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Inequalities and
Relationships
?
MODULE
13
LESSON 13.1
ESSENTIAL QUESTION
Writing Inequalities
How can you use inequalities
and relationships to solve
real-world problems?
6.9.A, 6.9.B, 6.10.B
LESSON 13.2
Addition and
Subtraction
Inequalities
6.9.B, 6.9.C, 6.10
LESSON 13.3
Multiplication and
Division Inequalities
with Positive Numbers
6.9.B, 6.9.C, 6.10
LESSON 13.4
Multiplication and
Division Inequalities
with Rational Numbers
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6.9.B, 6.10.A, 6.10.B
Real-World Video
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Some rides at amusement parks indicate a
minimum height required for riders. You can model
all the heights that are allowed to get on the ride
with an inequality.
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your
write-in student
edition, accessible on
any device.
Scan with your smart
phone to jump directly
to the online edition,
video tutor, and more.
Interactively explore
key concepts to see
how math works.
Get immediate
feedback and help as
you work through
practice sets.
345
Are YOU Ready?
Personal
Math Trainer
Complete these exercises to review skills you will need
for this chapter.
Understand Integers
EXAMPLE
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Online
Assessment and
Intervention
Decide whether the integer
is positive or negative:
into the ground → negative
Write the integer.
A water well was drilled
735 feet into the ground.
-735
Write an integer to represent each situation.
1. a loss of $75
2. a football player’s 3. spending $1,200 4. a climb of 2,400
gain of 9 yards
on a flat screen
feet
TV
Integer Operations
EXAMPLE
3 × 8 = 24
-30 ÷ (-5) = 6
The product or quotient of two integers is
positive if the signs of the integers are the same.
7 × (-4) = -28
-72 ÷ 9 = -8
The product or quotient of two integers is
negative if the signs of the integers are different.
Find the product or quotient.
9. 3 × (-7)
6. 15 ÷ (-5)
10. -64 ÷ 8
7. -8 × 6
11. -8 × (-2)
8. -100 ÷ (10)
12. 32 ÷ 2
Solve Multiplication Equations
EXAMPLE
3
_
h = 15
4
4 _
_
· 3 h = 15 · _43
3 4
·4
_____
h = 15
3
h = 20
Write the equation.
3
. Multiply both sides by the
h is multiplied by __
4
4
__
reciprocal, 3 , to isolate the variable.
Simplify.
Solve.
13. 9p = 108
346
Unit 4
14. _35 n = 21
15. _47 k = 84
3
16. __
e = 24
20
© Houghton Mifflin Harcourt Publishing Company
5. 6 × 9
Reading Start-Up
Visualize Vocabulary
Use the ✔ words to complete the graphic.
>, <
3x - 5
4x + 4 = 12; x = 2
Evaluating
Expressions
6×4
Vocabulary
Review Words
✔ algebraic expression
(expresión algebraica)
evaluating (evaluar)
✔ greater than (mayor que)
✔ less than (menor que)
like terms (términos
semejantes)
✔ numerical expression
(expresión numérica)
properties of operations
(propiedades de las
operaciones)
✔ solution (solución)
term (término, en una
expresión)
Preview Words
Understand Vocabulary
© Houghton Mifflin Harcourt Publishing Company
Match the term on the left to the correct expression on the right.
1. solution of
an inequality
A. A value or values that make the
inequality true.
2. coefficient
B. A specific number whose value
does not change.
3. constant
C. The number that is multiplied by the
variable in an algebraic expression.
coefficient (coeficiente)
constant (constante)
solution of an inequality
(solución de una
desigualdad)
variable (variable)
Active Reading
Two-Panel Flip Chart Create a two-panel
flip chart to help you understand the
concepts in this module. Label one flap
“Adding and Subtracting Inequalities.” Label
the other flap “Multiplying and Dividing
Inequalities.” As you study each lesson, write
important ideas under the appropriate flap.
Module 13
347
MODULE 13
Unpacking the TEKS
Understanding the TEKS and the vocabulary terms in the TEKS
will help you know exactly what you are expected to learn in this
module.
