ALGEBRA 1
Teacher’s Name:
Unit 1
Chapter 1
Chapter 2
This book belongs to:
FALL 2014
1
Algebra 1
Date
Name:
Section
Period:
Homework Assignment
1.1 Variables and Expressions
1.2 Order of Operations
1.3 Properties of Numbers
1.1 – 1.3 Review
----------
1.1 – 1.3 Quiz
1.5 Equations
1.6 Relations
1.7 Functions
1.5 – 1.7 Review
----------
1.5 – 1.7 Quiz
1.8 Interpreting Graphs of
Functions
2.1 Writing Equations
2.3 Solving Multi-step
Equations
2.3 Solving Multi-step
Equations
2.4 Solving Equations with the
Variable on Each Side
2.4 Solving Equations with the
Variable on Each Side
1.8, 2.1, 2.3, 2.4 Review
----------
1.8, 2.1, 2.3, 2.4 Quiz
2.8 Literal Equations and
Dimensional Analysis
2.8 Literal Equations and
Dimensional Analysis
Unit Review
----------
Unit 1 Test
TOTAL HW SCORE :
/
2
CHAPTER 1
3
Algebra 1
Section 1.1 Notes: Variables and Expressions
Algebraic Expression: an expression consisting of one or more
operations.
along with one or more arithmetic
Variables: symbols or letters used to represent
may be used as a variable.
Term: a
,a
numbers or values. Any
, or a
of numbers and variables.
Factors: in an algebraic expression, the
.
Product: in an algebraic expression, the
•
•
of quantities being multiplied.
In the equation 5𝑥 = 15,
are the factors. The product is
.
A raised dot or set of parentheses are often used to indicate a product. Here are several ways to represent the product of x
and y.
Power: An expression of the form
, read “x to the nth power”.
Exponent: In an expression of the form 𝑥 𝑛 , the exponent is
. It indicates the number of times x is
.
Base: In an expression of the form 𝑥 𝑛 , the base is
.
Example 1: Write a verbal expression for each algebraic expression.
a) 8𝑥 2
b) 𝑦 5 − 16𝑦
c) 16𝑢2 − 3
1
d) 𝑎 +
2
6𝑏
7
4
Example 2: Write an algebraic expression for each verbal expression.
a) a number c less than 5
b) 9 plus the product of 2 and a number d
c) one third of the area a
Variables can represent quantities that are known and quantities that are unknown. They are also used in
.
Example 3: Write an algebraic expression.
a) Mr. Nehru bought two adult tickets and three student tickets for the planetarium show. Write an algebraic expression that
represents the cost of the tickets.
b) Katie bakes 40 pastries and makes coffee for 200 people. Write an algebraic expression to represent this situation.
5
Algebra 1
Section 1.2 Notes: Order of Operations
Evaluate: to
or
of an expression
Example 1: Evaluate the expressions.
a) 26
b) 24
c) 45
d) 73
Order of Operations: the rule that lets you know
.
8-3
Example 2: Evaluate using the order of operations.
a) 48 ÷ 23 ∙ 3 + 5
b) 20 − 7 + 82 − 7 ∙ 11
When one or more grouping symbols are used, evaluate within the
Grouping symbols such as
operations.
( ),
[ ], and
.
{ } are used to clarify or change the order of
Example 3: Expressions with grouping symbols.
a) (8 − 3) ∙ 3(3 + 2)
b) 4[12 ÷ (6 − 2)]2
c)
25 −6 ∙ 2
33 −5 ∙ 3−2
To evaluate an algebraic expression,
the variables with their values. Then find the value of the numerical expression
using the
.
Example 4: Evaluate the algebraic expression.
a) 2(𝑥 2 − 𝑦) + 𝑧 2 if x = 4, y = 3, and z = 2
b) 5𝑑 + (6𝑓 − 𝑔) if d = 4, f = 3, and g = 12
6
Example 5:
Each side of the Great Pyramid of Giza, Egypt, is a triangle. The base of each triangle once measured 230 meters. The height of each
triangle once measured 187 meters. The area of a triangle is one-half the product of the base b and its height h.
a) Write an expression that represents the area of one side of the Great Pyramid.
b) Find the area of one side of the Great Pyramid.
According to the California Department of Forestry, an average of 539.2 fires each year are started by burning debris, while campfires
are responsible for an average of 129.1 each year.
a) Write an algebraic expression that represents the number of fires, on average, in d years of debris burning and c years of campfires.
b) How many fires would there be in 5 years?
7
Algebra 1
Section 1.3 Notes: Properties of Numbers
Equivalent expressions: Expressions that denote the same value for all values of the variable(s).
8
Example 1: Evaluate Using Properties
1
a) Evaluate (12 − 8) + 3(15 ÷ 5 − 2) Name the property used in each step.
4
b) Evaluate 2 ∙ 3 + (4 ∙ 2 − 8)
Name the property used in each step.
Example 2:
a) Magina made a list of trail lengths to find the total miles she rode. Find the total miles Magina rode her horse.
b) Rafael is buying furnishings for his first apartment. He buys a couch for $300, lamps for $30.50, a rug for $25.50, and a table for
$50. Find the total cost of these items.
Example 3: Evaluate 2 ∙ 8 ∙ 5 ∙ 7 using the properties of numbers. Name the property used in each step.
9
Algebra 1
Section 1.5 Notes: Equations
Open Sentence: a mathematical statement with one or more
.
Equation: a mathematical sentence that contains an
.
Solving an open sentence: Finding a
for the
that results in a true sentence or an
that results in a true statement when substituted into the equation.
Solution: a replacement value for the variable in an open sentence.
Replacement Set: a
from which replacements for a variable may be chosen.
Set: a
that is often shown using braces.
Element: Each
in the set.
Solution Set: the set of elements from the replacement set that make an open sentence
.
Example 1:
a) Find the solution set of the equation 4𝑎 + 7 = 23 if the replacement set is {2, 3, 4, 5, 6}.
b) Find the solution set of the equation 28 = 4(1 + 3𝑑) if the replacement set is {0, 1, 2, 3}.
You can often solve an equation by applying the order of operations.
Example 2: Standardized test practice
a) Solve 3 + 4(23 − 2) = 𝑏.
a. 19
b. 27
c. 33
d. 42
b. 6
c. 14.2
d. 27
b) Solve 𝑡 = 92 ÷ (5 − 2).
a. 3
10
Some equations have a
. Other equations
have a solution.
Example 3: Solve each equation.
a) 4 + (32 + 7) ÷ 𝑛 = 8
b) 4𝑛 − (12 + 2) = 𝑛(6 − 2) − 9
Identity: an equation that is true for every value of the variable.
Example 4: Solve (5 + 8 ÷ 4) + 3𝑘 = 3(𝑘 + 32) − 89
Example 5:
a) Dalila pays $16 per month for a gym membership. In addition, she pays $2 per Pilates class. Write and solve an equation to find the total amount
Dalila spent this month if she took 12 Pilates classes.
b) Amelia drives an average of 65 miles per hour. Write and solve an equation to find the time it will take her to drive 36 miles.
11
Algebra 1
Section 1.6 Notes: Relations
Coordinate System: The grid formed by the intersection of two number lines, the
Ordered pair: a set of
axis and the
used to locate any point on a coordinate plane, written in the form
x-coordinate: the
number in an ordered pair.
y-coordinate: the
number in an ordered pair.
axis.
.
Relation: a set of ordered pairs
Mapping: illustrates how each element of the domain is
with an element in the range.
Domain: the set of
numbers of the ordered pairs in a relation.
Range: the set of
numbers of the ordered pairs in a relation.
In the relation above, the domain is
and the range is
.
Example 1:
a) Express {(4, 3), (-2, -1), (2, -4), (0, -4)} as a table, a graph, and a mapping.
x
y
12
b) Determine the domain and range of the relation.
Independent variable: the variable in a relation with a value that is
.
Dependent variable: the variable in a relation with a value that
of the independent variable.
Example 3: Identify the independent and the dependent variable for each relation.
a) In warm climates, the average amount of electricity used rises as the daily average temperature increases and falls as the daily average temperature
decreases.
b) The number of calories you burn increases as the number of minutes that you walk increases.
A relation can be graphed without a scale on either axis. These graphs can be interpreted by analyzing their shape.
Example 3: Describe what is happening in each graph.
a)
b)
13
Algebra 1
Section 1.6 Worksheet
Name:
Period:
1. Express {(4, 3), (–1, 4), (3, –2), (–2, 1)} as a table, a graph, and a mapping. Then determine the domain and range.
Describe what is happening in each graph.
2. The graph below represents the height of a tsunami as it travels across an ocean.
3. The graph below represents a student taking an exam.
Express the relation shown in each table, mapping, or graph as a set of ordered pairs.
4.
5.
0
9
−8
3
2
1
6.
−6
4
7. BASEBALL The graph shows the number of home runs hit by Andruw Jones of the Atlanta Braves. Express the relation as a set of
ordered pairs. Then describe the domain and range.
14
8. HEALTH The American Heart Association recommends that your target heart rate during exercise should be between 50% and
75% of your maximum heart rate. Use the data in the table below to graph the approximate maximum heart rates for people of
given ages.
