Equations

7 and Inequalities
7.1
7.
7 1 Writing
Writing Equations in One Variable
7.2
7 2 S
Solving
l i Equations Using Addition or Subtraction
7.3 Solving Equations Using Multiplication
or Division
7.4 Writing Equations in Two Variables
7.5 Writing and Graphing Inequalities
7.6 Solving Inequalities Using Addition or Subtraction
7.7 Solving Inequalities Using Multiplication
or Division

elcome.”
“You’re w
Q?”
7Q plus 3
“What is

“With the help of your twi

n brother, I think I
have figured it out.” “You weigh 36 dog biscui
ts.”

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 What You
Learned Before
“Dear Sir:
I
Example 1 Evaluate 7x + 3y when the contest have the answer to
seconds a question, ‘How man
re in a yea y
x = 2 and y = 4. 12: Janua r?
ry 2nd, Feb ’ There are
ruary 2nd,
...”
⋅
7x + 3y = 7 2 + 3 4 ⋅ Substitute 2 for x and 4 for y.
= 14 + 12 Using order of operations, multiply from left to right.
= 26 Add 14 and 12.

Example 2 Evaluate 5x2 − 2( y + 1) + 9 when x = 2 and y = 1.

5x 2 − 2( y + 1) + 9 = 5(2)2 − 2(1 + 1) + 9 Substitute 2 for x and 1 for y.

⋅
= 5(2)2 − 2 2 + 9 Using order of operations, evaluate within
the parentheses.

⋅
=5 4−2 2+9 ⋅ Using order of operations, evaluate
the exponent.
Using order of operations, multiply from
= 20 − 4 + 9
left to right.
= 25 Subtract 4 from 20. Add the result to 9.

1
Evaluate the expression when a = — and b = 7.
2

1. 6ab 2. 16a − b 3. 3b − 2a − 9 4. b 2 − 16a + 5

Example 3 Write the phrase as an expression.

a. the sum of twice a number n and five b. twelve less than four times a number y
2n + 5 4y − 12

Write the phrase as an expression.

5. six more than three times a number w 6. the quotient of seven and a number p
7. two less than a number t 8. the product of a number x and five
9. five more than six divided by a number r 10. four less than three times a number b

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 7.1 Writing Equations in One Variable

How does rewriting a word problem help you

solve the word problem?

1 ACTIVITY: Rewriting a Word Problem

Work with a partner. Read the problem several times. Think about how you
could rewrite the problem. Leave out information that you do not need to
solve the problem.
Given Problem (63 words)

Your minivan has a flat,

rectangular area in the
back. When you fold
down the rear seats of
the van and move them
5 ft
forward, the width of the
rectangular area in the van
is increased by 2 feet, as
shown in the diagram. 2 ft 3 ft

By how many square feet does the rectangular area increase when
the rear seats are folded down and moved forward?

Rewritten Problem (28 words)

When you fold down the back seats of a minivan, the added area is a
5-foot by 2-foot rectangle. What is the area of this rectangle?

Can you make the problem even simpler?

Added Area â 2 ñ 5
Writing Equations Rewritten Problem ( words) â 10 ft2
In this lesson, you will
● write word sentences

as equations.

Explain why your rewritten problem is 5 ft

easier to read.

2 ft 3 ft

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 Math 2 ACTIVITY: Rewriting a Word Problem
Practice
Analyze Givens Work with a partner. Rewrite each problem using fewer words. Leave
What information out information that you do not need to solve the problem. Then solve
do you need to the problem.
solve the problem?
a. (63 words)

A supermarket is having its grand opening on Saturday morning. Every fifth customer
will receive a $10 coupon for a free turkey. Every seventh customer will receive a
$3 coupon for 2 gallons of ice cream. You are the manager of the store and you expect
to have 400 customers. How many of each type of coupon should you plan to give away?

b. (71 words)

You and your friend are at a football game. The stadium is 4 miles from your home.
You each brought $5 to spend on refreshments. During the third quarter of the game,
you say, “I read that the greatest distance that a baseball has been thrown is 445 feet
10 inches.” Your friend says, “That’s about one and a half times the length of the football
field.” Is your friend correct?

c. (90 words)

You are visiting your cousin who lives in the city. To get back home, you take a taxi.
The taxi charges $2.10 for the first mile and $0.90 for each additional mile. After riding
13 miles, you decide that the fare is going to be more than the $20 you have with you. So,
you tell the driver to stop and let you out. Then you call a friend and ask your friend to
come and pick you up. After paying the driver, how much of your $20 is left?

3. IN YOUR OWN WORDS How does rewriting a word problem help you solve
the word problem? Make up a word problem that has more than 50 words.
Then show how you can rewrite the problem using at most 25 words.

“Solving a math word problem is “You need to boil down 40 gallons

like making maple syrup.” of sap from a sugar maple tree to
get 1 gallon of syrup.”

Use what you learned about writing equations to complete

Exercises 4 and 5 on page 298.

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 7.1 Lesson
Lesson Tutorials

An equation is a mathematical sentence that uses an equal sign, =,

Key Vocabulary to show that two expressions are equal.
equation, p. 296
Expressions Equations
4+8 4 + 8 = 12
x+8 x + 8 = 12
To write a word sentence as an equation, look for key words or
phrases such as is, the same as, or equals to determine where to place
the equal sign.

EXAMPLE 1 Writing Equations

Write the word sentence as an equation.
a. The sum of a number n and 7 is 15.
The sum of a number n and 7 is 15.

n+7 = 15 Sum of means addition.

An equation is n + 7 = 15.

b. A number y decreased by 4 is 3.
A number y decreased by 4 is 3.

y−4 =3 Decreased by means subtraction.

An equation is y − 4 = 3.

c. 12 times a number p equals 48.

12 times a number p equals 48.

12p = 48 Times means multiplication.

An equation is 12p = 48.

Write the word sentence as an equation.

Exercises 6–13 1. 9 less than a number b equals 2.
2. The product of a number g and 5 is 30.
3. A number k increased by 10 is the same as 24.
4. The quotient of a number q and 4 is 12.

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 EXAMPLE 2 Writing an Equation
Ten servers decorate 25 tables for a wedding. Each table is decorated
as shown. Let c be the total number of white and purple candles.
Which equation can you use to find c ?
A
○ c = 25 + (4 × 6) B
○ c = 25(4 + 6)
C
○ c = 10(25 + 4 + 6) D
○ c = 10(4 + 6)

Words
Word The total is the number times the number
number of of tables of candles on
candles each table.

Variable
Varia Let c be the total number of candles.

Equation c = 25 × (4 + 6)

B .
The correct answer is ○

EXAMPLE 3 Real-Life Application

After two rounds, 24 students are eliminated from a spelling bee.
There are 96 students remaining. Write an equation you can use
to find the number of students that started the spelling bee.

Words The number minus the number is the number

of students of students of students
Reading that started eliminated remaining.
The word eliminated
means subtraction. Variable Let s be the number of students that started.

Equation s − 24 = 96

An equation is s − 24 = 96.

5. You enter an elevator and go down 7 floors. You exit on the

10th floor. Write an equation you can use to find the floor
where you entered the elevator.
6. Together you and a friend have $52. Your friend has $28. Write
an equation you can use to find how much money you have.
7. A typical person takes about 24,000 breaths each day. Write an
equation you can use to find the number of breaths a typical
person takes each minute.

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 7.1 Exercises
Help with Homework

1. VOCABULARY How are expressions and equations different?

2. DIFFERENT WORDS, SAME QUESTION Which is different? Write “both” equations.

4 less than a number n is 8. A number n is 4 less than 8.

A number n minus 4 equals 8. 4 subtracted from a number n is 8.

3. OPEN-ENDED Write a word sentence for the equation 28 − n = 5.

6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-

Rewrite the problem using fewer words. Leave out information that you do
not need to solve the problem. Then solve the problem.
4. In a cross-country race you run at a steady rate of 7 minutes per mile.
After 21 minutes, you finish in fourth place. How long is the race?
5. For a science project, you record the high temperature each day. The
high temperature on Day 1 was 6° less than on Day 4 and 4° less than
on Day 10. The high temperature on Day 10 was 62°F. What was the
high temperature on Day 1?

Write the word sentence as an equation.

1 6. The sum of a number x and 4 equals 12. 7. A number y decreased by 9 is 8.
8. 9 times a number b is 36. 9. A number w divided by 5 equals 6.
10. 54 equals 9 more than a number t. 11. 5 is one-fourth of a number c.
12. 11 is the quotient of a number y and 6. 13. 9 less than a number n equals 27.

14. ERROR ANALYSIS Describe the

error in writing the sentence
as an equation.
✗ A number n is 5 more than 12.
n + 5 = 12

15. FUNDRAISING Students and faculty raised $6042 for band

uniforms. The faculty raised $1780. Write an equation
you can use to find the amount a raised by
the students.

16. GOLF You hit a golf ball 90 yards. It travels 90 yd

three-fourths of the distance to the hole. d
Write an equation you can use to find the
distance d from the tee to the hole. tee

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 GEOMETRY Write an equation that you can use to find the value of x.
17. Perimeter of triangle: 16 in. 18. Perimeter of square: 30 mm

x x

x x

19. MUSIC You sell instruments at a Caribbean

music festival. You earn $326 by selling 12 sets
t off maracas,
6 sets of claves, and x djembe drums. Write an equation you
can use to find the number of djembe drums you sold.

20. SALES TAX Find a sales receipt from a store that

shows the total price and the total amount paid
including sales tax.
a. Write an equation you can use to find the sales tax rate r.
b. Can you use r to find the percent for the sales tax? Explain.

21. STRAWBERRIES You buy a basket of 24 strawberries. You eat them

as you walk to the beach. It takes the same amount of time to walk
each block. When you are halfway there, half of the berries are gone.
After walking 3 more blocks, you still have 5 blocks to go. You reach
the beach 28 minutes after you began. One-sixth of your strawberries
are left.
a. Is there enough information to find the time it takes to walk each
block? Explain.
b. Is there enough information to find how many strawberries you
ate while walking the last block? Explain.

14 in.
22. A triangle is cut from a rectangle. The height
of the triangle is half of the unknown side length s. The area
of the shaded region is 84 square inches. Write an equation s
you can use to find the side length s.

Evaluate the expression when a = 7. (Section 3.1)

35
23. 6 + a 24. a − 4 25. 4a 26. —
a

27. MULTIPLE CHOICE Which expression is equivalent to 8(x + 3)? (Section 3.4)
A 8x + 3
○ B 8x + 24
○ C 8x + 11
○ D x + 24
○

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 Solving Equations Using
7.2
Addition or Subtraction

How can you use addition or subtraction to

solve an equation?