Represent solutions for onevariable, one-step equations and
inequalities on number lines.
What It Means to You
You will learn to graph the solution of an
inequality on a number line.
Key Vocabulary
UNPACKING EXAMPLE 6.9.B
equation (ecuación)
A mathematical sentence that
shows that two expressions are
equivalent.
The temperature in a walk-in freezer must
stay under 5 °C. Write and graph an
inequality to represent this situation.
inequality (desigualdad)
A mathematical sentence that
shows the relationship between
quantities that are not equal.
solution of an
inequality (solución de una
desigualdad)
A value or values that make
the inequality true.
Write the inequality.
Let t represent the temperature in the freezer.
The temperature must be less than 5 °C.
6.10.A
Model and solve one-variable,
one-step equations and
inequalities that represent
problems, including geometric
concepts.
t<5
Graph the inequality.
0
5
10
What It Means to You
You can model and solve a one-variable, one-step inequality.
UNPACKING EXAMPLE 6.10.A
Donny buys 3 binders and spends more than $9. How much
did he spend on each binder?
Let x represent the cost of one binder.
Number of binders · Cost of a binder > Total cost of binders
x
>
Use algebra tiles to model 3x > 9
and solve the inequality.
x>3
Donny spent more than $3 on
each binder.
+
+
+
3
Visit my.hrw.com
to see all
the
unpacked.
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348
Unit 4
·
9
>
+ + +
+ + +
+ + +
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
Image Source/Corbis
6.9.B
LESSON
13.1 Writing Inequalities
?
ESSENTIAL QUESTION
Expressions,
equations, and
relationships—
6.9.A Write …
inequalities to represent
constraints or conditions
within problems.
Also 6.9.B, 6.10.B.
How can you use inequalities to represent real-world constraints
or conditions?
6.9.A
EXPLORE ACTIVITY
Using Inequalities to Describe Quantities
You can use inequality symbols with variables to describe quantities that can
have many values.
Symbol
Meaning
Word Phrases
<
Is less than
Fewer than, below
>
Is greater than
More than, above
≤
Is less than or equal to
At most, no more than
≥
Is greater than or equal to
At least, no less than
A The lowest temperature ever recorded in Florida
was -2 °F. Graph this temperature on the number line.
-8 -7 -6 -5 -4 -3 -2 -1
0 1 2 3 4 5 6 7 8
© Houghton Mifflin Harcourt Publishing Company
B The temperatures 0 °F, 3 °F, 6 °F, 5 °F, and -1 °F have also been
recorded in Florida. Graph these temperatures on the number line.
C How do the temperatures in B compare to -2? How can you see this
relationship on the number line?
D How many other numbers have the same relationship to -2 as the
temperatures in B ? Give some examples.
E Suppose you could graph all of the possible answers to
line. What would the graph look like?
F Let x represent all the possible answers to
Complete this inequality: x
D
D
on a number
.
-2
Lesson 13.1
349
Graphing the Solutions
of an Inequality
Math On the Spot
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A solution of an inequality that contains a variable is any value of the
variable that makes the inequality true. For example, 7 is a solution of
x > -2, since 7 > -2 is a true statement.
EXAMPLE 1
6.9.B
Graph the solutions of each inequality. Check the solutions.
A y ≤ -3
Math Talk
STEP 1
Draw a solid circle at -3 to show that -3 is a solution.
STEP 2
Shade the number line to the left of -3
to show that numbers less than -3 are
solutions.
Mathematical Processes
Is -4 _14 a solution of
y ≤ -3? Is -5.6?
-5 -4 -3 -2 -1
STEP 3
Use a solid circle
for an inequality
that uses ≥ or ≤.
0 1 2 3 4 5
Check your solution.
Choose a number that is on the shaded section of the number
line, such as -4. Substitute -4 for y.
-4 ≤ -3
-4 is less than -3, so -4 is a solution.
B 1<m
STEP 1
Draw an empty circle at 1 to show that 1 is not a solution.