Age (years)
20
25
30
35
40
Maximum Heart Rate
(beats per minute)
200
195
190
185
180
Source: American Heart Association
9. NATURE Maple syrup is made by collecting sap from sugar maple trees and boiling it down to remove excess water. The graph
shows the number of gallons of tree sap required to make different quantities of maple syrup. Express the relation as a set of
ordered pairs.
10. BAKING Identify the graph that best represents the relationship between the number of cookies and the equivalent number of
dozens.
11. DATA COLLECTION Margaret collected data to determine the number of books her schoolmates were bringing home each
evening. She recorded her data as a set of ordered pairs. She let x be the number of textbooks brought home after school, and y be
the number of students with x textbooks. The relation is shown in the mapping.
a. Express the relation as a set of ordered pairs.
b. What is the domain of the relation?
c. What is the range of the relation?
15
Algebra 1
Section 1.7 Notes: Functions
Function: a relationship between
. In a function there is
output for each input.
Example 1:Determine whether each relation is a function. Explain.
a)
Discrete function: a function of points that are
b)
connected.
Continuous function: a function that can be graphed with a
line or a
curve.
16
Example 2: There are three lunch periods at a school. During the first period, 352 students eat. During the second period 304 students
eat. During the third period, 391 students eat.
a) Make a table showing the number of students for each of the three lunch periods.
b) Determine the domain and range of the function.
c) Write the data set of ordered pairs then graph the data.
d) State whether the function is discrete or continuous. Explain your reasoning.
Vertical line test: if any
a function.
line passes through no more than one point of the graph of a relation, then the relation is
Example 3: Equations as functions
a) Determine whether 𝑥 = −2 represents a function.
b) Determine whether 4𝑥 = 𝑦 + 8 represents a function.
17
A function can be represented in different ways.
Function Notation: A way to
𝑦 = 3𝑥 − 8 is written 𝑓(𝑥) = 3𝑥 − 8.
a function that is defined by an equation. In function notation, the equation
It is said “f of x”. If a number is inside the parenthesis than that is the number you
Example 4: For 𝑓(𝑥) = 3𝑥 − 4, find each value.
a) 𝑓(4)
b) 𝑓(−5)
Nonlinear function: a function with a graph that is not a
Example 5: If ℎ(𝑡) = 1248 − 160𝑡 + 16𝑡 2 , find each value.
a) ℎ(3)
for x.
line.
b) ℎ(2𝑧)
18
Algebra 1
Section 1.8 Notes: Interpreting Graphs of Functions
y-intercept: the
of a point where a graph crosses the y – axis.
x-intercept: the
of a point where a graph crosses the x – axis .
Example 1: The graph shows the cost at a community college y as a function of the number of credit hours taken x. Identify the
function as linear or nonlinear. Then estimate and interpret the intercepts of the function.
Line Symmetry: if a vertical line is drawn and each half of the graph on either side of the line matches exactly.
Example 2: The graph shows the cost y to manufacture x units of product. Describe and interpret any symmetry.
19
Example 3: The graph shows the population y of deer x years after the animals are introduced on an island. Estimate and interpret
where the function is positive, negative, increasing and decreasing, the x – coordinate of any relative extrema, and the end behavior of
the graph.
20
Algebra 1
Section 1.8 Worksheet
Name:
Period:
Identify the function graphed as linear or nonlinear. Then estimate and interpret the intercepts of the graph, any symmetry,
where the function is positive, negative, increasing, and decreasing, the x–coordinate of any relative extrema, and the end
behavior of the graph.
1.
2.
3.
4.
5. HEALTH The graph shows the Calories y burned by a 130-pound person swimming freestyle laps as a function of time x. Identify
the function as linear or nonlinear. Then estimate and interpret the intercepts.
21
6. TECHNOLOGY The graph below shows the results of a poll that asks Americans whether they used the Internet yesterday.
Estimate and interpret where the function is positive, negative, increasing, and decreasing, the x-coordinates of any relative
extrema, and the end behavior of the graph.
7. GEOMETRY The graph shows the area y in square centimeters of a rectangle with perimeter 20 centimeters and width x
centimeters. Describe and interpret any symmetry in the graph.
8. EDUCATION Identify the function graphed as linear or nonlinear. Then estimate and interpret the intercepts of the graph, any
symmetry, where the function is positive, negative, increasing, and decreasing, the x-coordinate of any relative extrema, and the end
behavior of the graph.
22
CHAPTER 2
23
24
Algebra 1
Section 2.1 Notes: Writing Equations
To write an equation…
1) Identify the
for which you are looking and assign a variable to it.
2) Write the sentence as an
* Look for key words that indicate where to place the equal to sign.
Examples: is, is as much as, is the same as, is identical to
Example 1: Translate the sentence into an equation.
a) A number b divided by three is equal to six less than c.
b) Fifteen more than z times six is y times two minus eleven.
c) Two plus the quotient of a number and 8 is the same as 16.
d) Twenty-seven times k is h squared decreased by 9.
Example 2:
a) A jelly bean manufacturer produces 1,250,000 jelly beans per hour. How many hours does it take them to produce 10,000,000 jelly
beans?
b) There are 50 members in the North Carolina Senate. This is 70 fewer than the number in North Carolina House of Representatives.
How many members are in the North Carolina House of Representatives?
Formula: a rule for the relationship between certain quantities.
Example 3: Translate the sentence into a formula.
a) The perimeter of a square equals four times the length of a side.
b) In a right triangle, the square of the measure of the hypotenuse c is equal to the sum of the squares of the measures of the legs, a and
b.
If you are given an equation you can write a sentence or create your own word problem.
Example 4: Translate the equation into a verbal sentence.
a) 12 − 2𝑥 = −5
b) 𝑎2 + 3𝑏 =
𝑐
6
25
When given a set of information, you can create a problem that relates a story.
Example 5: Write a problem based on the given information.
a) 𝑓 = cost of fries
𝑓 + 1.50 = cost of burger
b) 𝑝 = Beth’s salary
0.1𝑝 = bonus
4(𝑓 + 1.50) − 𝑓 = 8.25
𝑝 + 0.1𝑝 = 525
26
Section 2.3 Notes: Solving Multi-Step Equations
Multi-Step Equation: an equation that requires
to solve.
To solve a multi-step equation, we must
.
Example 1: Solve and check your solution.
a) 2𝑞 + 11 = 3
Check
b)
𝑘+9
12
= −2
Check
Example 2: Write and solve a multi-step equation.
1
a) Susan had a $10 coupon for the purchase of any item. She bought a coat that was its original price. After using the coupon, Susan
2
paid $125 for the coat before taxes. What was the original price of the coat? Write an equation for the problem. Then solve the
equation.
3
b) Len read of a graphic novel over the weekend. Monday, he read 22 more pages. If he has read 220 pages, how many pages does
4
the book have. Write and solve a multi-step equation.
Consecutive integers: integers in counting order
Example: 4, 5, 6 or n, n + 1, n + 2.
Number theory: the study of numbers and the relationships between them.
Example 3: Write an equation for the problem below. Then solve the equation and answer the problem.
a) Find three consecutive odd integers whose sum is 57.
b) Find three consecutive integers with a sum of 21.
27
28
Algebra 1
Section 2.4 Notes: Solving Equations with the Variable on Each Side
To solve an equation that has variables on each side, use the
an equivalent equation with the variable terms on one side.
to write
Example 1: Solve the equation. Check your solution.
a) 8 + 5𝑐 = 7𝑐 − 2
𝑥
b) 5𝑎 + 2 = 6 − 7𝑎
1
c) + 1 = 𝑥 − 6
2
d) 1.3𝑐 = 3.3𝑐 + 2.8
4
If equation containing grouping symbols, such as parentheses or brackets, use the
to remove the grouping symbols.
Example 2: Solve each equation and check your solution.
1
a) (18 + 12𝑞) = 6(2𝑞 − 7)
3
Some equations have
Some equations are
b) 7(𝑛 − 1) = −2(3 + 𝑛)
meaning there is no value of the variable that will result in a true equation.
of the variable. We call these identities.
Example 3:Solve each equation.
a) 8(5𝑐 − 2) = 10(32 + 4𝑐)
1
b) 4(𝑡 + 20) = (20𝑡 + 400)
5
29
Example 4: Standardized test practice.
a) Find the value of h so that the figures have the same area.
A. 1
B. 3
C. 4
D. 5
b) Find the value of x so that the figures have the same perimeter.
F. 1.5
G. 2
H. 3.2
J. 4
30
Algebra 1
Section 2.8 Notes: Literal Equations and Dimensional Analysis
Some equations contain more than one
. At times, you will need to solve these equations for one of the variables.
Example 1: Solve for the specific variable
a) Solve 5𝑏 + 12𝑐 = 9 for b
c) Solve
𝑘−2
5
= 11𝑗 for k
b) Solve 15 = 3𝑛 + 6𝑝 for n
d) Solve 𝑎(𝑞 − 8) = 23 for q
Sometimes we need to solve equations for a variable that is on both sides of the equation.