When two sides of a scale weigh When you add or subtract the same amount
the same, the scale will balance. on each side of the scale, it will still balance.

1 ACTIVITY: Solving an Equation

Work with a partner.
a. Use a model to solve n + 3 = 7.
● Explain how the model represents the equation n + 3 = 7.

● How much does one weigh? How do you know?

The solution is n = .

Solving Equations b. Describe how you could check your answer in part (a).
In this lesson, you will c. Which model below represents the solution of n + 1 = 9?
● use addition or subtraction

to solve equations.
How do you know?
● use substitution to

check answers.
● solve real-life problems.

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 2 ACTIVITY: Solving Equations
Work with a partner. Solve the equation using the method in Activity 1.
a. nà5 â 10 b. xà2 â 11
Math
Practice
Understand
Quantities
What does the
variable represent c. 6 â yà3 d. 8 â mà8
in the equation?

3 ACTIVITY: Solving Equations Using Mental Math

Work with a partner. Write a question that represents the equation. Use
mental math to answer the question. Then check your solution.

Equation Question Solution Check

a. x + 1 = 5

b. 4 + m = 11

c. 8 = a + 3

d. x − 9 = 21

e. 13 = p − 4

4. REPEATED REASONING In Activity 3, how are parts (d) and (e) different from
parts (a)–(c)? Did your process to find the solution change? Explain.
5. Decide whether the statement is true or false. If false, explain your reasoning.
a. In an equation, you can use any letter as a variable.
b. The goal in solving an equation is to get the variable by itself.
c. In the solution, the variable must always be on the left side of the equal sign.
d. If you add a number to one side, you should subtract it from the other side.

6. IN YOUR OWN WORDS How can you use addition or subtraction to solve an
equation? Give two examples to show how your procedure works.
7. Are the following equations equivalent? Explain your reasoning.
x − 5 = 12 and 12 = x − 5

Use what you learned about solving equations to complete

Exercises 12–17 on page 305.

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 7.2 Lesson
Lesson Tutorials

Equations may be true for some values and false for others. A solution of
Key Vocabulary an equation is a value that makes the equation true.
solution, p. 302
inverse operations, Value of x x+3=7 Are both sides equal?
p. 303
?
3+3=7
3 no
6≠7 ✗
?
4+3=7
4 yes
Reading 7=7 ✓
?
5+3=7
The symbol ≠ means 5 no
is not equal to. 8≠7 ✗
So, the value x = 4 is a solution of the equation x + 3 = 7.

EXAMPLE 1 Checking Solutions

Tell whether the given value is a solution of the equation.
a. p + 10 = 38; p = 18
?
18 + 10 = 38 Substitute 18 for p.
28 ≠ 38 ✗ Sides are not equal.

28
38

So, p = 18 is not a solution.

b. 4y = 56; y = 14
?
4(14) = 56 Substitute 14 for y.
56 = 56 ✓ Sides are equal.

56 56

So, y = 14 is a solution.

Tell whether the given value is a solution of the equation.

Exercises 6 –11 1. a + 6 = 17; a = 9 2. 9 − g = 5; g = 3
q
3. 35 = 7n; n = 5 4. — = 28; q = 14
2

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 You can use inverse operations to solve equations. Inverse operations
“undo” each other. Addition and subtraction are inverse operations.

Addition Property of Equality

Words When you add the same number to each side of an equation,
the two sides remain equal.
Numbers 8= 8 Algebra x−4= 5
+5 +5 +4 +4
13 = 13 x= 9

Subtraction Property of Equality

Words When you subtract the same number from each side of an
equation, the two sides remain equal.
Numbers 8= 8 Algebra x+4= 5
−5 −5 −4 −4
3= 3 x= 1

EXAMPLE 2 Solving Equations Using Addition

a. Solve x − 2 = 6.
Check
x−2= 6 Write the equation.
x−2=6
Undo the subtraction. +2 +2 Addition Property of Equality ?
8−2=6
x= 8 Simplify.
6=6 ✓
The solution is x = 8.

b. Solve 18 = x − 7.
Study Tip Check
18 = x − 7 Write the equation.
You can check your 18 = x − 7
solution by substituting +7 +7 Addition Property of Equality ?
it for the variable in the 18 = 25 − 7
original equation. 25 = x Simplify.
18 = 18 ✓
The solution is x = 25.

Solve the equation. Check your solution.

Exercises 18–20 5. k − 3 = 1 6. n − 10 = 4 7. 15 = r − 6

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 EXAMPLE 3 Solving Equations Using Subtraction

a. Solve x + 2 = 9.
Check
x+2= 9 Write the equation.
x+2=9
Undo the addition. −2 −2 Subtraction Property of Equality
?
7+2=9
x= 7 Simplify.

The solution is x = 7.
9=9 ✓

b. Solve 26 = 11 + x.
Check
26 = 11 + x Write the equation.
26 = 11 + x
− 11 − 11 Subtraction Property of Equality
?
26 = 11 + 15
15 = x Simplify.

The solution is x = 15.

26 = 26 ✓

EXAMPLE 4 Real-Life Application

Your parents give you $20 to help buy the new pair of shoes shown.
After you buy the shoes, you have $5.50 left. Write and solve an equation
to find how much money you had before your parents gave you $20.

Words The starting plus the amount minus the cost is the amount
amount your parents of the left.
gave you shoes

Variable Let s be the starting amount.

Equation s + 20 − 59.95 = 5.50
Study Tip s + 20 − 59.95 = 5.50 Write the equation.
In Example 4, you can
s + 20 − 59.95 + 59.95 = 5.50 + 59.95 Addition Property of Equality
solve the problem
arithmetically by s + 20 = 65.45 Simplify.
working backwards
from $5.50. s + 20 − 20 = 65.45 − 20 Subtraction Property of Equality
5.50 + 59.95 − 20 s = 45.45 Simplify.
= 45.45
So, your answer is You had $45.45 before your parents gave you money.
reasonable.

Solve the equation. Check your solution.

Exercises 21–23 8. s + 8 = 17 9. 9=y+6 10. 13 + m = 20
11. You eat 8 blueberries and your friend eats 11 blueberries from a
package. There are 23 blueberries left. Write and solve an equation
to find the number of blueberries in a full package.

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 7.2 Exercises
Help with Homework

1. WRITING How can you check the solution of an equation?

Name the inverse operation you can use to solve the equation.
2. x − 8 = 12 3. n + 3 = 13 4. b + 14 = 33

5. WRITING When solving x + 5 = 16, why do you subtract 5 from the left side of
the equation? Why do you subtract 5 from the right side of the equation?

6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-

Tell whether the given value is a solution of the equation.

1 6. x + 42 = 85; x = 43 7. 8b = 48; b = 6
m
8. 19 − g = 7; g = 15 9. — = 16; m = 4
4
10. w + 23 = 41; w = 28 11. s − 68 = 11; s = 79

Use a scale to model and solve the equation.

12. n + 7 = 9 13. t + 4 = 5 14. c + 2 = 8

Write a question that represents the equation. Use mental math to answer
the question. Then check your solution.
15. a + 5 = 12 16. v + 9 = 18 17. 20 = d − 6

Solve the equation. Check your solution.

2 18. y − 7 = 3 19. z − 3 = 13 20. 8 = r − 14
3 21. p + 5 = 8 22. k + 6 = 18 23. 64 = h + 30

3 1
24. f − 27 = 19 25. 25 = q + 14 26. — = j − —
4 2
2 9
27. x + — = — 28. 1.2 = m − 2.5 29. a + 5.5 = 17.3
3 10

ERROR ANALYSIS Describe and correct the error in solving the equation.
30. 31.
✗ x + 7 = 13

x
+7 +7
= 20
✗ 34 = y − 12
− 12
22 = y
+ 12

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 32. PENGUINS An emperor penguin is 45 inches tall. It is
3
24 inches taller than a rockhopper penguin. Write and
solve an equation to find the height of a rockhopper
24 in.
penguin. Is your answer reasonable? Explain.
33. ELEVATOR You get in an elevator and go down 8 floors.
You exit on the 16th floor. Write and solve an equation
to find what floor you got on the elevator.

HAITI DOMINICAN
JAMAICA REPUBLIC
34. AREA The area of Jamaica Port-au-
Kingston Santo
is 6460 square miles less than Prince Domingo
the area of Haiti. Write and Area = 4181 mi2
solve an equation to find the
area of Haiti.

x + 3 = 12 Write the equation.

35. REASONING The solution of the
x + 3 − 3 = 12 − 3
equation x + 3 = 12 is shown.
Explain each step. Use a x+0=9
property, if possible. x=9

Write the word sentence as an equation. Then solve the equation.

36. 13 subtracted from a number w is 15. 37. A number k increased by 7 is 34.
38. 9 is the difference of a number n and 7. 39. 93 is the sum of a number g and 58.

Solve the equation. Check your solution.

40. b + 7 + 12 = 30 41. y + 4 − 1 = 18 42. m + 18 + 23 = 71
43. v − 7 = 9 + 12 44. 5 + 44 = 2 + r 45. 22 + 15 = d − 17

GEOMETRY Write and solve an addition equation to find x.

46. Perimeter = 48 ft 47. Perimeter = 132 in. 48. Perimeter = 93 ft
16 in.
18 ft 18 ft
20 ft
x 34 in. 34 in.
15 ft 15 ft

x
12 ft x

49. REASONING Explain why the equations x + 4 = 13 and 4 + x = 13 have the

same solution.
50. REASONING Explain why the equations x − 13 = 4 and 13 − x = 4 do not
have the same solution.

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 51. SIMPLIFYING AND SOLVING Compare and contrast the two problems.

Simplify the expression 2(x + 3) − 4. Solve the equation x + 3 = 4.

2(x + 3) − 4 = 2x + 6 − 4 x+3= 4
= 2x + 2 −3 −3
x= 1

52. PUZZLE In a magic square, the sum of the numbers in each row,
column, and diagonal is the same. Write and solve equations to a 37 16
find the values of a, b, and c.
19 25 b
53. FUNDRAISER You participate in a dance-a-thon fundraiser. After
your parents pledge $15.50 and your neighbor pledges $8.75, you 34 c 28
have $66.55. Write and solve an equation to find how much money
you had before your parents and neighbor pledged.

54. MONEY On Saturday, you spend $33, give $15 to a friend, and receive $20 for
mowing your neighbor’s lawn. You have $21 left. Use two methods to find
how much money you started with that day.

