LN Summer Math Requirement – Algebra Review
For students entering Algebra II
The purpose of this packet is to ensure that students are prepared for Algebra II. The Topics
contained in this packet are the core Algebra I concepts that students must understand to be
successful in Algebra II. There are 9 concepts addressed in this packet:
Topics
A. Simplifying Polynomial Expressions – Practice Set 1
B. Solving Equations - Practice Sets 2 & 3
C. Rules of Exponents - Practice Set 4
D. Binomial Multiplication - Practice Set 5
E. Factoring – Practice Set 6
F. Radicals – Practice Set 7
G. Lines – Practice Sets 8, 9, 10 & 11
H. Solving Systems of Equations – Practice Set 12
I. Absolute Values – Practice Sets 13
For each concept listed, there is an explanation with examples, a problem set, and a listing of
websites that deals with that particular topic. The websites include tutorials, videos, and extra
practice problems. An answer key is provided at the end of the packet.
Below is a list of websites that can help you with Algebra II concepts over the course of this
next year.
http://www.hippocampus.org/HippoCampus/Algebra%20%26%20Geometry;jsessionid=1304A407D3C4842D1BF226696
0BE6EB3
http://www.purplemath.com/modules/index.htm
http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/
http://www.khanacademy.org/#library-section
http://www.regentsprep.org/Regents/math/ALGEBRA/math-ALGEBRA.htm
http://patrickjmt.com/
1
Tutorials and Problems in this packet were compiled from the following sources:
Math teachers at Lawrence North High School
Prentice Hall Algebra 1 textbook
http://frankumstein.com/worksheets.htm
http://www.hcpss.org/parents/summer_enteringalgebra2_2010.pdf
http://www.docstoc.com/docs/124240773/ALGEBRA-II-%ef%bf%bd-SUMMER-PACKET---DOC
http://www.algebrafunsheets.com/algebratutorials/tutorials.php?name=SolvingAbsValueBasic.html
http://teacherspace.swindsor.k12.ct.us/staff/smazzonna/documents/Summeralg1review_000.pdf
2
3
Online Tutorials & Additional Practice – simplifying expressions
 http://www.purplemath.com/modules/polyadd.htm
 http://www.regentsprep.org/Regents/math/ALGEBRA/AV2/indexAV2.htm
 http://www.purplemath.com/modules/polymult.htm
 http://www.regentsprep.org/Regents/math/ALGEBRA/AV3/indexAV3.htm
 http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut27_addpoly.ht
m
 http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut28_multpoly.ht
m
4
5
6
Online tutorials & additional practice – solving equations
 http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/basicequation-practice/v/equations-3
 http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut7_lineq.htm
 http://www.purplemath.com/modules/solvelin4.htm
 http://www.montereyinstitute.org/courses/Algebra1/U02L1T2_RESOURCE/index.html
7
8
Online tutorials and extra practice – exponents
http://www.purplemath.com/modules/exponent.htm
http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_ExponentsRules.xml
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut23_exppart1.htm
http://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-arithmetic-operations/cc-6thexponents/v/introduction-to-exponents
9
10
Online tutorials and extra practice – multiplying polynomials:
 http://www.khanacademy.org/math/algebra/multiplying-factoring-expression/multiplyingbinomials/v/multiplication-of-polynomials
 http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut26_multpoly.htm
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12
FACTORING BASICS
Signs
+
+ ( + ) ( + )
-
+
( - )
-
-
( - ) ( + )
+
-
( + ) ( - )
Factoring without a leading coefficient
𝑥 2 − 2𝑥 − 63
___* ___ = -63 and ___ + ___ = -2
1.
2.
3.
4.
5.
Factoring with a leading coefficient
2𝑥 2 + 5𝑥 − 3
Use Borrow-Giveback
Factoring the Difference of Two
Squares
𝑥 2 − 63
( - )
Multiply and rewrite
Factor
Give Back
Simplify
Swing back
1. Signs are ALWAYS opposite
2. Take the square root of both terms
( - ) ( + )
1.
2.
3.
4.
Factoring by GROUPING
2𝑥³ + 𝑥² − 12𝑥 − 6
13
Group common terms
Pull out the GCF
Factor the groups
Rewrite
Practice Set 6
11. 2x2 + 7x – 4
12. 3x2 + 19x + 6
13. 6x2 – 5x – 4
14. 4x2 – 16x + 7
Online tutorials and extra practice - factoring
Factoring Trinomials (skip substitution method):
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut28_facttri.ht
m
Factoring a Trinomial: http://www.algebrahelp.com/lessons/factoring/trinomial/
Video:
http://www.khanacademy.org/math/algebra2/polynomial_and_rational/quad_factoring/v/factori
ng-quadratic-expressions
14
Practice Set 7
15
Online Tutorials & Additional Practice – Radicals
http://www.freemathhelp.com/Lessons/Algebra_1_Simplifying_Radicals_BB.htm
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut37_radical.htm
http://www.khanacademy.org/math/arithmetic/exponents-radicals/radical-radicals/v/simplifying-radicals
16
Practice Set 8 Find the slope of the line between that contains these points
17
18
19
20
Practice Set 10
21
22
Linear Equations in Two Variables
Examples:
a) Find the slope of the line passing through the points (-1, 2) and (3, 5).
slope = m =
y2 - y1
x2 - x1
m=

5-2
3 - (-1)

3
4
b) Graph y = 2/3 x - 4 with slope-intercept method.