Step 1: Get all terms with the variable onto
.
Step 2: Use the
to
the variable for which you are solving.
Example 2: Solve for the specific variable.
a) Solve 7𝑥 − 2𝑧 = 4 − 𝑥𝑦 for x
b) Solve 6𝑞 − 18 = 𝑞𝑟 + 𝑡 for q
Literal Equation: a formula or equation with several variables.
𝑚
Example 3: A car’s fuel economy E (miles per gallon) is given by the formula 𝐸 = , where m is the number of miles driven and g is
the number of gallons of fuel used.
𝑔
a) Solve the formula for m.
b) If Quanah’s car has an average fuel conception of 30 miles per gallon and she used 9.5 gallons, how far did she drive?
31
Dimensional Analysis or Unit Analysis: the process of carrying units throughout a computation.
Example 4:Use dimensional Analysis
a) The average weight of the chimpanzees at a zoo is 52 kilograms. If 1 gram ≈ 0.0353 ounce, use dimensional analysis to find the
average weight of the chimpanzees in pounds. (Hint: 1 lb = 16 oz)
b) A car travels a distance of 100 feet in about 2.8 seconds. What is the velocity of the car in miles per hour? Round to the nearest
whole number.
32
Algebra 1
Section 2.8 Worksheet
Name:
Period:
Solve each equation or formula for the variable indicated.
1. d = rt, for r
2. 6w – y = 2z, for w
3. mx + 4y = 3t, for x
4. 9s – 5g = –4u, for s
5. ab + 3c = 2x, for b
6. 2p = kx – t, for x
2
8. ℎ + g = d, for h
2
10. 𝑎 – q = k, for a
7. 𝑚 + a = a + r, for m
3
9. 𝑦 + v = x, for y
3
11.
𝑟𝑥 + 9
5
= h, for x
13. 2w – y = 7w – 2, for w
2
5
3
4
12.
3𝑏 − 4
2
= c, for b
14. 3ℓ + y = 5 + 5ℓ, for ℓ
15. ELECTRICITY The formula for Ohm’s Law is E = IR, where E represents voltage measured in volts, I represents current
measured in amperes, and R represents resistance measured in ohms.
a. Solve the formula for R.
b. Suppose a current of 0.25 ampere flows through a resistor connected to a 12-volt battery. What is the resistance in the circuit?
16. MOTION In uniform circular motion, the speed v of a point on the edge of a spinning disk is v =
disk and t is the time it takes the point to travel once around the circle.
2𝜋
𝑡
r, where r is the radius of the
a. Solve the formula for r.
b. Suppose a merry–go–round is spinning once every 3 seconds. If a point on the outside edge has a speed of 12.56 feet per
second, what is the radius of the merry-go-round? (Use 3.14 for π.)
17. HIGHWAYS Interstate 90 is the longest interstate highway in the United States, connecting the cities of Seattle, Washington and
Boston, Massachusetts. The interstate is 4,987,000 meters in length. If 1 mile = 1.609 kilometers, how many miles long is
Interstate 90?
18. INTEREST Simple interest that you may earn on money in a savings account can be calculated with the formula I = prt. I is the
amount of interest earned, p is the principal or initial amount invested, r is the interest rate, and t is the amount of time the money
is invested for. Solve the formula for p.
33
19. DISTANCE The distance d a car can travel is found by multiplying its rate of speed r by the amount of time t that it took to travel
the distance. If a car has already traveled 5 miles, the total distance d is found by the formula d = rt + 5 . Solve the formula for r.
20. ENVIRONMENT The United States released 5.877 billion metric tons of carbon dioxide into the environment through the
burning of fossil fuels in a recent year. If 1 trillion pounds = 0.4536 billion metric tons, how many trillions of pounds of carbon
dioxide did the United States release in that year?
𝐹
21. PHYSICS The pressure exerted on an object is calculated by the formula P = , where P is the pressure, F is the force, and A is
𝐴
the surface area of the object. Water shooting from a hose has a pressure of 75 pounds per square inch (psi). Suppose the surface
area covered by the direct hose spray is 0.442 square inch. Solve the equation for F and find the force of the spray.
22. GEOMETRY The regular octagon is divided into
8 congruent triangles. Each triangle has an area of 21.7 square centimeters. The perimeter of the octagon is 48 centimeters.
a. What is the length of each side of the octagon?
b. Solve the area of a triangle formula for h.
c. What is the height of each triangle? Round to the nearest tenth.
34
ALGEBRA 1
Linden
Unit 2
Chapter 3
Chapter 4
Chapter 5
This book belongs to:
FALL 2014
Algebra 1
Name:
Period:
35
Date
Section
Homework Assignment
3.3 Rate of Change and
Slope
3.3 Rate of Change and
Slope
4.1 Graphing Equations
in Slope-Intercept Form
4.2 Writing Equations in
Slope-Intercept From
3.3, 4.1, 4.2 Quiz
4.3 Writing Equations in
Point Slope Form
---------------
4.3 Day 2 (Standard
Form)
4.4 Parallel and
Perpendicular Lines
4.7 Inverse Linear
Functions
4.7 Lab
4.3, 4.4, 4.7 Quiz
---------------
5.3 Solving Multi-step
Inequalities
5.3 Solving Multi-step
Inequalities
5.5 Inequalities Involving
Absolute Value
5.5 Inequalities Involving
Absolute Value
5.6 Graphing Inequalities
in Two Variables
5.6 Graphing Inequalities
in Two Variables
No School
Unit 2 Review
Unit 2 Review
Unit 2 Test
TOTAL HW SCORE :
---------------
/
=
%
---------------
36
CHAPTER 3
37
Algebra 1
Section 3.3 Notes: Rate of Change and Slope
Rate of Change: a
quantity.
that describes, on average, how much a quantity changes with respect to a change in another
Example 1: Use the table to find the rate of change. Then explain its meaning.
In example 1 the rates of change have been constant. Many real-world situations involve rates of change that are _______ constant.
Example 2: The graph below shows the number of U.S. passports issued in 2002, 2004, and 2006.
a) find the rates of change for 2002- 2004 and 2004 – 2006.
b) Explain the meaning of the rate of change in each case.
c) How are the different rates of change shown on the graph?
A rate of change is constant for a function when the rate of change is the ____________ between any pair of points on the graph of the
function.
functions have a constant rate of change.
38
Example 3: Determine whether each function is linear.
a)
b)
Slope: the ratio of the change in the y – coordinates (rise) to the change in the x – coordinates (run) as you move from one point to
another.
Slope describes how steep a line is. The greater the absolute value of the slope, the steeper the line.
Because a linear function has a constant rate of change, any two points on a nonvertical line can be used to determine its slope.
The slope of a line can be positive, negative, zero, or undefined.
If the line is not horizontal or vertical, then the slope is either positive or negative.
Example 4: Find the slope of a line that passes through each pair of points.
a) (-3, 2) and (5, 5)
b) (-3, -4) and (-2, -8)
c) (-3, 4) and (4, 4)
Example 5: Find the slope of the line that passes through (-2, -4) and (-2, 3).
39
Example 6: Find the value of r so the line passes through each pair of points and has the given slope.
a) (6, 3), (r, 2); 𝑚 =
1
2
b) (–2, 6), (r, – 4); 𝑚 = −5
c) (r, –6), (5, –8); 𝑚 = −8
40
Algebra 1
Section 3.3 Worksheet
Find the slope of the line that passes through each pair of points.
1.
2.
3.
4. (6, 3), (7, –4)
5. (–9, –3), (–7, –5)
6. (6, –2), (5, –4)
7. (7, –4), (4, 8)
8. (–7, 8), (–7, 5)
9. (5, 9), (3, 9)
10. (15, 2), (–6, 5)
11. (3, 9), (–2, 8)
12. (–2, –5), (7, 8)
13. (12, 10), (12, 5)
14. (0.2, –0.9), (0.5, –0.9)
15. � , � , �− , �
7
3
4
3
1
3
2
3
Find the value of r so the line that passes through each pair of points has the given slope.
16. (–2, r), (6, 7), m =
1
17. (–4, 3), (r, 5), m =
19. (–5, r), (1, 3), m =
7
20. (1, 4), (r, 5), m undefined
2
6
22. (r, 7), (11, 8), m = –
1
5
1
4
18. (–3, –4), (–5, r), m = –
9
2
21. (–7, 2), (–8, r), m = –5
23. (r, 2), (5, r), m = 0
24. ROOFING The pitch of a roof is the number of feet the roof rises for each 12 feet horizontally. If a roof has a pitch of 8, what is
its slope expressed as a positive number?
25. SALES A daily newspaper had 12,125 subscribers when it began publication. Five years later it had 10,100 subscribers. What is
the average yearly rate of change in the number of subscribers for the five-year period?
26. HIGHWAYS Roadway signs such as the one below are used to warn drivers of an upcoming steep down grade that could lead to
a dangerous situation. What is the grade, or slope, of the hill described on the sign?