Bumper Cars: $1.75

Roller Coaster: $1.25 more than Ferris Wheel 55. AMUSEMENT PARK You have $15.
Giant Slide: $0.50 less than Bumper Cars
Ferris Wheel: $1.50 more than Giant Slide a. How much money do you
have left if you ride each
ride once?
b. Do you have enough
money to ride each ride
twice? Explain.

56. Consider the equation x + y = 15. The value of x increases by 3.

What has to happen to the value of y so that x + y = 15 remains true?

Find the value of the expression. Use estimation to check your answer. (Section 1.1)
57. 12 × 8 58. 13 × 16 59. 75 ÷ 15 60. 72 ÷ 3

61. MULTIPLE CHOICE What is the area of the parallelogram?

(Section 4.1) 5 in.
2 2
A 25 in.
○ B 30 in.
○
10 in.
C 50 in.2
○ D 100 in.2
○

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 Solving Equations Using
7.3
Multiplication or Division

How can you use multiplication or division to

solve an equation?

1 ACTIVITY: Finding Missing Dimensions

Work with a partner. Describe how you would find the value of x. Then find
the value and check your result.
a. rectangle b. parallelogram c. triangle
Area â 24 square units Area â 20 square units Area â 28 square units

6 x x

x 5
8

2 ACTIVITY: Using an Equation to Model a Story

Work with a partner.
a. Use a model to solve the problem.

Three people go out to lunch. They decide to share the $12 bill evenly.
How much does each person pay?

● What equation does the model represent? Explain how this represents
the problem.

Solving Equations
In this lesson, you will
● use multiplication or

division to solve equations.

● use substitution to

check answers.
● solve real-life problems.

● How much does one weigh? How do you know?

Each person pays .

b. Describe how you can check your answer in part (a).

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 3 ACTIVITY: Using Equations to Model a Story
ory
Work with a partner.
● What is the unknown?
● Write an equation that represents each problem.
m.
● What does the variable in your equation represent?
sent?
● Explain how you can solve the equation.
● Answer the question.

Problem Equati
Equa
Eq tio
ion
Equation
a.
a
Three robots go out to lunch. They decide to
share the $11.91 bill evenly. How much does each
robot pay?

b.
b
On Earth, objects weigh 6 times what they weigh on
the Moon. A robot weighs 96 pounds on Earth. What
does it weigh on the Moon?
Math
c.
Practice At maximum speed, a robot runs 6 feet in 1 second.
Interpret
How many feet does the robot run in 1 minute?
Results
What does the
solution represent? d.
Does the answer Four identical robots lie on the ground head-to-toe
e
make sense? and measure 14 feet. How tall is each robot?

4. Complete each sentence by matching.

● The inverse operation of addition ● is multiplication.
ulti
tipl
plic
icat
atio
ion.
● The inverse operation of subtraction ● is subtraction.
ubtr
trac
acti
t on.
● The inverse operation of multiplication ● is addition.
ddittio
dd ion
n
● The inverse operation of division ● is division.

5. IN YOUR OWN WORDS How can you use multiplication or division to solve
an equation? Give two examples to show how your procedure works.

Use what you learned about solving equations to complete

Exercises 15–18 on page 312.

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 7.3 Lesson
Lesson Tutorials

Multiplication Property of Equality

Remember Words When you multiply each side of an equation by the same
nonzero number, the two sides remain equal.
Inverse operations
“undo” each other. 8 x
Numbers —=2 Algebra —=2
Multiplication and 4 4
division are inverse
operations. 8
—
4 ⋅4 = 2 ⋅4 x
—
4 ⋅4 = 2 ⋅4
8=8 x=8

Multiplicative Inverse Property

1
Words The product of a nonzero number n and its reciprocal, — , is 1.
n
Numbers ⋅ 15
5 —=1 Algebra ⋅ n1 1
⋅
n — = — n = 1, n ≠ 0
n

EXAMPLE 1 Solving Equations Using Multiplication

w
a. Solve — = 12.
4
w Check
— = 12 Write the equation.
4 w
— = 12
Undo the division.
w
⋅
— 4 = 12
4 ⋅4 Multiplication Property
of Equality
4

48 ?
— = 12
w = 48 Simplify. 4

The solution is w = 48. 12 = 12 ✓

2
b. Solve — x = 6.
7
2
—x = 6 Write the equation.
7
Use the Multiplicative
Inverse Property.
7
—
2 ⋅ ( —27x ) = —72 ⋅ 6 Multiplication Property of Equality

x = 21 Simplify.

The solution is x = 21.

Solve the equation. Check your solution.

Exercises 7–10 a 2y
1. — = 6 2. 14 = — 3. 3z ÷ 2 = 9
8 5

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 Division Property of Equality
Words When you divide each side of an equation by the same
nonzero number, the two sides remain equal.

Numbers ⋅
8 4 = 32 Algebra 4x = 32

⋅
8 4 ÷ 4 = 32 ÷ 4
4x
—= —
4
32
4

8=8 x=8

EXAMPLE 2 Solving an Equation Using Division

Solve 5b = 65.
5b = 65 Write the equation. Check

5b 65
5b = 65
Undo the multiplication. —=— Division Property of Equality ?
5 5 5(13) = 65
b = 13 Simplify. 65 = 65 ✓
The solution is b = 13.

EXAMPLE 3 Real-Life Application

The area of the parallelogram-shaped courtyard is 2730 square feet.
What is the length of the sidewalk?
The height of the parallelogram represents the length of the sidewalk.
65 ft
A = bh Use the formula for area of a parallelogram.
2730 = 65h Substitute 2730 for A and 65 for b.

2730 65h
—=— Division Property of Equality
65 65

42 = h Simplify.

So, the sidewalk is 42 feet long.

Solve the equation. Check your solution.

Exercises 11–14
⋅
4. p 3 = 18 5. 12q = 60 6. 81 = 9r
7. You and four friends buy tickets to a baseball game. The total
cost is $70. Write and solve an equation to find the cost of
each ticket.

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 7.3 Exercises
Help with Homework

1. NUMBER SENSE What number divided by 12 equals 1?

x
2. WRITING What property of equality would you use to solve — = 7?
6
Explain how you would use the property.

Copy and complete the first step in the solution.

x
3. 4x = 24 4. — = 11 5. 8=n÷3
3
4x
—=—
24 x
—
3 ⋅ = 11 ⋅ 8 ⋅ = (n ÷ 3) ⋅
6. OPEN-ENDED Write an equation that can be solved using the Division
Property of Equality.

6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-

Solve the equation. Check your solution.

s t 3r
1 7. — = 7 8. 6 = — 9. 5x ÷ 6 = 20 10. 24 = —
10 5 4
2 11. 3a = 12 ⋅
12. 5 z = 35 13. 40 = 4y 14. 42 = 7k

15. 7x = 105 16. 75 = 6 w ⋅ 17. 13 = d ÷ 6 18. 9 = v ÷ 5

2c
19. — = 8.8
15
20. 7b ÷ 12 = 4.2 ⋅
21. 12.5 n = 32 22. 3.4m = 20.4

✗
23. ERROR ANALYSIS Describe and correct the error in x ÷ 4 = 28
solving the equation.
x÷4 28
—=—
4 4
24. ANOTHER WAY Show how you can solve the equation x=7
3x = 9 by multiplying each side by the reciprocal of 3.

25. BASKETBALL Forty-five basketball players

participate in a tournament. Write and
solve an equation to find the number of
3-person teams that they can form.

26. THEATER A theater has 1200 seats. Each

row has 20 seats. Write and solve an
equation to find the number of rows in
the theater.

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 Solve for x. Check your answer.
27. rectangle 28. rectangle 29. parallelogram

Area â 45 square units Area â 176 square units Area â 104 square units

5 x x

13
x 16

30. TEST SCORE On a test, you correctly answer six 5-point questions and eight
2-point questions. You earn 92% of the possible points on the test. How many
points p is the test worth?
Your Cards

31. CARD GAME You use index cards to play a homemade 48

12 mm
game. The object is to be the first to get rid of all your
Friend’s Cards
cards. How many cards are in your friend’s stack?
5 mm

32. SLUSH DRINKS A slush drink machine fills

1440 cups in 24 hours.
a. Write and solve an equation to find the
number c of cups each symbol represents.
Key:
â
c cups b. To lower costs, you replace the cups with
Slush drinks in 24
hours paper cones that hold 20% less. Write and solve
an equation to find the number n of paper cones
that the machine can fill in 24 hours.

STRUCTURE Solve the equation. Explain how you

found your answer.
33. 5x + 3x = 5x + 18 34. 8y + 2y = 2y + 40

35. The area of the picture

is 100 square inches. The length is
4 times the width. Find the length
and width of the picture.

Write the word sentence as an equation. (Section 7.1)

36. The sum of a number b and 8 is 17. 37. A number t divided by 3 is 7.

38. MULTIPLE CHOICE What is the value of a3 when a = 4? (Section 3.1)

A 12
○ B 43
○ C 64
○ D 81
○

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 Writing Equations in
7.4
Two Variables

How can you write an equation in

two variables?

1 ACTIVITY: Writing an Equation in Two Variables

Work with a partner. You earn $8 per hour
working part-time at a store.
a. Complete the table.

Hours Money Earned

Worked (dollars)
1

b. Use the values from the table to

complete the graph. Then answer 40

each question below.

32
● What does the horizontal axis
represent? What variable did 24

you use to identify it?

16
● What does the vertical axis
represent? What variable did 8

you use to identify it?

0
Writing Equations 0 1 2 3 4 5
● How are the ordered pairs in
In this lesson, you will
● identify independent and
the graph related to the values
dependent variables. in the table?
● write equations in

two variables. ● How are the horizontal and vertical distances shown on the graph
● use tables and graphs to
related to the values in the table?
analyze the relationship
between two variables.
c. How can you write an equation that shows how the two variables
are related?

d. What does the green line in the graph represent?

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 2 ACTIVITY: Describing Variables
Work with a partner. Use the equation you wrote in Activity 1.
a. How is this equation different from the equations earlier in this chapter?
b. One of the variables in this equation depends on the other variable.
Determine which variable is which by answering the following questions:
● Does the amount of money you earn depend on the number of hours
you work?
● Does the number of hours you work depend on the amount of money
you earn?
What do you think is the significance of having two types of variables?
How do you think you can use these types of variables in real life?

3 ACTIVITY: Describing a Formula in Two Variables

Work with a partner. Recall that the perimeter of a square is 4 times
Math its side length.
Practice
Look for a. Write the formula for the perimeter of a square.
Patterns Tell what each variable represents.
What pattern do b. Describe how the perimeter of a square changes as
you notice in the
table for the
its side length increases by 1 unit. Use a table and
perimeter of a graph to support your answer.
the square? c. In your formula, which variable depends on which?