STEP 2
Shade the number line to the right of
1 to show that numbers greater than 1
are solutions.
STEP 3
0 1 2 3 4 5
Check your answer.
Substitute 2 for m.
1<2
1 is less than 2, so 2 is a solution.
Reflect
1.
350
Unit 4
How is x < 5 different from x ≤ 5?
© Houghton Mifflin Harcourt Publishing Company
-5 -4 -3 -2 -1
Use an open circle for
an inequality that
uses > or <.
YOUR TURN
2.
Graph the solution of the inequality t ≤ -4.
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0 1 2 3 4 5 6 7 8 9 10
Personal
Math Trainer
Online Assessment
and Intervention
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Writing Inequalities
You can write an inequality to model the relationship between an algebraic
expression and a number. You can also write inequalities to represent certain
real-world situations.
Math On the Spot
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EXAMPL 2
EXAMPLE
6.9.A, 6.10.B
Write an inequality that represents the phrase the sum of y and 2 is greater
than 5. Draw a graph to represent the inequality.
STEP 1
Write the inequality.
Animated
Math
The sum of y and 2 is greater than 5.
y+2
STEP 2
>
Graph the solution.
For y + 2 to have a value greater than 5,
y must be a number greater than 3.
-5 -4 -3 -2 -1
© Houghton Mifflin Harcourt Publishing Company
STEP 3
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5
Use an open circle
at 3 and shade to
the right of 3.
0 1 2 3 4 5
Check your solution by substituting a number greater than 3,
such as 4, into the original inequality.
4+2>5
6>5
Substitute 4 for y.
6 is greater than 5, so 4 is a solution.
B To test the temperature rating of a coat, a scientist keeps the temperature
below 5 °C. Write and graph an inequality to represent this situation.
STEP 1
Write the inequality. Let t represent the temperature in the lab.
t<5
STEP 2
The temperature must be less than 5 °C.
Graph the inequality.
0 1 2 3 4 5 6 7 8 9 10
Lesson 13.1
351
YOUR TURN
Personal
Math Trainer
3.
Write an inequality that represents the phrase the sum of 1 and y is greater
than or equal to 3 . Check to see if y = 1 is a solution.
Online Assessment
and Intervention
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Write and graph an inequality to represent each situation.
4.
The highest temperature in February was 6 °F.
0 1 2 3 4 5 6 7 8 9 10 11 12
5.
Each package must weigh more than 2 ounces.
-2 -1
0 1 2 3 4 5 6 7 8 9 10 11 12
Guided Practice
-5 -4 -3 -2 -1
0 1 2 3 4 5
2. Graph -3 > z. Check the graph using substitution.
(Example 1)
-5 -4 -3 -2 -1
0 1 2 3 4 5
3. Write an inequality that represents the phrase “the sum
of 4 and x is less than 6.” Draw a graph that represents
the inequality, and check your solution. (Example 2)
-5 -4 -3 -2 -1
0 1 2 3 4 5
4. During hibernation, a garter snake’s body temperature
never goes below 3 °C. Write and graph an inequality
that represents this situation. (Example 2)
-5 -4 -3 -2 -1
0 1 2 3 4 5
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ESSENTIAL QUESTION CHECK-IN
5. Write an inequality to represent this situation: Nina wants to take at least
$15 to the movies. How did you decide which inequality symbol to use?
352
Unit 4
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1. Graph 1 ≤ x. Use the graph to determine which of these numbers are
solutions of the inequality: -1, 3, 0, 1 (Explore Activity and Example 1)
Name
Class
Date
13.1 Independent Practice
Personal
Math Trainer
6.9.A, 6.9.B, 6.10.B
my.hrw.com
Online
Assessment and
Intervention
6. Which of the following numbers are solutions to x ≥ 0?
-5, 0.03, -1, 0, 1.5, -6, _12
Graph each inequality.
7. t ≤ 8
8. -7 < h
9. x ≥ -9
10. n > 2.5
11. -4 _12 >x
-2 -1
0 1 2 3 4 5 6 7 8 9 10 11 12
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0
1
2
3
4
- 12 - 11 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0
1
2
-5
-4
-3
-2
-1
0
1
2
3
4
5
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
Write an inequality that matches the number line model.