41
27. AMUSEMENT PARKS The SheiKra roller coaster at Busch Gardens in Tampa, Florida, features a 138-foot vertical drop. What
is the slope of the coaster track at this part of the ride? Explain.
28. CENSUS The table shows the population density for the state of Texas in various years. Find the average annual rate of change in
the population density from 2000 to 2009.
Population Density
Year
People Per Square Mile
1930
22.1
1960
36.4
1980
54.3
2000
79.6
2009
96.7
Source: Bureau of the Census, U.S. Dept. of Commerce
29. REAL ESTATE A realtor estimates the median price of an existing single-family home in Cedar Ridge is $221,900. Two years
ago, the median price was $195,200. Find the average annual rate of change in median home price in these years.
30. COAL EXPORTS The graph shows the annual coal exports from U.S. mines in millions of short tons.
Source: Energy Information Association
a. What was the rate of change in coal exports between 2001 and 2002?
b. How does the rate of change in coal exports from 2005 to 2006 compare to that of 2001 to 2002?
c. Explain the meaning of the part of the graph with a slope of zero.
42
CHAPTER 4
43
Algebra 4.1
Section 4.1 Notes: Graphing Equations in Slope-Intercept Form
Slope-intercept form: an equation of the form 𝑦 = 𝑚𝑥 + 𝑏, where m is the slope and b is the y intercept.
The variables m and b are called parameters of the equation. Changing either value changes the equation’s graph.
When graphing an equation.
1) Plot the y – intercept
2) Use the slope to plot additional points.
3) Draw a line through the points.
1
Example 1: Write an equation in slope intercept form of the line with a slope of and a y-intercept of – 1. Then graph the equation.
4
When an equation is not written in slope-intercept form, it may be easier to rewrite it before graphing.
Example 2: Graph 5𝑥 + 4𝑦 = 8
44
Constant Functions: a linear function of the form 𝑦 = 𝑏.
Constant functions do not cross the x-axis except when 𝑦 = 0. Constant functions are horizontal lines therefore their slope is 0. Their
domain is all real numbers, and their range is b.
Vertical lines have no slope. So, equations of vertical lines cannot be written in slope-intercept form.
Example 3: Graph 𝑦 = −7
Writing an equation given a graph.
1) Locate the y-intercept
2) Find the slope by using rise and run to find another point on the graph.
3) Write the equation in slope-intercept form
Example 4: Write an equation in slope-intercept form for the line shown in the graph.
Example 5: The ideal maximum heart rate for a 25 year old exercising to burn fat is 117 beats per minute. For every five years older
than 25, that ideal rate drops three beats per minute.
a) Write a linear equation to find the ideal maximum heart rate for anyone over 25 who is exercising to burn fat.
b) Graph the equation.
c) Find the ideal maximum heart rate for a 55 year old person exercising to burn fat.
45
13. CAR CARE Suppose regular gasoline costs $2.76 per gallon. You can purchase a car wash at the gas station for $3. The graph of
the equation for the cost of x gallons of gasoline and a car wash is shown below. Write the equation in slope-intercept form for the
line.
14. ADULT EDUCATION Angie’s mother wants to take some adult education classes at the local high school. She has to pay a onetime enrollment fee of $25 to join the adult education community, and then $45 for each class she wants to take. The equation
y = 45x + 25 expresses the cost of taking x classes. What are the slope and y-intercept of the equation?
15. BUSINESS A construction crew needs to rent a trench digger for up to a week. An equipment rental company charges $40 per day
plus a $20 non-refundable insurance cost to rent a trench digger. Write and graph an equation to find the total cost to rent the trench
digger for d days.
16. ENERGY From 2002 to 2005, U.S. consumption of renewable energy increased an average of 0.17 quadrillion BTUs per year.
About 6.07 quadrillion BTUs of renewable power were produced in the year 2002.
a. Write an equation in slope-intercept form to find the amount of renewable power P (quadrillion BTUs) produced in year y
between 2002 and 2005.
b. Approximately how much renewable power was produced in 2005?
c. If the same trend continues from 2006 to 2010, how much renewable power will be produced in the year 2010?
46
Algebra 1
Section 4.2 Notes: Writing Equations in Slope-Intercept Form
Write an equation given slope and a point
1) Find the y-intercept by substituting the point and slope into 𝑦 = 𝑚𝑥 + 𝑏. Solve for b.
2) Write the equation with the given slope and the y-intercept.
Example 1:
1
a) Write an equation of a line that passes through (2, – 3) with a slope of .
2
b) Write an equation of a line that passes through (–2, 5) with a slope of 3.
If you are given two points through which a line passes, you can use them to find the slope first. Then follow the steps above.
Example 2: Write an equation of the line that passes through each pair of points.
a) (–3, –4) and (–2, –8)
b) (6, –2) and (3, 4)
Constraint: a condition that a solution must satisfy.
Equations can be viewed as constraints in a problem situation. The solutions of the equation meet the constraints of the problem.
Example 3: During one year, Malik’s cost for self-serve regular gasoline was $3.20 on the first of June and $3.42 on the first of July.
Write a linear equation to predict Malik’s cost of gasoline the first of any month during the year, using 1 to represent January.
Linear extrapolation: the use of a linear equation to predict values that our outside the range of data.
Example 4: On average, Malik uses 25 gallons of gasoline per month. He budgeted $100 for gasoline in October. Use the prediction
equation in Example 3 to determine if Malik will have to add to his budget. Explain.
47
Algebra 1
Section 4.2 Worksheet
Write an equation of the line that passes through the given point and has the given slope.
1.
7. (3, 7); slope
2.
2
7
5
8. �−2, � ; slope −
2
1
2
9. (5, 0); slope 0
Write an equation of the line that passes through each pair of points.
10.
11.
13. (0, –4), (5, –4)
15. (–2, –3), (4, 5)
19. DANCE LESSONS The cost for 7 dance lessons is $82. The cost for 11 lessons is $122. Write a linear equation to find the total
cost C for ℓ lessons. Then use the equation to find the cost of 4 lessons.
21. FUNDRAISING Yvonne and her friends held a bake sale to benefit a shelter for homeless people. The friends sold 22 cakes on
the first day and 15 cakes on the second day of the bake sale. They collected $88
on the first day and $60 on the second day. Let x represent the number of cakes sold and y represent the amount of money made.
Find the slope of the line that would pass through the points given.
48
Algebra 1
Section 4.3 Notes: Writing Equations in Point-Slope Form Day 1
Point-slope form: an equation that can be written in the form 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) where m is the slope and (𝑥1 , 𝑦1 ) is a given point.
3
Example 1: Write an equation in point-slope form for the line that passes through (– 2, 0) with a slope of − . Then graph the
2
equation.
If you are given the slope and the coordinates of one or two points, you can write the linear equations in the following ways.
���� and 𝐶𝐷
����
Example 4: The figure shows trapezoid ABCD, with bases 𝐴𝐵
49
���� .
a) Write an equation in point-slope form for the line containing the side 𝐵𝐶
b) Write an equation in standard form for the same line.
50
Algebra 1
Section 4.3 Notes: Writing Equations in Point-Slope Form Day 2 Focus Standard Form
Standard form: 𝐴𝑥 + 𝐵𝑦 = 𝐶 where 𝐴 ≥ 0, A and B are not both zero, and A, B, and C are integers with a greatest common factor
of 1.
Example 2: Write the following equation in standard form.
3
a) 𝑦 = 𝑥 − 5
4
b) 𝑦 − 1 = 7(𝑥 + 5)
To find the y-intercept of an equation, rewrite the equation in slope-intercept form.
Example 3: Write the given equation in slope-intercept form.
4
a) 𝑦 − 5 = (𝑥 − 3)
3
b) 𝑦 + 6 = −3(𝑥 − 4)
51
Algebra 1
Section 4.3 Worksheet
Write an equation in point-slope form for the line that passes through each point with the given slope.
4. (1, 3), m = −
3
5. (–8, 5), m = −
4
2
6. (3, –3), m =
5
1
3
Write each equation in standard form.
7. y – 11 = 3(x – 2)
3
11. y + 2 = − (x + 1)
4
3
8. y – 10 = –(x – 2)
10. y – 5 = (x + 4)
13. y + 4 = 1.5(x + 2)
14. y – 3 = –2.4(x – 5)
2
Write each equation in slope-intercept form.
16. y + 2 = 4(x + 2)
1
1
20. y – = – 3(x + )
4
4
18. y – 3 = –5(x + 12)
2
1
21. y – = –2(x – )
3
4
22. CONSTRUCTION A construction company charges $15 per hour for debris removal, plus a one-time fee for the use of a trash
dumpster. The total fee for 9 hours of service is $195.
a. Write the point-slope form of an equation to find the total fee y for any number of hours x.
b. Write the equation in slope-intercept form.
c. What is the fee for the use of a trash dumpster?
23. MOVING There is a daily fee for renting a moving truck, plus a charge of $0.50 per mile driven. It costs $64 to rent the truck on a
day when it is driven 48 miles.
a. Write the point-slope form of an equation to find the total charge y for a one-day rental with x miles driven.
b. Write the equation in slope-intercept form.
c. What is the daily fee?