4. IN YOUR OWN WORDS How can you write an equation in two variables?

5. The equation y = 7.75x shows how the number of movie tickets is related
to the total amount of money spent. Describe what each part of the
equation represents.

6. CHOOSE TOOLS In Activity 1, you want to know the amount of money you
earn after working 30.5 hours during a week. Would you use the table, the
graph, or the equation to find your earnings? What are your earnings?
Explain your reasoning.

7. Give an example of another real-life situation that you can model by an

equation in two variables.

Use what you learned about equations in two variables to

complete Exercises 4 and 5 on page 319.

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 7.4 Lesson
Lesson Tutorials

An equation in two variables represents two quantities that change in

relationship to one another. A solution of an equation in two variables is
an ordered pair that makes the equation true.

EXAMPLE 1 Identifying Solutions of Equations in Two Variables

Tell whether the ordered pair is a solution of the equation.
Key Vocabulary
a. y = 2x; (3, 6) b. y = 4x − 3; (4, 12)
equation in two
? ?
variables, p. 316 6 = 2(3) Substitute. 12 = 4(4) − 3
solution of an
equation in two
6=6 ✓ Compare. 12 ≠ 13 ✗
variables, p. 316
So, (3, 6) is a solution. So, (4, 12) is not a solution.
independent variable,
p. 316
You can use equations in two variables to represent situations involving
dependent variable,
p. 316 two related quantities. The variable representing the quantity that
can change freely is the independent variable. The other variable is
called the dependent variable because its value depends on the
independent variable.

EXAMPLE 2 Using an Equation in Two Variables

The equation y = 128 − 8x gives the amount y (in fluid ounces) of milk
remaining in a gallon jug after you pour x cups.
a. Identify the independent and dependent variables.
Because the amount y remaining depends on the number x
of cups you pour, y is the dependent variable and x is the
independent variable.
b. How much milk remains in the jug after you pour 10 cups?
Use the equation to find the value of y when x = 10.
y = 128 − 8x Write the equation.
= 128 − 8(10) Substitute 10 for x.
= 48 Simplify.
There are 48 fluid ounces remaining.

Tell whether the ordered pair is a solution of the equation.

Exercises 6–11
1. y = 7x; (2, 21) 2. y = 5x + 1; (3, 16)
and 13–17
3. The equation y = 10x + 25 gives the amount y (in dollars) in
your savings account after x weeks.
a. Identify the independent and dependent variables.
b. How much is in your savings account after 8 weeks?

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 Tables, Graphs, and Equations
Study Tip You can use tables and graphs to represent equations in two variables.
When you draw a line
The table and graph below represent the equation y = x + 2.
through the points, you y
graph all the solutions Independent Dependent Ordered Pair, 6
of the equation. Variable, x Variable, y (x, y) 5
(2, 4) (3, 5)
4
1 3 (1, 3)
3
(1, 3)
2 4 (2, 4) 2
1
3 5 (3, 5) 0
0 1 2 3 4 5 6 x

EXAMPLE 3 Writing and Graphing an Equation in Two Variables

An athlete burns 200 calories weight lifting. The athlete then works out
on an elliptical trainer and burns 10 calories for every minute. Write and
graph an equation in two variables that represents the total number of
calories burned during the workout.

Words The total equals calories plus calories times the number
Reading number burned burned per of minutes.
of calories weight minute
Make sure you read and
burned lifting
understand the context
of the problem. Because Variables Let c be the total number of calories burned, and let m be the
you cannot have a number of minutes on the elliptical trainer.
negative number of
minutes, use only whole
number values of m.
Equation c = 200 + 10 ⋅ m

To graph the equation, first make a table. Then plot the ordered pairs and
draw a line through the points.
c
Minutes, Calories, Ordered
c = 200 + 10m
m c Pair, (m, c) 600
Calories

(30, 500)
10 c = 200 + 10(10) 300 (10, 300) 400
(20, 400)
20 c = 200 + 10(20) 400 (20, 400) 200
(10, 300)

30 c = 200 + 10(30) 500 (30, 500)

0
0 20 40 60 m
Minutes

4. It costs $25 to rent a kayak plus $8 for each hour. Write

Exercises 22 and graph an equation in two variables that represents
and 23 the total cost of renting the kayak.

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 You can model many rate problems by using the distance formula d = rt,
where d is the distance traveled, r is the speed, and t is the time. When
you are given a speed, you can use the formula to write an equation in
two variables that represents the situation.

Distance Formula
Remember
Words To find the distance traveled d, multiply the speed r by the
Speed is an example of
time t.
a rate.
Algebra d = rt

EXAMPLE 4 Real-Life Application

A train averages 40 miles per hour between two cities. Use a graph to
show the relationship between the time and the distance traveled.
Method 1: Use a ratio table.
d You can use a ratio table and multiplication to find equivalent rates. Then
(6, 240)
Distance (miles)

240
plot the ordered pairs (time, distance) from the table and draw a line
through the points.
160
(4, 160)
×2 ×4 ×6
80
(2, 80)
(1, 40) Time (hours) 1 2 4 6
0
0 2 4 6 t
Time (hours) Distance (miles) 40 80 160 240

×2 ×4 ×6

Method 2: Use an equation in two variables.

Use the distance formula to write the equation d = 40t. Use the equation
to make a table. Then plot the ordered pairs and draw a line through the
points, as shown in the graph above.

Time (hours), t d = 40t Distance (miles), d Ordered Pair, (t, d )

1 d = 40(1) 40 (1, 40)

2 d = 40(2) 80 (2, 80)

4 d = 40(4) 160 (4, 160)

6 d = 40(6) 240 (6, 240)

5. WHAT IF? The train averages 50 miles per hour. Use a graph to show
Exercise 25 the relationship between the time and the distance traveled.

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 7.4 Exercises
Help with Homework

1. VOCABULARY How are independent variables and dependent variables different?

2. PRECISION Explain how to graph an equation in two variables.
3. WHICH ONE DOESN’T BELONG? Which one does not belong with the other three?
Explain your reasoning.

y = 12x + 25 c = 10t − 5 a = 7b + 11 n = 4n − 6

6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-

Write a formula for the given measure. Tell what each variable represents.
Identify which variable depends on which in the formula.
4. the perimeter of a rectangle with a length of 5 inches
5. the area of a trapezoid with base lengths of 7 feet and 11 feet

Tell whether the ordered pair is a solution of the equation.

1 6. y = 4x; (0, 4) 7. y = 3x; (2, 6) 8. y = 5x − 10; (3, 5)
9. y = x + 7; (1, 6) 10. y = 7x + 2; (2, 0) 11. y = 2x − 3; (4, 5)

12. ERROR ANALYSIS Describe and correct

the error in finding a solution of the
equation in two variables.
✗ y = 3x + 2; (5, 1)
?
5 = 3(1) + 2
5=5
So, (5, 1) is a solution.

Identify the independent and dependent variables.

2 13. The equation A = 25w gives the area A (in square feet) of a rectangular
dance floor with a width of w feet.

14. The equation c = 0.09s gives the amount c (in dollars) of commission
a salesperson receives for making a sale of s dollars.

15. The equation t = 12p + 12 gives the total cost t (in dollars)
of a meal with a tip of p percent (in decimal form).

16. The equation h = 60 − 4m gives the height h (in inches)

of the water in a tank m minutes after it starts to drain.

17. DRUM SET The equation b = 540 − 30m gives the

balance b (in dollars) that you owe on a drum set
after m monthly payments. What is the balance after
9 monthly payments?

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 OPEN-ENDED Complete the table by describing possible independent or
dependent variables.

Independent Variable Dependent Variable

18. The number of hours you study for a test

19. The speed you are pedaling a bike
20. Your monthly cell phone bill
21. The amount of money you earn

3 22. PIZZA A cheese pizza costs $5. Additional toppings cost $1.50 each. Write and
graph an equation in two variables that represents the total cost of a pizza.

23. GYM MEMBERSHIP It costs $35 to join a gym. The monthly fee is $25.
Write and graph an equation in two variables that represents the
total cost of a gym membership.

24. TEXTING The maximum size of a text message is 160 characters.

A space counts as one character.
a. Write an equation in two variables that represents the
Jack
remaining (unused) characters in a text message as you type. ie
Plea
se ca
b. Identify the independent and dependent variables. ll me
.
c. How many characters remain in the message shown?

4 25. CHOOSE TOOLS A car averages 60 miles per hour on a

road trip. Use a graph to show the relationship between
the time and the distance traveled. What method did
you use to create your graph?

Write and graph an equation in two variables that shows

the relationship between the time and the distance traveled.
26. 27.
Moves 2 meters
every 3 hours. Rises 5 stories
every 6 seconds.

28. 29.

Moves 660 feet

every 10 seconds.
Moves 960 kilometers
every 4 minutes.

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 Fill in the blank so that the ordered pair is a solution of the equation.
30. y = 8x + 3; (1, ) 31. y = 12x + 2; ( , 14) 32. y = 22 − 9x; ( , 4)

33. CRITICAL THINKING Can the dependent variable cause

a change in the independent variable? Explain.

34. OPEN-ENDED Write an equation in two variables that has

(3, 4) as a solution.

12 in. 35. WALKING You walk 5 city blocks in 12 minutes. How

many city blocks can you walk in 2 hours?

36. ANT How fast should the ant walk to go around the
16 in. rectangle in 4 minutes?

37. LIGHTNING To estimate how far you are from lightning (in miles), count the
number of seconds between a lightning flash and the thunder that follows.
Then divide the number of seconds by 5. Use a graph to show the relationship
between the time and the distance. Describe the method you used to create
your graph.

38. PROBLEM SOLVING You and a friend start biking in opposite directions from the
same point. You travel 108 feet every 8 seconds. Your friend travels 63 feet every
6 seconds.
a. How far apart are you and your friend after 15 minutes?
b. After 20 minutes, you take a 5-minute rest, but your friend does not. How far
apart are you and your friend after 40 minutes? Explain your reasoning.

c
39. The graph represents the cost c (in dollars)
of buying n tickets to a baseball game. 30
(3, 30)
Dollars

a. Should the points be connected with a line to

20
show all the solutions? Explain your reasoning. (2, 20)
10
b. Write an equation in two variables that represents (1, 10)
the graph.
0
0 2 4 6 n
Tickets

Write the fraction as a percent. (Section 5.5)

3 4 9 17
40. — 41. — 42. — 43. —
10 5 20 25

44. MULTIPLE CHOICE What is the area of the

triangle? (Section 4.2) 17 cm
8 cm
2 2
A 36 cm
○ B 68 cm
○
C 72 cm2
○ D 76.5 cm2
○ 9 cm

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 7 Study Help
Graphic Organizer

You can use an example and non-example chart to list examples and non-examples of a
vocabulary word or term. Here is an example and non-example chart for equations.