12.
© Houghton Mifflin Harcourt Publishing Company
13.
14.
15.
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0 1 2 3 4 5 6 7 8 9 10
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0 1 2 3 4 5 6 7 8 9 10
-5
-4
-3
-2
-1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
16. A child must be at least 48 inches tall to ride a roller coaster.
a. Write and graph an inequality
to represent this situation.
38 40 42 44 46 48 50 52 54 56 58
b. Can a child who is 46 inches tall ride the roller coaster? Explain.
Lesson 13.1
353
Write and graph an inequality to represent each situation.
17. The stock is worth at least $14.50.
10
11
12
13
14
15
16
17
18
19
20
0
1
2
3
4
5
18. The temperature is less than 3.5 °F.
-5
-4
-3
-2
-1
19. The goal of the fundraiser is to make more than $150.
0
50 100 150 200 250 300
Work Area
FOCUS ON HIGHER ORDER THINKING
20. Communicate Mathematical Ideas Explain how to graph
the inequality 8 ≥ y.
21. Represent Real-World Problems The number line shows an inequality.
Describe a real-world situation that the inequality could represent.
1
2
3
4
5
22. Critique Reasoning Natasha is trying to represent the following
situation with a number line model: There are fewer than 5 students in
the cafeteria. She has come up with two possible representations, shown
below. Which is the better representation, and why?
354
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0 1 2 3 4 5 6 7 8 9 10
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0 1 2 3 4 5 6 7 8 9 10
Unit 4
© Houghton Mifflin Harcourt Publishing Company
0
LESSON
13.2
?
Addition and
Subtraction Inequalities
Expressions,
equations, and
relationships—6.10.A
Model and solve one-variable,
one-step… inequalities that
represent problems. Also
6.9.B, 6.9.C, 6.10.B.
ESSENTIAL QUESTION
How can you solve an inequality involving addition
or subtraction?
6.10.A
EXPLORE ACTIVITY
Modeling One-Step Inequalities
You can use algebra tiles to model an inequality involving addition.
On a day in January in Watertown, NY, the temperature was 5 °F
at dawn. By noon it was at least 8 °F. By how many degrees did the
temperature increase?
A Let x represent the increase in temperature. Write an inequality.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Janusz Wrobel/
Alamy
Temperature
at dawn
+
Increase in
temperature
+
B The model shows 5 + x ≥ 8.
How many tiles must you remove
from each side to isolate x on
one side of the inequality?
≥
8
≥
8
+ + +
+ +
5
Circle these tiles.
+
+
≥
+ + + +
+ + + +
x
≥
8
C What values of x make this inequality true? Graph the solution of the
inequality on the number line.
x≥
-5 -4 -3 -2 -1
0 1 2 3 4 5
Reflect
1.
Analyze Relationships How is solving the inequality
5 + x ≥ 8 like solving the equation 5 + x = 8? How is it different?
Math Talk
Mathematical Processes
Could the temperature have
increased by 2 degrees
by noon? Could it have
increased by 5 degrees?
Explain.
Lesson 13.2
355
Using Properties of Inequalities
Addition and Subtraction Properties of Inequality
Math On the Spot
Addition Property of Inequality
Subtraction Property of Inequality
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You can add the same number to You can subtract the same number
both sides of an inequality and the from both sides of an inequality
inequality will remain true.
and the inequality will remain true.
EXAMPLE 1
6.9.B, 6.10.B
Solve each inequality. Graph and check the solution.
A x + 5 < -12
STEP 1
Solve the inequality.
x + 5 < -12
5 ____
-5
____
x < -17
STEP 2
Use the Subtraction Property of Inequality.
Subtract 5 from both sides.
Graph the solution.
-20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10
STEP 3
Math Talk
Mathematical Processes
-13 < -12
B 8≤ y-3
STEP 1
Solve the inequality.