24. BICYCLING Harvey rides his bike at an average speed of 12 miles per hour. In other words, he rides
12 miles in 1 hour, 24 miles in 2 hours, and so on.
Let h be the number of hours he rides and d be distance traveled. Write an equation for the relationship between distance and time
in point-slope form.
52
Algebra1 1
Section 4.4 Notes: Parallel and Perpendicular Lines
Parallel lines: lines in the same plane that do not intersect.
Finding the equation of a line given the equation of a parallel line and a point on the line.
1. Find the slope of the given line.
2. Substitute the point provided and the slope from the given line into the point-slope form.
Example 1:
1
a) Write an equation in slope-intercept form for the line that passes through (4, – 2) and is parallel to 𝑦 = 𝑥 − 7.
2
1
b) Write an equation in point-slope form for the line that passes through (4, – 1) and is parallel to 𝑦 = 𝑥 + 7.
4
Perpendicular lines: lines that intersect at right angles. The slopes of nonvertical perpendicular lines are opposite reciprocals.
You can use slope to determine whether two lines are perpendicular.
���� and ����
Example 2: The height of a trapezoid is the length of a segment that is perpendicular to both bases. In trapezoid ARTP, 𝐴𝑃
𝑅𝑇
are bases.
���� be used to measure the height of the trapezoid? Explain.
a) Can 𝐸𝑍
b) Are the bases parallel?
53
You can determine whether the graphs of two linear equations are parallel or perpendicular by comparing the slopes of the lines.
Example 3:
1
a) Determine whether the graphs of 3𝑥 + 𝑦 = 12, 𝑦 = 𝑥 + 2, and 2𝑥 − 6𝑦 = −5 are parallel or perpendicular. Explain.
3
b) Determine whether the graphs of 6𝑥 − 2𝑦 = −2, 𝑦 = 3𝑥 − 4, and 𝑦 = 4 are parallel or perpendicular. Explain.
You can write the equation of a line perpendicular to a given line if you know a point on the line and the equation of the given line.
Example 4:
a) Write an equation in slope-intercept form for the line that passes through (4, – 1) and is perpendicular to the graph of 7𝑥 − 2𝑦 = 3.
2
b) Write an equation in slope-intercept form for the line that passes through (4, 7) and is perpendicular to the graph of 𝑦 = 𝑥 − 1.
3
54
Algebra 1
Section 4.4 Worksheet
Write an equation in slope-intercept form for the line that passes through the given point and is parallel to the graph of the
given equation.
1. (3, 2), y = x + 5
4
4. (12, 3), y = x + 5
3
3
2. (–2, 5), y = –4x + 2
3. (4, –6), y = − x + 1
6. (3, 1), 2x + y = 5
7. (–3, 4), 3y = 2x – 3
8. (–1, –2), 3x – y = 5
4
9. (–8, 2), 5x – 4y = 1
Write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the graph of
the given equation.
1
13. (–2, –2), y = − x + 9
14. (–6, 5), x – y = 5
15. (–4, –3), 4x + y = 7
16. (0, 1), x + 5y = 15
17. (2, 4), x – 6y = 2
18. (–1, –7), 3x + 12y = –6
3
���� and ����
25. GEOMETRY Quadrilateral ABCD has diagonals 𝐴𝐶
𝐵𝐷.
����
����
Determine whether 𝐴𝐶 is perpendicular to 𝐵𝐷 . Explain.
26. GEOMETRY Triangle ABC has vertices A(0, 4), B(1, 2), and C(4, 6).
Determine whether triangle ABC is a right triangle. Explain.
28. ARCHITECTURE The front view of a house is drawn on graph paper. The left side of the roof of the house is represented by the
equation y = x. The rooflines intersect at a right angle and the peak of
the roof is represented by the point (5, 5). Write the equation in slope-intercept form for the line that creates the right side of the roof.
55
Algebra 1
Section 4.7 Notes: Inverse Linear Functions
Inverse relation: the set of ordered pairs obtained by exchanging the x-coordinates of each ordered pair in a relation.
Notice the domain of a relation becomes the range of its inverse, and the range of the relation becomes the domain of its inverse.
Example 1: Find the inverse of each relation.
a) {(– 3, 26), (2, 11), (6, –1), (–1, 20)}
b)
The graphs of relations can be used to find and graph inverse relations.
Example 2: Graph the inverse of each relation.
a)
b)
Inverse function: A linear relation that is described by a function has an inverse function that can generate ordered pairs of the
inverse relation. The inverse of the function 𝑓(𝑥) can be written as 𝑓 −1 (𝑥) and is read 𝑓 of 𝑥 inverse or the inverse of 𝑓 of 𝑥.
56
Example 3: Find the inverse of each function.
a) 𝑓(𝑥) = −3𝑥 + 27
b) 𝑓(𝑥) =
5
4
𝑥−8
Example 4: Carter sells paper supplies and makes a base salary of $2200 each month. He also earns 5% commission on his total
sales. His total earnings 𝑓(𝑥) for a month which he compiled 𝑥 dollars in total sales is 𝑓(𝑥) = 2200 + 0.05𝑥.
a) Find the inverse function.
b) What do 𝑥 and 𝑓 −1 (𝑥) represent in the context of the inverse function?
c) Find Carter’s total sales for last month if his earnings for that month were $3450.
57
Algebra 1
Section 4.7 Worksheet
Find the inverse of each relation.
1. {(–2, 1), (–5, 0), (–8, –1), (–11, 2)}
2. {(3, 5), (4, 8), (5, 11), (6, 14)}
3. {(5, 11), (1, 6), (–3, 1), (–7, –4)}
4. {(0, 3), (2, 3), (4, 3), (6, 3)}
Graph the inverse of each function.
5.
6.
7.
Find the inverse of each function.
6
8. f (x) = x – 3
5
11. f (x) = 3(3x + 4)
9. f (x) =
4𝑥 + 2
3
12. f (x) = –5(–x – 6)
Write the inverse of each equation in 𝒇−𝟏 (x) notation.
10. f (x) =
3𝑥 − 1
13. f (x) =
2𝑥 − 3
6
7
13. 4x + 6y = 24
14. –3y + 5x = 18
15. x + 5y = 12
16. 5x + 8y = 40
17. –4y – 3x = 15 + 2y
18. 2x – 3 = 4x + 5y
19. CHARITY Jenny is running in a charity event. One donor is paying an initial amount of $20.00 plus an extra $5.00 for every mile
that Jenny runs.
a. Write a function D(x) for the total donation for x miles run.
b. Find the inverse function, 𝐷−1 (x).
c. What do x and 𝐷 −1 (x) represent in the context of the inverse function?
58
22. SERVICE A technician is working on a furnace. He is paid $150 per visit plus $70 for every hour he works on the furnace.
a. Write a function C(x) to represent the total charge for every hour of work.
b. Find the inverse function, 𝐶 −1 (x).
c. How long did the technician work on the furnace if the total charge was $640?
23. FLOORING Kara is having baseboard installed in her basement. The total cost C(x) in dollars is given by C(x) = 125 + 16x,
where x is the number of pieces of wood required for the installation.
a. Find the inverse function 𝐶 −1 (x).
b. If the total cost was $269 and each piece of wood was 12 feet long, how many total feet of wood were used?
24. BOWLING Libby’s family went bowling during a holiday special. The special cost $40 for pizza, bowling shoes, and unlimited
drinks. Each game cost $2. How many games did Libby bowl if the total cost was $112 and the six family members bowled an
equal number of games?
59
CHAPTER 5
60
Algebra 1
Section 5.3 Notes: Solving Multi-Step Inequalities
Multi-step inequalities can be solved by undoing the operations in the same way you would solve a multi-step equation.
Example 1:
a) Adriana has a budget of $115 for faxes. The fax service she uses charges $25 to activate an account and $0.08 per page to send
faxes. How many pages can Adriana fax and stay within her budget? Use the inequality 25 + 0.08𝑝 ≤ 115.
b) The Print Shop advertises a special to print 400 flyers for less than the competition. The price includes a $3.50 set-up fee. If the
competition charges $35.50, what does the Print Shop charge for each flyer?
When multiplying or dividing by a negative number, the direction of the inequality symbol changes.
Example 2: Solve the inequality.
a) 23 ≥ 10 − 2𝑤
b) 13 − 11𝑑 ≥ 79
c) 43 > −4𝑦 + 11
You can translate sentences into multi-step inequalities and then solve them using the Properties of Inequalities.
Example 3: Define a variable, write an inequality, and solve the problem. Then check your solution.
a) Four times a number plus twelve is less than the number minus three.
b) Two more than half of a number is greater than twenty-seven.
61
When solving inequalities that contain grouping symbols, use the Distributive Property to remove the grouping symbols first. Then
use the order of operations to simplify the resulting inequality.
Example 4: Solve each inequality. Graph the solution on a number line.
a) 6𝑐 + 3(2 − 𝑐) ≥ −2𝑐 + 1
b) 6(5𝑧 − 3) ≤ 36𝑧
c) 2(ℎ + 6) > −3(8 − ℎ)
If solving an inequality results in a statement that is always true, the solution set is the set of all real numbers. This solution set is
written as {x| x is a real number.}. If solving an inequality results in a statement that is never true, the solution set is the empty set,
which is written as the symbol ᴓ. The empty set has no members.