Equations
Examples Non-Examples
x=5 5

2a = 16 2a

x + 4 = 19 x+4

5=x+3 x+3

12 − 7 = 5 12 − 7
3 3
y=6
4 4

Make example and non-example charts to help you study these topics.
1. inverse operations
2. equations solved using addition
or subtraction
3. equations solved using multiplication
or division
4. equations in two variables

After you complete this chapter, make

example and non-example charts for the
following topics.
5. inequalities
6. graphs of inequalities
7. inequalities solved using addition
or subtraction
“I need a good non-example of a cool animal
8. inequalities solved using multiplication for my example and non-example chart.”
or division

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 7.1– 7.4 Quiz
Progress Check

Write the word sentence as an equation. (Section 7.1)

1. A number x decreased by 3 is 5. 2. A number a divided by 7 equals 14.

Solve the equation. Check your solution. (Section 7.2 and Section 7.3)

3. 4 + k = 14 4. 3.5 = m − 2.2

5. 8 = —
4w
3 ⋅
6. 31 = 6.2 y

Tell whether the ordered pair is a solution of the equation. (Section 7.4)
7. y = 6x; (3, 24) 8. y = 3x + 4; (4, 16)

Write and graph an equation in two variables that shows the relationship between
the time and the distance traveled. (Section 7.4)
9. 10. Moves 900 feet
every 10 seconds.

Rises 4 feet
in 9 seconds.

r
11. RIBBON The length of the blue ribbon is two-thirds the
length of the red ribbon. Write an equation you can use to
find the length r of the red ribbon. (Section 7.1) 30 cm

12. BRIDGES The main span of the Sunshine Skyway Bridge is 360 meters long.
The Skyway’s main span is 30 meters shorter than the main span of the
Dames Point Bridge. Write and solve an equation to find the length ℓ of the
main span of the Dames Point Bridge. (Section 7.2)

13. SHOPPING At a farmer’s market,

you buy 4 pounds of tomatoes and
2 pounds of sweet potatoes. You
spend 80% of the money in your
wallet. Write and solve an equation
to find how much money is in your
wallet before you pay. (Section 7.3)

14. SUNDAE A sundae costs $2. Additional

toppings cost $0.50 each. Write and
graph an equation in two variables
that represents the total cost of
a sundae. (Section 7.4)

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 7.5 Writing and Graphing Inequalities

How can you use a number line to represent

solutions of an inequality?

1 ACTIVITY: Understanding Inequality Statements

Work with a partner. Read the statement. Circle each
number that makes the statement true, and then
answer the questions.
a. “Your friend is more than 3 minutes late.”

−3 −2 −1 0 1 2 3 4 5 6
● What do you notice about the numbers that you circled?
● Is the number 3 included? Why or why not?
● Write four other numbers that make the statement true.

b. “The temperature is at most 2 degrees.”

−5 −4 −3 −2 −1 0 1 2 3 4
● What do you notice about the numbers that you circled?
● Can the temperature be exactly 2 degrees? Explain.
● Write four other numbers that make the statement true.

c. “You need at least 4 pieces of paper for your math homework.”

−3 −2 −1 0 1 2 3 4 5 6
● What do you notice about the numbers that you circled?
● Can you have exactly 4 pieces of paper? Explain.
Writing Inequalities ● Write four other numbers that make the
In this lesson, you will statement true.
● write word sentences

as inequalities.
● use a number line to

graph the solution set

d. “After playing a video game for 20 minutes, you
of inequalities. have fewer than 6 points.”
● use inequalities to

represent real-life −2 −1 0 1 2 3 4 5 6 7
situations.
● What do you notice about the numbers that you circled?
● Is the number 6 included? Why or why not?
● Write four other numbers that make the statement true.

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 2 ACTIVITY: Understanding Inequality Symbols
Work with a partner.
a. Consider the statement “x is a number such that x < 2.”
● Can the number be exactly 2? Explain.
● Circle each number that makes the statement true.
Math −5 −4 −3 −2 −1 0 1 2 3 4
Practice ● Write four other numbers that make the statement true.
State the
Meaning of
Symbols b. Consider the statement “x is a number such that x ≥ 1.”
What do the ● Can the number be exactly 1? Explain.
symbols < and
≥ mean? ● Circle each number that makes the statement true.
−5 −4 −3 −2 −1 0 1 2 3 4
● Write four other numbers that make the statement true.

3 ACTIVITY: How Close Can You Come to 0?

Work with a partner.
a. Which number line shows x > 0? Which number line shows x ≥ 0?
Explain your reasoning.

Ź8 Ź6 Ź4 Ź2 0 2 4 6 8

Ź8 Ź6 Ź4 Ź2 0 2 4 6 8

b. Write the least positive number you can think of that is still a
solution of the inequality x > 0. Explain your reasoning.

4. IN YOUR OWN WORDS How can you use a number line to represent
solutions of an inequality?
5. Write an inequality. Graph all solutions of your inequality on a
number line.
6. Graph the inequalities x > 9 and 9 < x on different number lines.
What do you notice?

Use what you learned about graphing inequalities to complete

Exercises 17–20 on page 329.

Section 7.5 Writing and Graphing Inequalities 325

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 7.5 Lesson
Lesson Tutorials

An inequality is a mathematical sentence that compares expressions.

Key Vocabulary It contains the symbols <, >, ≤, or ≥. To write an inequality, look for the
inequality, p. 326 following phrases to determine where to place the inequality symbol.
solution of an
inequality, p. 327
Inequality Symbols
solution set, p. 327
graph of an Symbol < > ≤ ≥
inequality, p. 328
● is less ● is greater ● is less than or ● is greater than
Key than than equal to or equal to
Phrases ● is fewer ● is more ● is at most ● is at least
than than ● is no more than ● is no less than

EXAMPLE 1 Writing Inequalities

Write the word sentence as an inequality.

a. A number c is less than −4.
A number c is less than −4.

c < −4
An inequality is c < −4.

b. A number k plus 5 is greater than or equal to 8.

A number k plus 5 is greater than or equal to 8.

k+5 ≥ 8
An inequality is k + 5 ≥ 8.

c. Four times a number q is at most 16.

Four times a number q is at most 16.

4q ≤ 16
An inequality is 4q ≤ 16.

Write the word sentence as an inequality.

Exercises 5–10 1. A number n is greater than 1.
2. Twice a number p is fewer than 7.
3. A number w minus 3 is less than or equal to 10.
4. A number z divided by 2 is at least −6.

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 A solution of an inequality is a value that makes the inequality true. An
inequality can have more than one solution. The set of all solutions of an
inequality is called the solution set.

Value of x x+3≤7 Is the inequality true?

?
3 3+3 ≤ 7 yes
6≤7 ✓
?
4+3 ≤ 7
Reading 4
7≤7 ✓
yes

The symbol ≤ means

?
is not less than or
5 5+3 ≤ 7 no
equal to. 8 ≤/ 7 ✗

EXAMPLE 2 Checking Solutions

Tell whether the given value is a solution of the inequality.

a. x + 1 > 7; x = 8
x+1 > 7 Write the inequality.
?
8+1 > 7 Substitute 8 for x.
9>7 ✓ Add. 9 is greater than 7.
So, 8 is a solution of the inequality.
b. 7y < 27; y = 4
7y < 27 Write the inequality.
?
7(4) < 27 Substitute 4 for y.
28 </ 27 ✗ Multiply. 28 is not less than 27.
So, 4 is not a solution of the inequality.
z
c. — ≥ 5; z = 15
3
z
— ≥ 5 Write the inequality.
3

15 ?
— ≥ 5 Substitute 15 for z.
3

5≥5 ✓ Divide. 5 is greater than or equal to 5.

So, 15 is a solution of the inequality.

Tell whether 3 is a solution of the inequality.

Exercises 11–16 5. b + 4 < 6 6. 9−n ≥ 6 7. 18 ÷ x ≤ 10

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 The graph of an inequality shows all the solutions of the inequality on a
number line. An open circle ○ is used when a number is not a solution.
A closed circle ● is used when a number is a solution. An arrow to the
left or right shows that the graph continues in that direction.

EXAMPLE 3 Graphing an Inequality

Graph g > 2.
Use an open circle because
Reading 2 is not a solution.
The inequality g > 2 is
the same as 2 < g.
Ź5 Ź4 Ź3 Ź2 Ź1 0 1 2 3 4 5

Test a number to the left of 2. Test a number to the right of 2.

g â 0 is not a solution. g â 3 is a solution.

Shade the number line on the side where you

found the solution. The graph shows there are
infinitely many solutions.

Ź5 Ź4 Ź3 Ź2 Ź1 0 1 2 3 4 5

EXAMPLE 4 Real-Life Application

The NASA Solar Probe Plus can withstand temperatures up to and

including 2600°F. Write and graph an inequality that represents
the temperatures the probe can withstand.
Words temperatures up to and including 2600°F
Variable Let t be the temperatures the probe can withstand.
Inequality t ≤ 2600 V I D E O

An inequality is t ≤ 2600.

2000 2200 2400 2600 2800 3000

Graph the inequality on a number line.

Exercises 25–36 8. a < 4 9. f ≤ 7 10. n > 0 11. p ≥ −3

Write and graph an inequality for the situation.

12. A cruise ship can carry at most 3500 passengers.
13. A board game is designed for ages 12 and up.

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 7.5 Exercises
Help with Homework

1. VOCABULARY How are greater than and greater than or equal to similar?
How are they different?
2. DIFFERENT WORDS, SAME QUESTION Which is different? Write “both” inequalities.

A number n is at most 3. A number n is at least 3.

A number n is less than or equal to 3. A number n is no more than 3.

3. WRITING Explain
l h
how the h off x ≤ 6 is different ffrom the
h graph h graph
h off x < 6.
4. WRITING Are the graphs of x ≤ 5 and 5 ≥ x the same or different? Explain.

6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-

Write the word sentence as an inequality.

1 5. A number k is less than 10. 6. A number a is more than 6.
3
7. A number z is fewer than —. 8. A number b is at least −3.
4

9. One plus a number y is no more than −13.

10. A number x divided by 3 is at most 5.

Tell whether the given value is a solution of the inequality.

2 11. x − 1 ≤ 7; x = 6 12. y + 5 < 13; y = 17
b
13. 3z > 6; z = 3 14. — ≥ 6; b = 10
2

15. c + 2.5 < 4.3; c = 1.8 16. a ≤ 0; a = −5

Match the inequality with its graph.