8 ≤y - 3
+
3
+
3
_
_
11 ≤ y
356
Unit 4
The inequality is true.
Use the Addition Property of Inequality.
Add 3 to both sides.
You can rewrite 11 ≤ y as y ≥ 11.
STEP 2
Graph the solution.
STEP 3
Check the solution. Substitute a solution from the shaded
part of your number line into the original inequality.
5 6 7 8 9 10 11 12 13 14 15
?
8 ≤ 12 - 3
Substitute 12 for y in 8 ≤ y - 3
8≤ 9
The inequality is true.
© Houghton Mifflin Harcourt Publishing Company
What would it tell you if the
inequality is false when you
check the solution?
Check the solution. Substitute a solution from the shaded
part of your number line into the original inequality.
?
-18 + 5 < -12 Substitute -18 for x into x + 5 < -12
YOUR TURN
Solve each inequality. Graph and check the solution.
2. y - 5 ≥ -7
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Math Trainer
3. 21 > 12 + x
Online Assessment
and Intervention
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-5 -4 -3 -2 -1
0 1 2
3 4 5
0 1 2 3 4 5 6 7
8 9 10
Interpreting Inequalities as Comparisons
You can write a real-world problem for a given inequality. Examine each
number and mathematical operation in the inequality.
EXAMPL 2
EXAMPLE
Math On the Spot
6.9.C
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Write a real-world problem for the inequality 60 ≥ w + 5.
Then solve the inequality.
STEP 1
Examine each part of the inequality.
w is the unknown quantity.
5 is added to w.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Lew
Robertson/Corbis
60 is greater than or equal to a number added to 5.
STEP 2
Write a comparison that the inequality could describe. June’s
dog will travel to a dog show in a pet carrier. The pet carrier
weighs 5 pounds. The total weight of the pet carrier and the
dog must be no more than 60 pounds. What inequality describes
the weight of June’s dog?
STEP 3
Solve the inequality.
60 ≥ w + 5
-5
____
55 ≥ w
-5
____
June’s dog currently weighs ≤ 55 pounds.
Reflect
4. If you were to graph the solution, would all points on the graph make
sense for the situation?
Lesson 13.2
357
YOUR TURN
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Math Trainer
5. Write a real-world problem that can be modeled by x - 13 > 20. Solve
your problem and tell what values make sense for the situation.
Online Assessment
and Intervention
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Guided Practice
1. Write the inequality shown on the model. Circle the tiles you would
remove from each side and give the solution. (Explore Activity)
Inequality:
+ + +
Solution:
+
≤
+ + + + +
Solve each inequality. Graph and check the solution. (Example 1)
2. x + 4 ≥ 9
0 1 2 3 4 5 6 7
3. 5 > z - 3
8 9 10
4. t + 5 > 12
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7
8 9 10
5. y - 4 < 2
0 1 2 3 4 5 6 7 8 9 10
?
?
ESSENTIAL QUESTION CHECK-IN
7. Explain how to solve 7 + x ≥ 12. Tell what property of inequality you
would use.
358
Unit 4
© Houghton Mifflin Harcourt Publishing Company
6. Write a real-world problem that can be represented by the inequality
y - 4 < 2. Solve the inequality and tell whether all values in the solution
make sense for the situation. (Example 2)
Name
Class
Date
13.2 Independent Practice
Personal
Math Trainer
6.9.B, 6.9.C, 6.10.A, 6.10.B
my.hrw.com
Online
Assessment and
Intervention
Solve each inequality. Graph and check the solution.
8. x - 35 > 15
0 10 20 30 40 50 60 70 80 90 100
9. 193 + y ≥ 201
0 1 2 3 4 5 6 7
8 9 10
10. y - 5 ≥ -15
- 12 - 11 - 10 - 9
-8
-7
-6
-5
-4
-3
-2
- 15 - 14 - 13 - 12 - 11 - 10 - 9
-8
-7
-6
-5
11. 15 ≥ z + 26
Write an inequality to solve each problem.