Example 5: Solve each inequality. Check your solution.
a) −7(𝑘 + 4) + 11𝑘 ≥ 8𝑘 − 2(2𝑘 + 1)
b) 2(4𝑟 + 3) ≤ 22 + 8(𝑟 − 2)
c) 18 − 3(8𝑐 + 4) ≥ −6(4𝑐 − 1)
d) 46 ≤ 8𝑚 − 4(2𝑚 + 5)
62
Algebra 1
Section 5.3 Worksheet
Justify each indicated step
1.
3
4
3
4
t – 3 ≥ –15
2. 5(k + 8) – 7 ≤ 23
t – 3 + 3 ≥ –15 +3
3
4
5k + 40 – 7 ≤ 23
a. ?
t ≥ –12
4
3
3
5k + 33 ≤ 23
4
� �t ≥ (–12)
4
a. ?
5k + 33 – 33 ≤ 23 – 33
b. ?
3
t ≥ –16
b. ?
5k ≤ –10
5𝑘
5
≤
−10
5
k ≤ –2
21. BEACHCOMBING Jay has lost his mother’s favorite necklace, so he will rent a metal detector to try to find it. A rental company
charges a one-time rental fee of $15 plus $2 per hour to rent a metal detector. Jay has only $35 to spend. What is the maximum
amount of time he can rent the metal detector?
22. AGES Bobby, Billy, and Barry Smith are each one year apart in age. The sum of their ages is greater than the age of their father,
who is 60. How old can the oldest brother can be?
1
23. TAXI FARE Jamal works in a city and sometimes takes a taxi to work. The taxicabs charge $1.50 for the first mile and $0.25
5
1
for each additional mile. Jamal has only $3.75 in his pocket. What is the maximum distance he can travel by taxi if he does not tip
5
the driver?
25. MEDICINE Clark’s Rule is a formula used to determine pediatric dosages of over-the-counter medicines.
weight of child ( lb)
150
× adult dose = child dose
a. If an adult dose of acetaminophen is 1000 milligrams and a child weighs no more than 90 pounds, what is the recommended
child’s dose?
b. This label appears on a child’s cold medicine.
Weight (lb)
Age (yr)
Dose
under 48
under 6
call a doctor
48-95
6-11
2 tsp or 10 mL
c. What is the maximum adult dosage in milliliters?
What is the adult minimum dosage in milliliters?
Algebra 1
63
Section 5.5 Inequalities Involving Absolute Value
LESS THAN:
When solving absolute value inequalities, there are two cases to consider.
Case 1: The expression inside the absolute value symbols is nonnegative.
Case 2: The expression inside the absolute value symbols is negative.
**The solution is the intersection of the solutions of these two cases.
The inequality |𝑥| < 3 means that the distance between x and 0 is less than 3.
So, 𝑥 > 3 and 𝑥 < 3. The solution set is {𝑥| − 3 < 𝑥 < 3}.
Example 1: Solve the inequality. Then graph the solution set.
a) |𝑛 − 3| ≤ 12
b) |𝑥 + 6| < −8
Example 2:
a) The average annual rainfall in California for the last 100 years is 23 inches. However, the annual rainfall can differ by 10 inches
from the 100 year average. What is the range of annual rainfall for California?
b) The melting point of ice is 0°C. During a chemistry experiment, Jill observed ice melting within °C of this measurement. Write the
range of temperatures that Jill observed.
GREATER THAN:
64
Again, we must consider both cases.
Case 1: The expression inside the absolute value symbols is nonnegative.
Case 2: The expression inside the absolute value symbols is negative.
The inequality |𝑥| > 3 means that the distance between x and 0 is greater than 3.
So, 𝑥 < −3 or 𝑥 > 3. The solution set is {𝑥|𝑥 < −3 𝑜𝑟 𝑥 > 3}.
Example 3: Solve the inequality. Then graph the solution set.
a) |3𝑦 − 3| > 9
b) |2𝑥 + 7| ≥ −11
65
Algebra 1
Section 5.5 Worksheet
14. RESTAURANTS The menu at Jeanne’s favorite restaurant states that the roasted chicken with vegetables entree typically
contains 480 Calories. Based on the size of the chicken, the actual number of Calories in the entree can vary by as many as 40
Calories from this amount.
a. Write an absolute value inequality to represent the situation.
b. What is the range of the number of Calories in the chicken entree?
15. SPEEDOMETERS The government requires speedometers on cars sold in the United States to be accurate within ±2.5% of the
actual speed of the car. If your speedometer reads 60 miles per hour while you are driving on a highway, what is the range of
possible actual speeds at which your car could be traveling?
16. BAKING Pete is making muffins for a bake sale. Before he starts baking, he goes online to research different muffin recipes. The
recipes that he finds all specify baking temperatures between 350°F and 400°F, inclusive. Write an absolute value inequality to
represent the possible temperatures t called for in the muffin recipes Pete is researching.
18. CATS During a recent visit to the veterinarian’s office, Mrs. Van Allen was informed that a healthy weight for her cat is
approximately 10 pounds, plus or minus one pound. Write an absolute value inequality that represents unhealthy weights w for her cat.
19. STATISTICS The most familiar statistical measure is the arithmetic mean, or average. A second important statistical measure is
the standard deviation, which is a measure of how far the individual scores deviate from the mean. For example, in a recent year the
mean score on the mathematics section of the SAT test was 515 and the standard deviation was 114. This means that people within
one deviation of the mean have SAT math scores that are no more than 114 points higher or 114 points lower than the mean.
a. Write an absolute value inequality to find the range of SAT mathematics test scores within one standard deviation of the mean.
b. What is the range of SAT mathematics test scores ±2 standard deviation from the mean?
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Algebra 1
Section 5.6 Notes: Graphing Inequalities in Two Variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality. An equation defines a
boundary, which divides the coordinate plane into two half planes.
The boundary may or not be included in the solution. When it is included, the solution is a closed half-plane (solid line).
When not included, the solution is an open half-plane (dashed line).
Example 1: Graph the inequality
a) 2𝑦 − 4𝑥 > 6
b) 𝑥 − 1 > 𝑦
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Example 2: Graph the inequality.
a) 𝑥 + 4𝑦 ≥ 2
b) 2𝑥 + 3𝑦 ≥ 18
You can use a coordinate plane to solve inequalities with one variable.
Example 3: Use a graph to solve.
a) 2𝑥 + 3 ≤ 7
b) −2𝑥 + 6 > 12
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An inequality can be viewed as a constraint in a problem situation. Each solution of the inequality represents a combination that
meets the constraint. In real-world problems, the domain and range are often restricted to nonnegative or whole numbers.
Example 4: Write, Solve and Graph an Inequality
a) Ranjan writes and edits short articles for a local newspaper. It takes him about an hour to write an article and about a half-hour to
edit an article. If Ranjan works up to 8 hours a day, how many articles can he write and edit in one day?
b) Neil wants to run a marathon at a pas of at least 6 miles per hour. Write and graph an inequality for the miles y he will run in x
hours.
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Algebra 1
Section 5.6 Worksheet
Determine which ordered pairs are part of the solution set for each inequality.
1. 3x + y ≥ 6, {(4, 3), (–2, 4), (–5, –3), (3, –3)}
2. y ≥ x + 3, {(6, 3), (–3, 2), (3, –2), (4, 3)}
3. 3x – 2y < 5, {(4, –4), (3, 5), (5, 2), (–3, 4)}
Graph each inequality.
4. 2y – x < –4
5. 2x – 2y ≥ 8
6. 3y > 2x – 3
Use a graph to solve each inequality.
7. –5 ≤ x – 9
2
8. 6 > x + 5
3
1
9. > –2 x +
2
7
2
10. MOVING A moving van has an interior height of 7 feet (84 inches). You have boxes in 12 inch and 15 inch heights, and want to
stack them as high as possible to fit. Write an inequality that represents this situation.
11. BUDGETING Satchi found a used bookstore that sells pre-owned DVDs and CDs. DVDs cost $9 each, and CDs cost $7 each.
Satchi can spend no more than $35.
a. Write an inequality that represents this situation.
b. Does Satchi have enough money to buy 2 DVDs and 3 CDs?
12. FAMILY Tyrone said that the ages of his siblings are all part of the solution set of y > 2x, where x is the age of a sibling and y is
Tyrone’s age. Which of the following ages is possible for Tyrone and a sibling?
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Tyrone is 23; Maxine is 14.
Tyrone is 18; Camille is 8.
Tyrone is 12; Francis is 4.
Tyrone is 11; Martin is 6.
Tyrone is 19; Paul is 9.
13. FARMING The average value of U.S. farm cropland has steadily increased in recent years. In 2000, the average value was $1490
per acre. Since then, the value has increased at least an average of $77 per acre per year. Write an inequality to show land values
above the average for farmland.