17. x ≥ 2 18. x < 2 19. x > −2 20. x ≤ −2

A.
Ź3 Ź2 Ź1 0 1 2 3

B.
Ź3 Ź2 Ź1 0 1 2 3

C.
Ź3 Ź2 Ź1 0 1 2 3

D.
Ź3 Ź2 Ź1 0 1 2 3

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 Write an inequality and a word sentence that represent the graph.
21. 22.
Ź3 Ź2 Ź1 0 1 2 3 Ź2 Ź1 0 1 2 3 4

23. 24.
Ź6 Ź4 Ź2 0 2 4 6 Ź5 0 5 10 15 20 25

Graph the inequality on a number line.

1
3 25. a > 4 26. n ≥ 8 27. 3 ≥ x 28. y < —
2

2
29. x < — 30. −3 ≥ c 31. m > −5 32. 0 ≤ b
9

1 7
33. 1.5 > f 34. t ≥ −— 35. p > −1.6 36. — ≥ z
2 3

ERROR ANALYSIS Describe and correct the error in graphing the inequality.
37. 38.
x≥1 x > −1

✗ 0 1 2 3
✗ Ź3 Ź2 Ź1 0

39. FISHING You are fishing and are allowed

to catch at most 3 striped bass. Each
striped bass must be no less than
18 inches long.
a. Write and graph an inequality to
represent the number of striped
bass you are allowed to catch.
18 in.
b. Write and graph an inequality to
represent the length of each striped
bass you are allowed to catch.

40. MODELING For a food to be labeled low sodium,

there must be no more than 140 milligrams of sodium
per serving.
a. Write and graph an inequality to represent the
amount of sodium in a low-sodium serving.
b. Write and graph an inequality to represent the
amount of sodium in a serving that does not qualify
as low sodium.
c. Does the food represented by the nutrition facts label
qualify as a low-sodium food? Explain.

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 41. SHOPPING You have $33. You want to buy a necklace
and one other item from the list. Item Price (with tax)
a. Write an inequality to represent the situation. T-shirt $ 15
Book $ 20
b. Can the other item be a T-shirt? Explain.
DVD $ 13
c. Can the other item be a book? Explain. Necklace $ 16

Determine whether the statement is sometimes, always, or never true.

Explain your reasoning.
42. A number that is a solution of the inequality x > 5 is also a solution of the
inequality x ≥ 5.
43. A number that is a solution of the inequality 5 ≤ x is also a solution of the
inequality x > 5.

44. BUS RIDE A bus ride costs $1.50. A 30-day bus pass costs $36. Write an
inequality to represent the number of bus rides you would need to take
for the bus pass to be a better deal.

45. MOVIE THEATER Fifty people are seated in a movie theater. The maximum
capacity of the theater is 425 people. Write an inequality to represent the
number of additional people who can still be seated.

46. The map shows the elevations above sea level for an area of land.
a. Graph the possible elevations of A. Write
A
the set of elevations as two inequalities.
b. Graph the possible elevations of C. How
B Key can you write this set of elevations as a
C 0–100 ft
100–200 ft single inequality? Explain.
200–300 ft c. What is the elevation of B ? Explain.
300–400 ft
400–500 ft

Solve the equation. Check your solution. (Section 7.2)

47. x + 3 = 12 48. x − 6 = 8 49. 16 + x = 44 50. 7.6 = x − 6.5
3
51. MULTIPLE CHOICE A stack of boards is 24 inches high. Each board is — of an
8
inch thick. How many boards are in the stack? (Section 2.2)
1 1
A
○ — B
○ — C 9
○ D 64
○
9 6

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 Solving Inequalities Using
7.6
Addition or Subtraction

How can you use addition or subtraction to

solve an inequality?

1 ACTIVITY: Writing an Inequality

Work with a partner. In 3 years,
your friend will still not be old
enough to vote.
a. Which of the following represents
your friend’s situation? What does
x represent? Explain your reasoning.

x + 3 < 18 x + 3 ≤ 18

x + 3 > 18 x + 3 ≥ 18

b. Graph the possible ages of your friend on a number line. Explain how you
decided what to graph.

2 ACTIVITY: Writing an Inequality

Work with a partner. Baby manatees
are about 4 feet long at birth. They
grow to a maximum length of
13 feet.
a. Which of the following can
represent a baby manatee’s
Solving Inequalities growth? What does x represent?
In this lesson, you will Explain your reasoning.
● use addition or subtraction

to solve inequalities.
● use a number line to x + 4 < 13 x + 4 ≤ 13
graph the solution set
of inequalities.
● solve real-life problems.
x − 4 > 13 x − 4 ≥ 13

b. Graph the solution on a number line. Explain how you decided

what to graph.

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 3 ACTIVITY: Solving Inequalities
Work with a partner. Complete the following steps for Activity 1. Then repeat
the steps for Activity 2.
Math ● Use your inequality from part (a). Replace the inequality symbol with an
Practice equal sign.
Interpret
Results ● Solve the equation.
What does ● Replace the equal sign with the original inequality symbol.
the solution of
the inequality ● Graph this new inequality.
represent?
● Compare the graph with your graph in part (b). What do you notice?

4 ACTIVITY: The Triangle Inequality

Work with a partner. Draw different triangles whose
sides have lengths 10 cm, 6 cm, and x cm.
a. Which of the following describes how small x can be? x
10 cm
Explain your reasoning.

6 + x < 10 6 + x ≤ 10

6 + x > 10 6 + x ≥ 10
6 cm

b. Which of the following describes how large x can be?

Explain your reasoning.

x − 6 < 10 x − 6 ≤ 10 x − 6 > 10 x − 6 ≥ 10

c. Graph the possible values of x on a number line.

5. IN YOUR OWN WORDS How can you use addition or subtraction to solve
an inequality?
6. Describe a real-life situation that you can represent with an inequality.
Write the inequality. Graph the solution on a number line.

Use what you learned about solving inequalities to complete

Exercises 5 – 7 on page 336.

Section 7.6 Solving Inequalities Using Addition or Subtraction 333

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 7.6 Lesson
Lesson Tutorials

Study Tip
Addition Property of Inequality
You can solve
inequalities the Words When you add the same number to each side of an inequality,
same way you solve the inequality remains true.
equations. Use inverse
Numbers 3< 5 Algebra x−4 > 5
operations to get the
variable by itself. +2 +2 +4 +4
5< 7 x > 9
Graph x9

7 8 9 10 11 12

Subtraction Property of Inequality

Words When you subtract the same number from each side of an
inequality, the inequality remains true.
Numbers 3< 5 Algebra x+4 > 5
−2 −2 −4 −4
1< 3 x > 1
Graph x1

Ź1 0 1 2 3 4

These properties are also true for ≤ and ≥.

EXAMPLE 1 Solving an Inequality Using Addition

Solve x − 3 > 1. Graph the solution.

x−3 > 1 Write the inequality.

Undo the subtraction. +3 +3 Addition Property of Inequality
x> 4 Simplify.
Check:
? The solution is x > 4.
x = 3: 3 − 3 > 1
0>1 ✗ x4
?
x = 5: 5 − 3 > 1 Ź2 Ź1 0 1 2 3 4 5 6 7 8

2>1 ✓ x â 3 is not a solution. x â 5 is a solution.

Solve the inequality. Graph the solution.

1. x − 2 < 3 2. x−6 ≥ 4 3. 10 ≥ x − 1

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 EXAMPLE 2 Solving an Inequality Using Subtraction

Solve 15 ≥ 6 + x. Graph the solution.

15 ≥ 6+x Write the inequality.
Undo the addition. −6 −6 Subtraction Property of Inequality
9≥x Simplify.

Reading The solution is x ≤ 9.

The inequality x ≤ 9 is xb9
the same as 9 ≥ x.
Ź15 Ź12 Ź9 Ź6 Ź3 0 3 6 9 12 15

Solve the inequality. Graph the solution.

Exercises 5–16 4. x + 3 > 7 5. y + 2 < 17 6. 16 ≤ m + 9

EXAMPLE 3 Real-Life Application

A flea market advertises that it has more than 250 vending booths.
Of these, 184 are currently filled. Write and solve an inequality to
represent the number of vending booths still available.

Words The number plus the number is greater the total

of booths of remaining than number
filled booths of booths.

Variable Let b be the number of remaining booths.

Inequality 184 + b > 250

184 + b > 250 Write the inequality.

− 184 − 184 Subtraction Property of Inequality
b> 66 Simplify.

More than 66 vending booths are still available.

7. You have already spent $24

shopping online for clothes.
Write and solve an inequality
to represent the additional
amount you must spend to
get free shipping.

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 7.6 Exercises
Help with Homework

1. OPEN-ENDED Write an inequality that can be solved by subtracting 7 from

each side.
2. WRITING Explain how to solve the inequality x − 6 > 3.
3. WRITING Describe the graph of the solution of x + 3 ≤ 4.
4. OPEN-ENDED Write an inequality that the graph represents. Then use the
Subtraction Property of Inequality to write another inequality that the
graph represents.

Ź2 Ź1 0 1 2 3 4 5 6 7 8

6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-

Solve the inequality. Graph the solution.

1 2 5. x − 4 < 5 6. 5 + h > 7 7. 3 ≥ y − 2
8. 9 ≤ c + 1 9. 18 > 12 + x 10. 37 + z ≤ 54
11. y − 21 < 85 12. g − 17 ≥ 17 13. 7.2 < x + 4.2
3 1 1 3
14. 12.7 ≥ s − 5.3 15. — ≤ — + n 16. — + b > —
4 2 3 4

17. ERROR ANALYSIS Describe and correct the error in

solving the inequality.
✗ 28 ≥ t − 9
−9 −9
19 ≥ t
ZERO

ZERO UNIT POINT 18. AIR TRAVEL Your carry-on bag can weigh at most
40 pounds. Write and solve an inequality to represent
how much more weight you can add to the bag and still
meet the requirement.

19. SHOPPING It costs $x for a round-trip

bus ticket to the mall. You have $24.
Write and solve an inequality to represent
how much money you can spend for
the bus fare and still have enough
to buy the baseball cap.

Write the word sentence as an inequality. Then solve the inequality.

20. Five more than a number is less than 17.
21. Three less than a number is more than 15.

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 Solve the inequality. Graph the solution.
22. x + 9 − 3 ≤ 14 23. 44 > 7 + s + 26 24. 6.1 − 0.3 ≥ c + 1

25. VIDEO GAME The high score for a video game is 36,480. Your current score is
34,280. Each dragonfly you catch is worth 1 point. You also get a 1000-point
bonus for reaching 35,000 points. Write and solve an inequality to represent
the number of dragonflies you must catch to earn a new high score.