Reminder: y = mx + b is slope-intercept form where m =. slope and b = y-intercept.
Therefore, slope is 2/3 and the y-intercept is – 4.
Graph accordingly.
c) Graph 3x - 2y - 8 = 0 with slope-intercept method.
Put in Slope-Intercept form: y = -3/2 x + 4
m = 3/2
b = -4
d) Write the equation of the line with a slope of 3 and passing through the point (2, -1)
y = mx + b
-1 = 3(2) + b
-7 = b
Equation:
y = 3x – 7
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Practice Set 11
Write an equation, in slope-intercept form using the given information.
1) (5, 4) m =
2) (-2, 4)
2
3
m = -3
3) (-6, -3) (-2, -5)
Online tutorials and extra practice
Using the slope and y-intercept to graph lines: http://www.purplemath.com/modules/slopgrph.htm
Straight-line equations (slope-intercept form): http://www.purplemath.com/modules/strtlneq.htm
Slopes and Equations of Lines: http://www.regentsprep.org/Regents/math/ALGEBRA/AC1/indexAC1.htm
List of videos to watch: http://www.khanacademy.org/search?page_search_query=lines
Video: http://www.montereyinstitute.org/courses/Algebra1/U06L1T1_RESOURCE/index.html
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H. Solving Systems of Equations
Solve for x and y:
x = 2y + 5
Solve for x and y:
3x + 7y = 2
3x + 5y = 1
2x + 3y = 0
Using linear combination (addition/
subtraction) method:
Using substitution method:
3(3x + 5y = 1)
3(2y + 5) + 7y = 2
-5(2x + 3y = 0)
6y + 15 + 7y = 2
9x + 15y = 3
13y = -13
-l0x - 15y = 0
y = -1
-1x = 3
x = -3
x = 2(-1) + 5
x=3
2(-3) + 3y = 0
y=2
Solution: (3, -1)
Solution: (-3, 2)
Solve each system of equations by either the substitution method or the linear combination (addition/
subtraction) method. Write your answer as an ordered pair.
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Practice Set 12
1.
y = 2x + 4
-3x + y = -9
2. 2x + 3y = 6
-3x + 2y = 17
3.
x – 2y = 5
3x – 5y = 8
4.
3x + 7y = -1
6x + 7y = 0
Online tutorials and extra practice – solving systems of equations
http://www.purplemath.com/modules/systlin1.htm
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut19_systwo.htm
http://www.khanacademy.org/math/algebra2/systems_eq_ineq/systems_tutorial_precalc/v/trolls--tolls-and-systems-of-equations
http://www.montereyinstitute.org/courses/Algebra1/U06L1T2_RESOURCE/index.html
http://www.montereyinstitute.org/courses/Algebra1/U06L1T3_RESOURCE/index.html
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J. Absolute Values
Absolute Value Equations
Example 1:
|x - 3| = 9
The absolute value term is already by itself on the left side.
x - 3 = 9 and x - 3 = -9
We need to create 2 equations. The first equation is exactly like
the original problem, but without the absolute value. The second
equation is created by keeping everything the same on the variable
side, but just changing the sign of the expression on the other side
(in this problem, 9 becomes -9).
x - 3 = 9 and x - 3 = -9
+3 +3
+3 +3
x = 12 and
x = -6
Solve each equation separately. Since we have 2 equations, most of
the time there will be 2 answers (you won't always have 2 answers).
Answer: x = 12 and x = -6
Example 2:
|x + 4| - 2 = 11
Our first goal in absolute value equations is to isolate the absolute
value. In this case, on the absolute value side of the equation, we
have the terms |x + 4| and -2. In other words, we have to get rid
of that -2.
|x + 4| - 2 = 11
+2 +2
|x + 4| = 13
Adding 2 to both sides gets the absolute value by itself.
Now we have to create our 2 equations to solve.
x + 4 = 13 and x + 4 = -13
The first equation is the same as our isolated equation in the
previous step, but without the absolute value. For the second
equation, we'll just change the sign of the other side of the
expression so that 13 becomes -13.
x + 4 = 13 and x + 4 = -13
-4 -4
-4 -4
x = 9 and
x = -17
Solve each equation separately.
Answer: x = 9 and x = -17
TIP
Remember, absolute value means distance from zero on a number line, and distance is always
positive. So, the absolute value of something can NEVER be negative. (I'm not saying your
solutions can't be negative, we had lots of negative solutions in the above problems). Just that
the answer to the question "what is the distance from zero" is always positive. Look at the
examples below.
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Example 3:
|x| = -3
The absolute value is already isolated. This problem is asking "what
number is -3 away from zero on a number line?" We know the absolute
value of anything HAS to be positive, so this problem has no solutions
because nothing is -3 away from zero.