14. SHIPPING An international shipping company has established size limits for packages with all their services. The total of the
length of the longest side and the girth (distance completely around the package at its widest point perpendicular to the length) must
be less than or equal to 419 centimeters. Write and graph an inequality that represents this situation.
15. FUNDRAISING Troop 200 sold cider and donuts to raise money for charity. They sold small boxes of donut holes for $1.25 and
cider for $2.50 a gallon. In order to cover their expenses, they needed to raise at least $100. Write and graph an inequality that
represents this situation.
16. INCOME In 2006 the median yearly family income was about $48,200 per year. Suppose the average annual rate of change since
then is $1240 per year.
a. Write and graph an inequality for the annual family incomes y that are less than the median for x years after 2006.
b. Determine whether each of the following points is part of the solution set.
(2, 51,000)
(8, 69,200)
(5, 50,000)
(10, 61,000)
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ALGEBRA 1
UNIT 3
WORKBOOK
CHAPTER 6
FALL 2014
73
74
Algebra 1
Section 6.1 Notes: Graphing Systems of Equations
System of Equations: a set of two or more equations with the same variables, graphed in the same coordinate plane
The ordered pair that is a solution of both equations is the solution of the system. A system of two linear equations can have one
solution, an infinite number of solutions, or no solution.
Consistent: a system of equations that has at least one solution
Independent: a consistent system of equations that has exactly one solution
Dependent: a consistent system of equations that has an infinite number of solutions; this means that there are an unlimited solutions
that satisfy both equations
Inconsistent: a system of equations that has no solution; the graphs are parallel
Example 1: Use the graph to determine whether the system is consistent or inconsistent and if it is independent or dependent.
a)
y = –x + 1
y = –x + 4
b)
y=x–3
y = –x + 1
Solve by Graphing: One method of solving a system of equations is to graph the equations on the same coordinate plane and find their
point of intersection. This point is the solution of the system.
Example 2: Graph the system of equations. Then determine whether the system has no solution, one solution, or infinitely many
solutions. If the system has one solution, name it.
a)
y = 2x + 3
8x – 4y = –12
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b)
x – 2y = 4
x – 2y = –2
c) x – y = 2
3y + 2x = 9
We can use what we know about systems of equations to solve many real-world problems involving constraints that are modeled by
two or more different functions.
Example 3: Naresh and Diego are having a bicycling competition. Naresh is able to ride 20 miles at the start of the competition and
plans to ride 35 more miles than the previous week each upcoming week. Diego is able to ride 50 miles at the start of the competition
and plans to ride 25 more miles than the previous week each upcoming week. Predict the week in which Naresh and Diego will have
ridden the same number of miles.
1) Write a system of equations to represent the system
3)
2) Graph the system to determine the solution.
Use substitution to check your answer.
Example 4: Alex and Amber are both saving money for a summer vacation. Alex has already saved $100 and plans to save $25 per
week until the trip. Amber has $75 and plans to save $30 per week. In how many weeks will Alex and Amber have the same amount
of money?
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Algebra 1
Section 6.1 Worksheet
Use the graph at the right to determine whether
each system is consistent or inconsistent and
if it is independent or dependent.
1. x + y = 3
x + y = –3
2. 2x – y = –3
4x – 2y = –6
3. x + 3y = 3
x + y = –3
4. x + 3y = 3
2x – y = –3
Graph each system and determine the number of solutions that it has. If it has one solution, name it.
5. 3x – y = –2
3x – y = 0
6. y = 2x – 3
4x = 2y + 6
7. x + 2y = 3
3x – y = –5
8. BUSINESS Nick plans to start a home-based business
producing and selling gourmet dog treats. He figures it
will cost $20 in operating costs per week plus $0.50 to
produce each treat. He plans to sell each treat for $1.50.
a. Graph the system of equations y = 0.5x + 20 and
y = 1.5x to represent the situation.
b. How many treats does Nick need to sell per week to break even?
9. SALES A used book store also started selling used
CDs and videos. In the first week, the store sold 40
used CDs and videos, at $4.00 per CD and $6.00 per
video. The sales for both CDs and videos totaled $180.00
a. Write a system of equations to represent the situation.
b. Graph the system of equations.
c. How many CDs and videos did the store sell in the first week?
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Algebra 1
Section 6.2 Notes: Substitution
In the previous lesson, we learned how to solve a system of equations by graphing. Another method for solving a system of equation
is called substitution.
Example 1: Use substitution to solve the system of equations.
a) y = –4x + 12
b) y = 4x – 6
2x + y = 2
5x + 3y = -1
If a variable is not isolated in one of the equations in the system, solve an equation for a variable first. Then you can use substitution
so solve the system.
Example 2: Use substitution to solve the system of equations.
a) x – 2y = –3
b) 3x – y = –12
3x + 5y = 24
–4x + 2y = 20
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Generally, if you solve a system of equations and the result is a false statement such as 3 = -2, there is no solution. If the result is an
identity, such as 3 = 3, then there are an infinite number of solutions.
Example 3: Use substitution to solve the system of equations.
a) 2x + 2y = 8
b) 3x – 2y = 3
x + y = –2
–6x + 4y = –6
Example 4:
a) A nature center charges $35.25 for a yearly membership and $6.25 for a single admission. Last week it sold a combined
total of 50 yearly memberships and single admissions for $660.50. How many memberships and how many single
admissions were sold?
b)
As of 2009, the New York Yankees and the Cincinnati Reds together had won a total of 32 World Series. The Yankees
had won 5.4 times as many as the Reds. How many World Series had each team won?
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Algebra 1
6.2 Worksheet
Use substitution to solve each system of equations.
1. y = 6x
2x + 3y = –20
4. y = 2x – 2
y=x+2
7. x + 2y = 13
–2x – 3y = –18
10. 2x – 3y = –24
x + 6y = 18
13. 0.5x + 4y = –1
x + 2.5y = 3.5
2. x = 3y
3x – 5y = 12
5. y = 2x + 6
2x – y = 2
8. x – 2y = 3
4x – 8y = 12
11. x + 14y = 84
2x – 7y = –7
14. 3x – 2y = 11
1
x– 𝑦=4
2
16. 1 – 3 x – y = 3
17. 4x – 5y = –7
2x + y = 25
y = 5x
3. x = 2y + 7
x=y+4
6. 3x + y = 12
y = –x – 2
9. x – 5y = 36
2x + y = –16
12. 0.3x – 0.2y = 0.5
x – 2y = –5
1
15. 𝑥 + 2y = 12
2
x – 2y = 6
18. x + 3y = –4
2x + 6y = 5
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19. EMPLOYMENT Kenisha sells athletic shoes part-time at a department store. She can earn either $500 per month plus a 4%
commission on her total sales, or $400 per month plus a 5% commission on total sales.
a. Write a system of equations to represent the situation.
b. What is the total price of the athletic shoes Kenisha needs to sell to earn the same income from each pay scale?
c. Which is the better offer?
20. MOVIE TICKETS Tickets to a movie cost $7.25 for adults and $5.50 for students. A group of friends purchased
8 tickets for $52.75.
a. Write a system of equations to represent the situation.
b. How many adult tickets and student tickets were purchased?
21. BUSINESS Mr. Randolph finds that the supply and demand for gasoline at his station are generally given by the following
equations.
x – y = –2
x + y = 10
Use substitution to find the equilibrium point where the supply and demand lines intersect.
22. GEOMETRY The measures of complementary angles have a sum of 90 degrees. Angle A and angle B are complementary, and
their measures have a difference of 20°. What are the measures of the angles?
23. MONEY Harvey has some $1 bills and some $5 bills. In all, he has 6 bills worth $22. Let x be the number of $1 bills and let y be
the number of $5 bills. Write a
system of equations to represent the information and use substitution to determine how many bills of each
denomination Harvey has.
24. POPULATION Sanjay is researching population trends in South America. He found that the population of Ecuador to increased
by 1,000,000 and the population of Chile to increased by 600,000 from 2004 to 2009. The table displays the information he found.
Ecuador
13,000,000
5-Year
Population
Change
+1,000,000
Chile
16,000,000
+600,000
Country
2004
Population
Source: World Almanac
If the population growth for each country continues at the same rate, in what year are the populations of Ecuador and Chile
predicted to be equal?
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Algebra 1
Section 6.3 Notes: Elimination Using Addition and Subtraction
You have learned about solving a system of equations using the graphing method and the substitution method. A third way to solve a
system of equations is called elimination. Elimination involves using addition or subtraction to solve a system.
Example 1: Use elimination to solve the system of equations.
a)
–3x + 4y = 12
3x – 6y = 18
b) 3x – 5y = 1
2x + 5y = 9
Example 2: Four times one number minus three times another number is 12. Two times the first number added to three times the
second number is 6. Write a system of linear equations and then use elimination to solve it and find the numbers.
Example 3: Use elimination to solve the system of equations.
a)
4x + 2y = 28
4x – 3y = 18
b) 9x – 2y = 30
x – 2y = 14
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Example 4:
a) A hardware store earned $956.50 from renting ladders and power tools last week. The store charged 36 days for ladders
and 85 days for power tools. This week the store charged 36 days for ladders, 70 days for power tools, and earned $829.