26. PICKUP TRUCKS You can register a pickup truck as a passenger vehicle if the
truck is not used for commercial purposes and the weight of the truck with
its contents does not exceed 8500 pounds.
a. Your pickup truck weighs 4200 pounds. Write an inequality to represent the
number of pounds your truck can carry and still qualify as a passenger vehicle.
Then solve the inequality.
b. A cubic yard of sand weighs about
1600 pounds. How many cubic yards of
sand can you haul in your truck and still
qualify as a passenger vehicle? Explain
your reasoning.

27. TRIATHLON You complete two events of a triathlon. Your goal is to finish with
an overall time of less than 100 minutes.
a. Write and solve an inequality to represent how Triathlon
many minutes you can take to finish the running
Your Time
event and still meet your goal. Event
(minutes)
b. The running event is 3.1 miles long. Estimate how
Swimming 18.2
many minutes it would take you to run 3.1 miles.
Biking 45.4
Would this time allow you to reach your goal?
Explain your reasoning. Running ?

28. The possible values of x are given by x − 3 ≥ 2. What is

the least possible value of 5x?

Solve the equation. Check your solution. (Section 7.3)

t 2s
29. — = 4 30. 6 = — 31. 8x = 72 32. 9 = 1.5z
12 9

33. MULTIPLE CHOICE Which brand of Brand A B C D

turkey is the best buy? (Section 5.4)
Cost (dollars) 10.38 13.47 21.45 34.93
A Brand A
○ B Brand B
○ Pounds 2 3 5 7
C Brand C
○ D Brand D
○

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 Solving Inequalities Using
7.7
Multiplication or Division

How can you use multiplication or division to

solve an inequality?

1 ACTIVITY: Writing an Inequality

Work with a partner. A store has a clearance rack
of shirts that each cost the same amount. You buy
2 shirts and have money left after paying with
a $20 bill.
a. Which of the following represents your purchase?
What does x represent? Explain your reasoning.

2x < 20 2x ≤ 20

2x > 20 2x ≥ 20

b. Graph the possible values of x on a number

line. Explain how you decided what to graph.
c. Can you buy a third shirt? Explain
your reasoning.

2 ACTIVITY: Writing an Inequality

Work with a partner. One of your favorite stores is having a 75% off sale.
You have $20. You want to buy a pair of jeans.
a. Which of the following represents your
ability to buy the jeans with $20? What
does x represent? Explain your reasoning.
Solving Inequalities
1 1
In this lesson, you will — x < 20 — x ≤ 20
● use multiplication 4 4
or division to solve
inequalities.
● use a number line to 1 1
graph the solution set — x > 20 —x ≥ 20
4 4
of inequalities.
● solve real-life problems.

b. Graph the possible values of x on a number line.

Explain how you decided what to graph.
c. Can you afford a pair of jeans that originally costs $100?
Explain your reasoning.

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 3 ACTIVITY: Solving Inequalities
Work with a partner. Complete the following steps for Activity 1.
Then repeat the steps for Activity 2.
● Use your inequality from part (a). Replace the inequality symbol with an
equal sign.
● Solve the equation.
● Replace the equal sign with the original inequality symbol.
● Graph this new inequality.
● Compare the graph with your graph in part (b). What do you notice?

4 ACTIVITY: Matching Inequalities

Work with a partner. Match the inequality with its graph. Explain
your method.
x
a. 3x < 9 b. 3x ≤ 9 c. — ≥ 1
2

x
d. 6 < 2x e. 12 ≤ 4x f. — < 2
2

A.
Math Ź8 Ź6 Ź4 Ź2 0 2 4 6 8
Practice B.
Make a Plan
Ź8 Ź6 Ź4 Ź2 0 2 4 6 8
What strategy will
you use to choose C.
the correct graph? Ź8 Ź6 Ź4 Ź2 0 2 4 6 8

D.
Ź8 Ź6 Ź4 Ź2 0 2 4 6 8

E.
Ź8 Ź6 Ź4 Ź2 0 2 4 6 8

F.
Ź8 Ź6 Ź4 Ź2 0 2 4 6 8

5. IN YOUR OWN WORDS How can you use multiplication or division to solve
an inequality?

Use what you learned about solving inequalities to complete

Exercises 8–11 on page 342.

Section 7.7 Solving Inequalities Using Multiplication or Division 339

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 7.7 Lesson
Lesson Tutorials

Multiplication Property of Inequality

Remember
Words When you multiply each side of an inequality by the same
Multiplication and positive number, the inequality remains true.
division are inverse
operations. x
Numbers 8>6 Algebra — < 2
4
8×2 > 6×2
x
—
4 ⋅4 < 2 ⋅4
16 > 12 x<8

Division Property of Inequality

Words When you divide each side of an inequality by the same
positive number, the inequality remains true.
Numbers 8>6 Algebra 4x < 8
4x 8
8÷2 > 6÷2 — < —
4 4
4>3 x<2

These properties are also true for ≤ and ≥.

EXAMPLE 1 Solving an Inequality Using Multiplication

x
Solve — ≤ 2. Graph the solution.
5
x
— ≤ 2 Write the inequality.
5

Undo the division.

x
—
5 ⋅5 ≤ 2 ⋅5 Multiplication Property of Inequality

x ≤ 10 Simplify.

The solution is x ≤ 10.

x b 10

5 6 7 8 9 10 11 12 13 14 15

x â 5 is a solution. x â 15 is not a solution.

Solve the inequality. Graph the solution.

3 s
Exercises 6–9 1. p ÷ 3 > 2 2. —q ≤ 6 3. 1< —
5 7

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 EXAMPLE 2 Solving an Inequality Using Division
Solve 4n > 32. Graph the solution.
4n > 32 Write the inequality.
4n 32
Undo the multiplication. — > — Division Property of Inequality
4 4
n>8 Simplify.

The solution is n > 8.

n8

Ź4 Ź2 0 2 4 6 8 10 12 14 16

EXAMPLE 3 Real-Life Application

A one-way bus ride costs $1.75. A 30-day bus pass costs $42.
a. Write and solve an inequality to find the least number of one-way
rides you must take for the 30-day pass to be a better deal.
b. You ride the bus an average of 20 times each month. Is the pass
a better deal? Explain.

a. Words The price of a times the number of is more $42.

one-way ride one-way rides than
Variable Let r be the number of one-way rides.

Inequality 1.75 ⋅ r > 42

1.75r > 42 Write the inequality.

1.75r 42
— > — Division Property of Inequality
1.75 1.75
r > 24 Simplify.

So, you need to take more than 24 one-way rides for the pass
to be a better deal.
b. No. The cost of 20 one-way rides is less than $42. So, the pass is
not a better deal.

Solve the inequality. Graph the solution.

Exercises 10–13 4. 11k ≤ 33 5. ⋅
5 j > 20 6. 50 ≤ 2m
7. The sign shows the toll for driving on Alligator Alley.
Write and solve an inequality to represent the number of
times someone can drive on Alligator Alley with $15.

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 7.7 Exercises
Help with Homework

1. REASONING How is the graph of the solution of 2x ≥ 10 different from the

graph of the solution of 2x = 10?

Name the property you should use to solve the inequality.

x
2. 3x ≤ 27 3. 7x > 49 4. — < 36
2
5. OPEN-ENDED Write two inequalities that have the same solution set: one that
you can solve using division and one that you can solve using multiplication.

6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-

Solve the inequality. Graph the solution.

m t 1
1 6. — < 4 7. n ÷ 6 > 2 8. — ≥ 15 9. —c ≥ 9
8 3 11

2 10. 12x < 96 11. 5x ≥ 25 ⋅

12. 8 w ≤ 72 13. 7p ≤ 42

3 5
14. —b > 15 15. 6x < 90 16. 3s ≥ 36 17. —v ≤ 45
4 9

3 5x
18. 4t > 72 19. —w ≤ 24 20. 12m < 132 21. — ≥ 30
4 8

22. ERROR ANALYSIS Describe and correct the error in

✗
x
solving the inequality. — ≤ 30
6

23. GEOMETRY The length of a rectangle is 8 feet, and its

x
—
6 ⋅6 ≤ —
30
6
area is less than 168 square feet. Write and solve an x≤5
inequality to represent the width of the rectangle.

24. PLAYGROUND Students at a playground are divided into 5 equal

groups with at least 6 students in each group. Write and solve an
inequality to represent the number of students at the playground.

Write the word sentence as an inequality.

Then solve the inequality.
25. Eight times a number n is less than 72.
26. A number t divided by 32 is at most 4.25.
27. 225 is no less than 12 times a number w.

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 Graph the numbers that are solutions to both inequalities.
28. x + 7 > 9 and 8x ≤ 64 29. x − 3 ≤ 8 and 6x < 72

30. THRILL RIDE A thrill ride at an amusement park holds

a maximum of 12 people per ride.
a. Write and solve an inequality to find the least
number of rides needed for 15,000 people.
b. Do you think it is possible for 15,000 people to
ride the thrill ride in 1 day? Explain.
31. FOOTBALL A winning football team more than doubled
the offensive yards gained by its opponent. The opponent
gained 272 offensive yards. The winning team had
80 offensive plays. Write and solve an inequality to find
the possible number of yards per play for the winning team.

Park Hours
32. LOGIC Explain how you know that 7x < 7x has no solution.
10:00 A.M.–10:00 P.M.

33. OPEN-ENDED Give an example of a real-life situation in which you can list
all the solutions of an inequality. Give an example of a real-life situation in
which you cannot list all the solutions of an inequality.

34. FUNDRAISER You are selling items from a

catalog for a school fundraiser. Write and
solve two inequalities to find the range
of sales that will earn you between $40
and $50.

Let a > b and x > y. Tell whether the statement is always true.
Explain your reasoning.
a y
35. a + x > b + y 36. a − x > b − y 37. ax > by 38. — > —
x b

Classify the quadrilateral. (Skills Review Handbook)

39. 40. 41.

42. MULTIPLE CHOICE On a normal day, 12 airplanes arrive at an airport every

15 minutes. Which rate does not represent this situation? (Section 5.3)
A 24 airplanes every 30 minutes
○ B 4 airplanes every 5 minutes
○
C 6 airplanes every 5 minutes
○ D 48 airplanes each hour
○

Section 7.7 Solving Inequalities Using Multiplication or Division 343

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 7.5–7.7 Quiz
Progress Check

Write the word sentence as an inequality. (Section 7.5)

1. A number x is greater than 0. 2. Twice a number c is at least −8.

Tell whether the given value is a solution of the inequality. (Section 7.5)
3. 2n > 16; n = 9 4. x − 1 ≤ 9; x = 10

Graph the inequality on a number line. (Section 7.5)

3
5. y > −4 6. m ≤ —
5

Solve the inequality. Graph the solution. (Section 7.6)

7. x + 4 ≤ 8 8. 18 > 16 + g

Write the word sentence as an inequality. Then solve the inequality. (Section 7.6)
9. Two less than a number is more than 15.