Answer: No Solution
-3|x + 2| = 9
To isolate our absolute value, we have to divide both sides by -3, which
transforms our problem to: |x + 2| = -3. Now that the absolute value is
isolated, we can clearly see that this problem has no solutions.
Answer: No Solution
-2|x + 5| = -10
TIP Just because an equation has negative numbers in it does not mean
it is automatically a No Solution problem. You cannot make this
determination until the absolute value has been isolated! To isolate this
equation, you have to divide both sides by -2. This gives us the equation:
|x + 5| = 5. I won't finish out this problem, but you can see now that
this problem will have solutions.
(which are -10 and 0, in case you are curious).
Problem Set 13
Solve.
1. |𝑥| = 3
2.
|𝑥 − 4| = −3
3. |𝑥| − 10 = −3
4. −3|𝑥| = −6
5. 12 = −4|𝑥|
6. |𝑥 + 2| = 6
Online tutorials and extra practice:
http://www.algebrafunsheets.com/algebratutorials/tutorials.php?name=SolvingAbsValueBasic.html
http://www.purplemath.com/modules/solveabs.htm
http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/absolute-valueequations/v/absolute-value-equations
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ANSWERS TO PRACTICE SETS
PRACTICE SET 1
1. 24𝑥 + 3𝑦
2. −15𝑦 2 + 37𝑦 + 22
3. 9𝑛 − 3
4. −22𝑏 + 6
5. 160𝑞𝑥 + 110𝑞
6. −5𝑥 + 6
7. 74𝑧 − 24𝑤
8. 56𝑐 − 117
9. −27𝑥 2 + 54𝑥 − 9
10. 31𝑥 − 𝑦 + 42
PRACTICE SET 2
1. 7
2. 26
3. 9.5
4. 13
5. 3.5
6. 16
7. 19
8. -2.8
9. 9.5
10. 0
PRACTICE SET 3
1. V = W - Y
4. 𝑥 =
10−𝑡
𝑑
2. 𝑤 =
81
5. 𝑔 =
𝑃+1620
3. 𝑓 = −3 +
9𝑟
6. 𝑥 =
180
2
3
𝑑
9𝑦+5ℎ+𝑢
4
PRACTICE SET 4
1. 𝑐 8
2. 𝑚12
3. 𝑘 20
4. 1
5. 𝑝11 𝑞 7
6. 9𝑧 9
7. −𝑡 21
8. 3𝑓 3
9. 60ℎ8 𝑘 5
10.
11. 81𝑚8 𝑛4
12. 1
13. 30𝑎3 𝑏 4 𝑐
14. 4𝑥
𝑎3 𝑏 4
3𝑐
15. 24𝑥 4 𝑦 7
PRACTICE SET 5
1. x2 + x - 90
2. x2 - 5x - 84
3. x2 - 12x + 20
4. x2 + 73x - 648
5. 8x2 + 2x - 3
6. 18x2 - 100x + 50
7. -6x2 - 20x - 16
8. x2 + 20x + 100
9. x2 - 10x + 25
10. 4x2 - 12x + 9
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PRACTICE SET 6
2. 4ab2(a – 4b + 2c)
3. (x – 5)(x + 5)
4. (n + 5)(n + 3)
5. (g – 5)(g – 4)
6.
7. (z – 10)(z + 3)
8. (m + 9)2
9. 4y(y – 3)(y + 3)
10. 5(x + 9)(x – 3)
11. (x + 4)(2x – 1)
12. (x + 6)(3x + 1)
13. (3x – 4)(2x + 1)
14. (2x – 7)(2x – 1)
1.
3x ( x + 2)
(d + 7)(d – 4)
PRACTICE SET 7
2.
3.
11
5.
9 6
6.
8
7.
60 5
8.
21 3
9.
40 19
10.
5
5 5
or
5
3
3
3 10
4.
12 2
1.
5 7
PRACTICE SET 8
1.
-3
4.
0
2.
2
3
3.
5. 1
1
2
6. Undefined
PRACTICE SET 9
1.
Slope: 2;
y-intercept: (0,5)
2.
Slope:
30
1
; y-intercept: (0, -3)
2
3.
Slope:
-2
; y-intercept: (0, 4)
5
5. Slope: -1; y-intercept: (0, 2)
4.
Slope: -3;
6. Slope: 1;
y-intercept: (0, 0)
PRACTICE SET 10
1.
3x + y = 3
2. 5x + 2y = 10
31
y-intercept: (0, 0)
3.
4. 4x – 3y = 9
y=4
5. -2x + 6y = 12
6. x = -3
PRACTICE SET 11
1. y =
-2
22
x+
3
3
2. y = -3x - 2
PRACTICE SET 12
1.
(13, 30)
2.
(– 3, 4)
3.
(– 9, – 7)
4.
1 2
 , 
3 7
32
3. y =
-1
x-6
2
PRACTICE SET 13
1.
-3, 3
2.
1, 7
3.
-7, 7
4.
-2, 2
5.
No solution
6.
-8, 4
33