number is decreased
by 54. Find the number. 93
16. The sum of the digits of a two-digit number is 7.
If the digits are reversed, the number is decreased
by 27. Find the number. 52
17. Separate 90 into two parts such that the second part
is 30 less than twice the first part. 40, 50
18. Separate 80 into two parts such that the first part is
5 more than twice the second part. 55, 25
19. Separate 100 into two parts such that the larger part
is 5 less than 6 times the smaller part. 85, 15
20. Separate 35 into two parts such that the second part
is 3 more than 7 times the first part. 4, 31
21. Separate 64 into two parts so that 1/9 of the larger
part added to 1/4 of the smaller part is 11. 36, 28
22. Separate 40 into two parts so that 3/5 of the larger
part added to 1/2 of the smaller part is 29. 40, 10
23. Separate 81 into two parts so that 2/3 of the larger
part added to 1/6 of the smaller part is 39. 51, 30
24. Separate 50 into two parts so that 1/9 of the larger
part added to 1/5 of the smaller part is 6. 45, 5
Perimeter
25. The perimeter of a rectangle is 44 in. If the length
is increased by twice the width, the result is 31 in.
Find the length and width of the rectangle. 13, 9 in.
26. The perimeter of a rectangle is 56 m. If the width is
increased by half the length, the result is 18 m. Find
the length and width of the rectangle. 20, 8 m
27. The perimeter of a rectangle is 74 cm. If the width is
doubled and the length is halved, the new rectangle
will have a perimeter of 76 cm. Find the dimensions
of the original rectangle. 13 × 24 cm
28. The perimeter of a rectangle is 76 ft. If the width is
doubled and the length is halved, the new rectangle
will have a perimeter of 62 ft. Find the dimensions
of the original rectangle. 8 × 30 ft
ALG catalog ver. 2.6 – page 346 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
QE
29. The width of a rectangle is half the length. If
the width is increased by 2 cm and the length is
decreased by 3 cm, the new perimeter is 58 cm. What
are the dimensions of the original rectangle?
30. The length of a rectangle is twice its width. If the
length is increased by 4 ft and the width is decreased
by 1 ft, the new perimeter is 42 ft. What are the
dimensions of the original rectangle? 12 × 6 ft
20 × 10 cm
31. The perimeter of a rectangle is 156 meters. When
the length is decreased by 13 meters and the width
is increased by 13 meters, the resulting figure is a
square. Find the length and width of the rectangle.
52, 26 m
32. The perimeter of a rectangle is 88 inches. When
the length is decreased by 10 inches and the width
is increased by 10 inches, the resulting figure is a
square. Find the length and width of the rectangle.
32, 12 in.
33. A square and an equilateral triangle have the same
perimeter. Each side of the triangle is 5 m less than
twice the length of each side of the square. How long
is each side of the square? 7.5 m
34. A square and an equilateral triangle have the same
perimeter. Each side of the triangle is 7 in. less than
twice the length of each side of the square. How long
is each side of the square? 10.5 m
35. The sides of a square are half as long as the sides of
an equilateral triangle. The sum of their perimeters
is 45 ft. How long is each side of the triangle? 9 ft
36. The sides of a square are one-third as long as the
sides of an equilateral triangle. The difference
between their perimeters is 20 cm. How long is each
side of the triangle? 12 cm
Time, distance, rate
37. An airplane flies 500 km against the wind in 2 hours.
It flies the same distance with the wind in 1 hour.
What is the airplane’s speed in still air? 375 km/hr
38. A canoe goes 8 miles down the river in 2 hours and
the same distance up the river in 4 hours. What is
the speed of the canoe in stillwater? 3 mph
39. A boat travels 4 miles downstream in half an hour,
and the same distance upstream in 2 hours. What is
the speed of the boat in stillwater? 5 mph
40. With a tail wind, a helicopter flies 30 kilometers
in half an hour. When the helicopter flies in the
opposite direction, it takes 1 hour to go the same
distance. What is the speed of the helicopter in still
air? 45 km/hr
41. A motorboat can travel 48 mi down the river in
3 hours. It takes the boat 4 hours to travel the same
distance up the river. Find the speed of the boat in
stillwater. 14 mph
42. A steamboat traveling against a current took 3 hours
to go 36 kilometers. Returning the same distance
with the current took 2 hours. Find the rate of the
steamboat in still water. 15 km/h
43. An airplane flew 570 kilometers in 3 hours with a
tailwind. It took 5 hours for the return trip against
the wind. Find the rate of the wind and the rate of
the airplane in still air. airplane, 152 km/hr; wind, 38 km/hr
44. A jet flies 1400 miles in 4 hours against the wind.
The return trip with the wind takes 5 hours. Find
the wind speed and the jet’s speed in still air.
45. A barge went 18 miles down river in 1 12 hours. The
return trip up the river took 45 minutes longer. Find
the barge’s speed in still water and the speed of the
current. barge, 10 mph; current, 2 mph
46. A motor boat went 14 kilometers downstream in
30 minutes. The return trip upstream took 6 minutes
longer. Find the rate of the boat in still water and
the rate of the current. boat, 24 km/hr; current, 4 km/hr
47. A boat takes half an hour to go 7 12 miles
downstream. The return trip upstream takes 34 hour.
Find the rate of the boat in still water. 12.5 mph
48. Traveling down river, a fishing boat goes 45 km in
2 hours and 30 minutes. The return trip up the river
takes 3 hours and 45 minutes. Find the boat’s speed
in still water. 25 km/hr
49. A bicyclist travels 1 kilometer uphill and 2 kilometers
downhill in 24 minutes. The same trip in the
opposite direction takes 30 minutes. Find the
bicyclist’s rates uphill and downhill.
50. A backpacker walks 2 miles uphill and 5 miles
downhill in 3 hours. The same trip in the opposite
direction takes 4 hours. Find the person’s rates
uphill and downhill. up, 1.5 mph; down, 3 mph
jet, 315 mph; wind, 35 mph
up, 5 km/hr; down, 10 km/hr
ALG catalog ver. 2.6 – page 347 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.
QE
51. Andy can ride his bicycle over a route in 4 hours,
and Betsy can ride her bike over the same route in
3.5 hours. If they started riding toward each other
at the same time, from oppposite ends of the route,
how long would it take them to meet? (Give answer
in hours and minutes, and assume all speeds are
constant.) 1 hr 52 min
52. Frances can walk a certain distance in 1.5 hours, and
Enrique can walk the same distance in 1 hour. If
they started walking toward each other at the same
time, from oppposite ends of the route, how long
would it take them to meet? (Give answer in hours
and minutes, and assume all speeds are constant.)
53. Two cars, initially 153 miles apart, head toward
each other and meet in 1 21 hours. Three hours and
20 minutes later, they are 340 miles apart. How fast
is each car traveling? not enough info.
54. Two cars pass each other on the highway, traveling
in opposite directions. One hour later, they are
120 km apart. Three and a half hours later, they are
420 km apart. How fast is each car traveling?
36 min
not enough info.
55. The sum of two numbers is 20. Five more than
twice the sum of the numbers is 45. What are the
numbers? not enough info.
56. The sum of two numbers is 32. Ten more than half
of the sum of the numbers is 26. What are the
numbers? not enough info.
ALG catalog ver. 2.6 – page 348 Catalog copyright (c) 1989–1994 by EAS EducAide Software, Inc. All rights reserved.