10. Seven more than a number is less than or equal to 27.

Solve the inequality. Graph the solution. (Section 7.7)

3a
11. — < 24 12. 121 ≥ 11s
2

Write the word sentence as an inequality. Then solve the inequality. (Section 7.7)
13. Three times a number x is more than 18.

14. 84 is no less than 7 times a number k.

15. WATER PARK Each visit to a water park costs $19.95.

An annual pass to the park costs $89.95. Write an
inequality to represent the number of times you
would need to visit the park for the pass to be a
better deal. (Section 7.5)

16. GARDEN You want to use a square section of your

yard for a garden. You have at most 52 feet of
fencing to surround the garden. Write and solve
an inequality to represent the possible lengths of
each side of the garden. (Section 7.7)

17. DELIVERY You were planning to spend $12 on a

pizza. Write and solve an inequality to represent
the additional amount you must spend to get
free delivery. (Section 7.6)

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 7 Chapter Review
Vocabulary Help

Review Key Vocabulary

equation, p. 296 solution of an equation in inequality, p. 326
solution, p. 302 two variables, p. 316 solution of an inequality, p. 327
inverse operations, p. 303 independent variable, p. 316 solution set, p. 327
equation in two variables, p. 316 dependent variable, p. 316 graph of an inequality, p. 328

Review Examples and Exercises

7.1 Writing Equations in One Variable (pp. 294–299)

Write the word sentence “The quotient of a number b and 6 is 9” as

an equation.

The quotient of a number b and 6 is 9.

b÷6 = 9 Quotient of means division.

An equation is b ÷ 6 = 9.

Write the word sentence as an equation.

1. The product of a number m and 2 is 8.
2. 6 less than a number t is 7.
3. A number m increased by 5 is 7.
4. 8 is the quotient of a number g and 3.

7.2 Solving Equations Using Addition or Subtraction (pp. 300–307)

Solve z + 5 = 13.
Check
z + 5 = 13 Write the equation.
z + 5 = 13
Undo the addition. −5 −5 Subtraction Property of Equality ?
8 + 5 = 13
z = 8 Simplify.
13 = 13 ✓
The solution is z = 8.

Solve the equation. Check your solution.

5. x − 1 = 8 6. m + 7 = 11 7. 21 = p − 12

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 7.3 Solving Equations Using Multiplication or Division (pp. 308–313)

Solve 4c = 32. Check

4c = 32 Write the equation. 4c = 32
Undo the 4c 32 ?
—=— Division Property of Equality 4(8) = 32
multiplication. 4
c=8
4
Simplify.
32 = 32 ✓

Solve the equation. Check your solution.

⋅
8. 7 q = 42 9. 7k ÷ 3 = 21
5a
10. — = 25
7

7.4 Writing Equations in Two Variables (pp. 314–321)

Tell whether (6, 16) is a solution of the equation y = 3x − 4.

?
16 = 3(6) − 4 Substitute.
16 ≠ 14 ✗ Compare.

So, (6, 16) is not a solution.

Tell whether the ordered pair is a solution of the equation.

11. y = 3x + 1; (2, 7) 12. y = 7x − 4; (4, 22)
13. TAXI A taxi ride costs $3 plus $2.50 per mile. Write and graph an equation in
two variables that represents the total cost of a taxi ride.

7.5 Writing and Graphing Inequalities (pp. 324–331)

Write the word sentence as an inequality.

a. A number x is more than −9. b. A number r divided by 2 is at most 4.
A number x is more than −9. A number r divided by 2 is at most 4.
r
x > −9 — ≤ 4
2
r
An inequality is x > −9. An inequality is — ≤ 4.
2

Write the word sentence as an inequality.

14. A number m is less than 5. 15. A number h is at least −12.

Graph the inequality on a number line.

16. x < 0 17. a ≥ 3 18. n ≤ −1

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 7.6 Solving Inequalities Using Addition or Subtraction (pp. 332–337)

Solve 1 ≤ x − 4. Graph the solution.

1 ≤ x−4 Write the inequality.

Undo the subtraction. +4 +4 Addition Property of Inequality
5≤x Simplify.

The inequality 5 ≤ x is the same as x ≥ 5.

The solution is x ≥ 5.
xr5

Ź2 Ź1 0 1 2 3 4 5 6 7 8 9

Solve the inequality. Graph the solution.

19. x + 1 > 3 20. k − 7 ≤ 0 21. y + 8 ≥ 9
22. 24 < 11 + x 23. 4 ≤ n − 4 24. x − 20 > 24
1 1
25. b + 12 ≤ 26 26. s − 1.5 < 2.5 27. — + m ≤ —
4 2

7.7 Solving Inequalities Using Multiplication or Division (pp. 338–343)

Solve 7n < 42. Graph the solution.

7n < 42 Write the inequality.

7n 42
Undo the multiplication. — < — Division Property of Inequality
7 7
n<6 Simplify.

The solution is n < 6.

n6

Ź2 Ź1 0 1 2 3 4 5 6 7 8 9

Solve the inequality. Graph the solution.

5
28. x ÷ 2 < 4 29. 9n ≥ 63 30. — x ≤ 10
3
3
31. 9 ≥ 3b 32. 10p > 40 33. — k < 15
11

34. TICKETS The cost of three tickets to a movie is at least $20. Write and
solve an inequality that represents the situation.

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 7 Chapter Test
Test Practice

Write the word sentence as an equation.

1. 7 times a number s is 84. 2. 13 is one-third of a number m.

Solve the equation. Check your solution.

3. 15 = 7 + b 4. v − 6 = 16 5. 5x = 70
6m 8k
6. 3b = 45 7. — = 30 8. — = 32
7 3

Tell whether the ordered pair is a solution of the equation.

9. y = 9x; (3, 27) 10. y = 4x + 2; (8, 36)

Write an inequality for the situation.

11. An MP3 player holds up to 300 songs. 12. Riders must be at least 48 inches tall.

Graph the inequality on a number line.

13. x ≥ 5 14. m ≤ −2

Solve the inequality. Graph the solution.

15. x − 3 < 7 16. 12 ≥ n + 6
4
17. — b ≤ 12 18. 72 > 12p
3

19. SCHOOL DANCE Each ticket to a school dance is $4. The total amount
collected in ticket sales is $332. Write and solve an equation to find
the number of students attending the dance.

20. T-SHIRTS A soccer team will sell T-shirts for a fundraiser.

The company that makes the T-shirts charges $10 per shirt
plus a $20 shipping fee per order.
a. Write and graph an equation in two variables that represents
the total cost of ordering the shirts.
b. Choose an ordered pair that lies on your graph in part (a).
Interpret it in the context of the problem.

21. HURRICANE A hurricane has wind speeds that are greater than or
equal to 74 miles per hour. Write an inequality to represent the
possible wind speeds during a hurricane.

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 7 Cumulative Assessment
Test-Takin
g Strateg
y
1. What is the area of the balcony shown below? Work Bac
kwards
4.25 ft

4 ft

8.75 ft

A. 9 ft 2 C. 26 ft 2

B. 18 ft 2 D. 52 ft 2

2. You are making identical fruit baskets using

16 apples, 24 pears, and 32 bananas. What “Work b
a
and 9. Y ckwards by try
is the greatest number of baskets you can ou will s ing 6, 7
e ,8
So, C is e that 3(8)ä
24 ,
make using all the fruit? correct. .
”

F. 2 H. 8

G. 4 I. 16

3. Which equation represents the word sentence below?

The sum of 18 and 5 is equal to 9 less than a number y.

A. 18 − 5 = 9 − y C. 18 + 5 = y − 9

B. 18 + 5 = 9 − y D. 18 − 5 = y − 9

4. Which number line is a graph of the solution of the inequality below?

x ≥5

F.
1 2 3 4 5 6 7 8 9

G.
1 2 3 4 5 6 7 8 9

H.
1 2 3 4 5 6 7 8 9

I.
1 2 3 4 5 6 7 8 9

Cumulative Assessment 349

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 5. The steps your friend took to divide two mixed numbers are shown below.

3 1 18 3
3— ÷ 1— = — × —
5 2 5 2
27
=—
5
2
= 5—
5

What should your friend change in order to divide the two mixed
numbers correctly?

A. Find a common denominator of 5 and 2.

18
B. Multiply by the reciprocal of —.
5
3
C. Multiply by the reciprocal of —.
2
3 8
D. Rename 3 — as 2 —.
5 5

6. An inequality is graphed on the number line below.

0 1 2 3 4 5 6 7 8 9 10 11 12

What is the least whole number value that is a solution of the inequality?

7. A company ordering parts receives a charge of $25 for shipping and handling
plus $20 per part. Which equation represents the cost c of ordering p parts?

F. c = 25 + 20p H. p = 25 + 20c

G. c = 20 + 25p I. p = 20 + 25c

8. Which property is illustrated by the statement below?

5(3 + 6) = 5(3) + 5(6)

A. Associative Property of Multiplication

B. Commutative Property of Multiplication

C. Commutative Property of Addition

D. Distributive Property

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 9. What is the value of the expression below?

46.8 ÷ 0.156

10. In a fish tank, 75% of the fish are goldfish. How many fish are in the tank if
there are 24 goldfish?

F. 6 H. 32

G. 18 I. 96

11. What are the coordinates of point P in the coordinate plane below?
y
4
3
2
1

Ź4 Ź3 Ź2 Ź1 1 2 3 4 x
Ź1
Ź2
P
Ź3
Ź4

A. (−3, −2) C. (−2, −3)

B. (3, −2) D. (−2, 3)

12. What is the first step in evaluating the expression below?

⋅
3 (5 + 2)2 ÷ 7

F. Multiply 3 and 5. H. Evaluate 52.

G. Add 5 and 2. I. Evaluate 22.

13. Jeff wants to save $4000 to buy a used car. He has already saved $850. He
plans to save an additional $150 each week.

Part A Write and solve an equation to represent the number of weeks

remaining until he can afford the car.
3
Jeff saves $150 per week by saving — of what he earns at his job each week.
4
He works 20 hours per week.

Part B Write an equation to represent the amount per hour that Jeff must
earn to save $150 per week. Explain your reasoning.

Part C What is the amount per hour that Jeff must earn? Show your work
and explain your reasoning.

Cumulative Assessment 351

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