© Houghton Mifflin Harcourt Publishing Company
12. The water level in the aquarium’s shark tank is always greater than 25 feet.
If the water level decreased by 6 feet during cleaning, what was the water
level before the cleaners took out any water?
13. Danny has at least $15 more than his big brother. Danny’s big brother has
$72. How much money does Danny have?
14. The vet says that Ray’s puppy will grow to be at most 28 inches tall. Ray’s
puppy is currently 1 foot tall. How much more will the puppy grow?
15. Pierre’s parents ordered some pizzas for a party. 4.5 pizzas were eaten at
the party. There were at least 5_12 whole pizzas left over. How many pizzas
did Pierre’s parents order?
16. To get a free meal at his favorite restaurant, Tom needs to spend $50 or
more at the restaurant. He has already spent $30.25. How much more
does Tom need to spent to get his free meal?
Lesson 13.2
359
17. Multistep The table shows Marco’s checking account
activity for the first week of June.
a. Marco wants his total deposits for the month of June
to exceed $1,500. Write and solve an inequality to find
how much more he needs to deposit to meet this goal.
Deposit – Paycheck
$520.45
Purchase – Grocery Store
$46.50
Purchase – Movie Theatre
$24.00
Purchase – Water bill
$22.82
b. Marco wants his total purchases for the month to be less than $450.
Write and solve an inequality to find how much more he can spend
and still meet this goal.
c. There are three weeks left in June. If Marco spends the same amount in
each of these weeks that he spent during the first week, will he meet his
goal of spending less than $450 for the entire month? Justify your answer.
FOCUS ON HIGHER ORDER THINKING
Work Area
19. Critical Thinking José solved the inequality 3 > x + 4 and got x < 1.
Then, to check his solution, he substituted -2 into the original inequality
to check his solution. Since his check worked, he believes that his answer
is correct. Describe another check José could perform that will show his
solution is not correct. Then explain how to solve the inequality.
20. Look for a Pattern Solve x + 1 > 10, x + 11 > 20, and x + 21 > 30.
Describe a pattern. Then use the pattern to predict the solution of
x + 9,991 > 10,000.
360
Unit 4
© Houghton Mifflin Harcourt Publishing Company
18. Critique Reasoning Kim solved y - 8 ≤ 10 and got y ≤ 2. What might
Kim have done wrong?
LESSON
13.3
?
Multiplication and
Division Inequalities
with Positive Numbers
ESSENTIAL QUESTION
Expressions,
equations, and
relationships—6.10.A
Model and solve one-variable,
one-step inequalities that
represent problems. Also
6.9.B, 6.9.C, 6.10.B.
How can you solve an inequality involving multiplication
or division with positive numbers?
6.10.A
EXPLORE ACTIVITY
Modeling One-Step Inequalities
You can use algebra tiles to solve inequalities that involve multiplying
positive numbers.
Dominic is buying school supplies. He buys 3 binders and spends
more than $9. How much did he spend on each binder?
A Let x represent the cost of one binder. Write an inequality.
Number of binders
•
Cost of a binder
•
B The model shows the inequality from
There are
9
>
9
.
+ + +
x-tiles, so draw circles
to separate the tiles into
© Houghton Mifflin Harcourt Publishing Company
A
>
>
equal groups.
+ + +
+ + +
+ + +
How many units are in each group?
C What values make the inequality you wrote in
A
-5 -4 - 3 - 2 - 1
0 1 2 3 4 5
true? Graph the solution of the inequality.
Reflect
1. Analyze Relationships Is 3.25 a solution of the inequality you wrote
in A ? If so, does that solution make sense for the situation?
2. Represent Real-World Problems Rewrite the situation in
represent the inequality 3x < 9.
A
to
Lesson 13.3
361
Solving Inequalities Involving
Multiplication and Division
Math On the Spot
You can use properties of inequality to solve inequalities involving
multiplication and division with positive integers.
my.hrw.com
Multiplication and Division Properties of Inequality
• You can multiply both sides of an inequality by the same positive
number and the inequality will remain true.
• You can divide both sides of an inequality by the same positive
number and the inequality will remain true.