How much does the store charge per day for ladders and for power tools?
b)
For a school fundraiser, Marcus and Anisa participated in a walk-a-thon. In the morning, Marcus walked 11 miles and
Anisa walked 13. Together they raised $523.50. After lunch, Marcus walked 14 miles and Anisa walked 13. In the
afternoon they raised $586.50. How much did each raise per mile of the walk-a-thon?
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Algebra 1
6.3 Worksheet
Use elimination to solve each system of equations.
1. x – y = 1
2. p + q = –2
3. 4x + y = 23
x + y = –9
p–q=8
4. 2x + 5y = –3
5. 3x + 2y = –1
6. 5x + 3y = 22
2x + 2y = 6
4x + 2y = –6
5x – 2y = 2
7. 5x + 2y = 7
8. 3x – 9y = –12
9. –4c – 2d = –2
3x – 15y = –6
2c – 2d = –14
–2x + 2y = –14
10. 2x – 6y = 6
11. 7x + 2y = 2
2x + 3y = 24
7x – 2y = –30
13. 2x + 4y = 10
14. 2.5x + y = 10.7
x – 4y = –2.5
16. 4a + b = 2
4a + 3b = 10
2.5x + 2y = 12.9
1
4
17. – x – = –2
1
3
3
2
3
x– 𝑦 =4
3
3x – y = 12
12. 4.25x – 1.28y = –9.2
x + 1.28y = 17.6
15. 6m – 8n = 3
2m – 8n = –3
3
1
18. x – 𝑦 = 8
4
3
2
2
1
x – 𝑦 = 19
2
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19. The sum of two numbers is 41 and their difference is 5. What are the numbers?
20. Four times one number added to another number is 36. Three times the first number minus the other number is 20. Find the
numbers.
21. One number added to three times another number is 24. Five times the first number added to three times the other number is 36.
Find the numbers.
22. LANGUAGES English is spoken as the first or primary language in 78 more countries than Farsi is spoken as the first
language. Together, English and Farsi are spoken as a first language in 130 countries. In how many countries is English spoken
as the first language? In how many countries is Farsi spoken as the first language?
23. DISCOUNTS At a sale on winter clothing, Cody bought two pairs of gloves and four hats for $43.00. Tori bought two pairs of
gloves and two hats for $30.00. What were the prices for the gloves and hats?
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Algebra 1
Section 6.4 Notes: Elimination Using Multiplication
Two systems don’t always have to have the same or opposite coefficients for a variable to use elimination. You can use multiplication
and elimination to solve a system when this is the case.
Example 1: Use elimination to solve the system of equations.
a)
2x + y = 23
3x + 2y = 37
b) x + 7y = 12
3x – 5y = 10
Example 2: Use elimination to solve the system of equations.
a)
4x + 3y = 8
3x – 5y = –23
b) 3x + 2y = 10
2x + 5y = 3
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Example 3:
a) A fishing boat travels 10 miles downstream in 30 minutes. The return trip takes the boat 40 minutes. Find the rate in
miles per hour of the boat in still water.
b)
A helicopter travels 360 miles with the wind in 3 hours. The return trip against the wind takes the helicopter 4 hours.
Find the rate of the helicopter in still air.
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Algebra 1
6.4 Worksheet
Use elimination to solve each system of equations.
1. 2x – y = –1
3x – 2y = 1
4. 2x – 4y = –22
3x + 3y = 30
7. 3x + 4y = 27
5x – 3y = 16
10. 6x – 3y = 21
2x + 2y = 22
2. 5x – 2y = –10
3x + 6y = 66
5. 3x + 2y = –9
5x – 3y = 4
8. 0.5x + 0.5y = –2
x – 0.25y = 6
11. 3x + 2y = 11
2x + 6y = –2
3. 7x + 4y = –4
5x + 8y = 28
6. 4x – 2y = 32
–3x – 5y = –11
3
9. 2x – 𝑦 = –7
4
1
x+ 𝑦=0
2
12. –3x + 2y = –15
2x – 4y = 26
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13. Eight times a number plus five times another number is –13. The sum of the two numbers is 1. What are the numbers?
14. Two times a number plus three times another number equals 4. Three times the first number plus four times the other number is 7.
Find the numbers.
15. FINANCE Gunther invested $10,000 in two mutual funds. One of the funds rose 6% in one year, and the other rose 9% in one
year. If Gunther’s investment rose a total of $684 in one year, how much did he invest in each mutual fund?
16. CANOEING Laura and Brent paddled a canoe 6 miles upstream in four hours. The return trip took three hours. Find the rate at
which Laura and Brent paddled the canoe in still water.
17. NUMBER THEORY The sum of the digits of a two-digit number is 11. If the digits are reversed, the new number is 45 more
than the original number. Find the number.
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Algebra 1
Section 6.5 Notes: Applying Systems of Linear Equations
You have learned five methods for solving systems of linear equations. The table summarizes the methods and the types of systems
for which each method works best.
Example 1:
a) Determine the best method to solve the system of equations. Then solve the system.
2x + 3y = 23
4x + 2y = 34
b) POOL PARTY At the school pool party, Mr. Lewis bought 1 adult ticket and 2 child tickets for $10.
Mrs. Vroom bought 2 adult tickets and 3 child tickets for $17. Write a system of linear equations for this situation and
then determine the best method to solve the system of equations. Then solve the system.
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Example 2:
a)
CAR RENTAL Ace Car Rental rents a car for $45 and $0.25 per mile. Star Car Rental rents a car for $35 and $0.30 per
mile. How many miles would a driver need to drive before the cost of renting a car at Ace Car Rental and renting a car at Star
Car Rental were the same?
b)
VIDEO GAMES The cost to rent a video game from Action Video is $2 plus $0.50 per day. The cost to rent a video game
at TeeVee Rentals is $1 plus $0.75 per day. After how many days will the cost of renting a video game at Action Video be
the same as the cost of renting a video game at TeeVee Rentals?
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Algebra 1
6.5 Worksheet
Determine the best method to solve each system of equations. Then solve the system.
1. 5x + 3y = 16
3x – 5y = –4
3. y = 3x – 24
2. 3x – 5y = 7
2x + 5y = 13
4. –11x – 10y = 17
5x – y = 8
5x – 7y = 50
5. 4x + y = 24
6. 6x – y = –145
5x – y = 12
x = 4 – 2y
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7. VEGETABLE STAND A roadside vegetable stand sells pumpkins for $5 each and squashes for $3 each. One day they sold 6 more
squash than pumpkins, and their sales totaled $98. Write and solve a system of equations to find how many pumpkins and quash
they sold?
8. INCOME Ramiro earns $20 per hour during the week and $30 per hour for overtime on the weekends. One week Ramiro earned a
total of $650. He worked 5 times as many hours during the week as he did on the weekend. Write
and solve a system of equations to determine how many hours of overtime Ramiro worked on the weekend.
9. BASKETBALL Anya makes 14 baskets during her game. Some of these baskets were worth 2-points and others were worth 3points. In total, she scored 30 points. Write and solve a system of equations to find how 2-points baskets she made.
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Algebra 1
Section 6.6 Notes: Systems of Inequalities
A set of two or more inequalities with the same variable is called a system of inequalities.
The solution of a system of inequalities with two variables is the set of ordered pairs that satisfy all of the inequalities in the system.
The solution set is represented by the overlap, or intersection, of the graphs of the inequalities.
Example 1:
a) Solve the system of inequalities by graphing.
y < 2x + 2
y≥–x–3
b)
Choose the correct solution to the system: 2x + y ≥ 4 and x + 2y > –4.
A.
C.
B.
D.
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Example 2: Solve the system of inequalities by graphing.
y ≥ –3x + 1
y ≤ –3x – 2
Example 3:
a) SERVICE A college service organization requires that its members maintain at least a
3.0 grade point average, and volunteer at least
10 hours a week. Define the variables and write a system of inequalities to represent this situation. Then graph the system.
b) SERVICE A college service organization requires that its members maintain at least a
3.0 grade point average, and volunteer at least 10 hours a week. Name one possible solution.
\
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Algebra 1
6.6 Worksheet
Solve each system of inequalities by graphing.
1. y > x – 2
y≤x
2. y ≥ x + 2
y > 2x + 3
3. x + y ≥ 1
x + 2y > 1
4. y < 2x – 1
y>2–x
5. y > x – 4
2x + y ≤ 2
6. 2x – y ≥ 2
x – 2y ≥ 2
7. FITNESS Diego started an exercise program in which each week he works out at the
gym between 4.5 and 6 hours and walks between 9 and 12 miles.
a. Make a graph to show the number of hours Diego works out at the gym and the
number of miles he walks per week.
b. List three possible combinations of working out and walking that meet Diego’s goals.
8. SOUVENIRS Emily wants to buy turquoise stones on her trip to New Mexico to give to
at least 4 of her friends. The gift shop sells stones for either $4 or $6 per stone. Emily
has no more than $30 to spend.
a. Make a graph showing the numbers of each price of stone Emily can purchase.
b. List three possible solutions.
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