EXAMPLE 1
6.9.B, 6.10.B
Solve each inequality. Graph and check the solution.
A 12x < 24
Solve the inequality.
12x __
___
< 24
12
12
Math Talk
Mathematical Processes
Are all negative
numbers solutions to
12x < 24? Explain.
Divide both sides by 12.
x<2
STEP 2
Graph the solution.
-5 -4 - 3 - 2 - 1
STEP 3
Use an open circle to
show that 2 is not a
solution.
0 1 2 3 4 5
Check the solution by substituting a solution from the shaded
part of the graph into the original inequality.
?
Substitute 0 for x in the original inequality.
12(0) < 24
0 < 24
The inequality is true.
y
B _3 ≥ 5
STEP 1
Solve the inequality.
y
3 ( _3 ) ≥ 3(5)
Multiply both sides by 3.
y ≥ 15
STEP 2
Use a closed circle
to show that 15 is
a solution.
Graph the solution.
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
STEP 3
Check the solution by substituting a solution from the shaded
part of the graph into the original inequality.
18 ?
__
≥5
Substitute 18 for x in the original inequality.
3
6≥5
362
Unit 4
The inequality is true.
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STEP 1
YOUR TURN
Solve each inequality. Graph and check the solution.
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3. 5x ≥ 100
15 16 17 18 19 20 21 22 23 24 25
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4. _4z < 11
40 41 42 43 44 45 46 47 48 49 50
Solving Real-World Problems
You can use multiplication and division inequalities to model and solve
real-world problems.
EXAMPL 2
EXAMPLE
Problem
Solving
Math On the Spot
6.10.A
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Cy is making a square flag. He wants the perimeter to be at least 22 inches.
Write and solve an inequality to find the possible side lengths.
Analyze Information
Find the possible lengths of 1 side of a square that has a perimeter
of at least 22 inches.
Formulate a Plan
Write and solve a multiplication inequality. Use the fact that the
perimeter of a square is 4 times its side length.
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Justify and Evaluate
Solve
4x ≥ 22
Let x represent a side length.
4x __
__
≥ 22
4
4
Divide both sides by 4.
x ≥ 5.5
The side lengths must be greater than or equal to 5.5 in.
Cy’s flag should have a side length of 5.5 inches or more.
Justify and Evaluate
Check the solution by substituting a value in the solution set
in the original inequality. Try x = 6.
?
4(6) ≥ 22
24 ≥ 22
Substitute 6 for x.
The statement is true.
Cy’s flag could have a side length of 6 inches.
Lesson 13.3
363
Reflect
5. Represent Real-World Problems Write and solve a real-world problem
for the inequality 4x ≤ 60.
YOUR TURN
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6. A paperweight must weigh less than 4 ounces. Brittany wants to make
6 paperweights using sand. Write and solve an inequality to find the
possible weight of the sand she needs.
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Guided Practice
1. Write the inequality shown on the model. Circle groups of tiles
to show the solution. Then write the solution. (Explore Activity)
Inequality:
+ +
<
+ + + +
+ + + +
Solution:
Solve each inequality. Graph and check the solution. (Example 1)
35 36 37 38 39 40 41 42 43 44 45
3. _3r ≥ 11
30 31 32 33 34 35 36 37 38 39 40
4. Karen divided her books and put them on 6 shelves. There were at least
14 books on each shelf. How many books did she have? Write and solve
an inequality to represent this situation. (Example 2)
?
?
ESSENTIAL QUESTION CHECK-IN
5. Explain how to solve and check the solution to 5x < 40 using properties
of inequalities.
364
Unit 4
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2. 8y < 320
Name
Class
Date
13.3 Independent Practice
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6.9.B, 6.9.C, 6.10.A, 6.10.B
Write and solve an inequality for each
problem.
6. Geometry The perimeter of
a regular hexagon is at most
42 inches. Find the possible
side lengths of the hexagon.
7. Tamar needs to make at least $84 at work
on Tuesday to afford dinner and a movie
on Wednesday night. She makes $14 an
hour at her job. How many hours does she
need to work on Tuesday?
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8. In a litter of 7 kittens, each kitten weighs
more than 3.5 ounces. Find the possible
total weight of the litter.
9. To cover his rectangular backyard, Will
needs at least 170.5 square feet of sod. The
length of Will’s yard is 15.5 feet. What are
the possible widths of Will’s yard?
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12. Steve pays less than $32 per day to rent his
apartment. August has 31 days. What are
the possible amounts Steve could pay for
rent in August?
13. If you were to graph the solution for
exercise 12, would all points on the graph
make sense for the situation? Explain.
14. Multistep Lina bought 4 smoothies at a
health food store. The bill was less than $16.
a. Write and solve an inequality to
represent the cost of each smoothie.
b. What values make sense for this
situation? Explain.
c. Graph the values that make sense for
this situation on the number line.
Solve each inequality. Graph and check the
solution.
0 1 2 3 4 5
Solve each inequality.
10. 10x ≤ 60
0 1 2 3 4 5 6 7 8 9 10
p
15. __
≤ 30
13
16. 2t > 324
11. _2t > 0
-5 -4 - 3 - 2 - 1
-1
0 1 2 3 4 5
17. 12y ≥ 1
x
18. ___
< 11
9.5
Lesson 13.3
365
The sign shows some prices at a produce stand.
19. Tom has $10. What is the greatest amount of spinach
he can buy?
Price per Pound
Produce
$1.25
Onions
$0.99
Yellow Squash
$3.00
Spinach
$0.50
Potatoes
20. Gary has enough money to buy at most 5.5 pounds
of potatoes. How much money does Gary have?
21. Florence wants to spend no more than $3 on onions. Will she be able to
buy 2.5 pounds of onions? Explain.
22. The produce buyer for a local restaurant wants to buy more than 30 lb
of onions. The produce buyer at a local hotel buys exactly 12 pounds of
spinach. Who spends more at the produce stand? Explain.
FOCUS ON HIGHER ORDER THINKING
Work Area
24. Represent Real-World Problems Write and solve a word problem that
can be represented with 240 ≤ 2x.
25. Persevere in Problem Solving A rectangular prism has a length of
13 inches and a width of _12 inch. The volume of the prism is at most
65 cubic inches. Find all possible heights of the prism. Show your work.
366
Unit 4
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2
23. Critique Reasoning A student solves _5r ≤ _25 and gets r ≤ __
. What is the
25
correct solution? What mistake might the student have made?
LESSON
13.4
?
Multiplication and
Division Inequalities
with Rational Numbers
ESSENTIAL QUESTION
Expressions,
equations, and
relationships—6.9.B
Represent solutions for
one-step inequalities on
number lines. Also 6.10.A,
6.10.B
How do you solve inequalities that involve multiplication and
division of integers?
EXPLORE ACTIVITY
6.10.A
Investigating Inequality Symbols
You have seen that multiplying or dividing both sides of an inequality by the
same positive number results in an equivalent inequality. How does multiplying
or dividing both sides by the same negative number affect an inequality?
A Complete the tables.
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Inequality
Multiply each
side by:
3<4
2
2 ≥ -3
3
5>2
-1
-8 > -10
-8
Inequality
Divide each
side by:
4<8
4
12 ≥ -15
New inequality
New inequality is
true or false?
New inequality
New inequality is
true or false?
3
-16 ≤ 12
-4
15 > 5
-5
B What do you notice when you multiply or divide both sides of an
inequality by the same negative number?
C How could you make each of the multiplication and division
inequalities that were not true into true statements?
Lesson 13.4
367
Multiplication and Division
Properties of Inequality
Math On the Spot
Recall that you can multiply or divide both sides of an inequality by the same
positive number, and the statement will still be true.
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Multiplication and Division Properties of Inequality
• If you multiply or divide both sides of an inequality by the same
negative number, you must reverse the inequality symbol for the
statement to still be true.
EXAMPLE 1
6.9.B, 6.10.B
Solve each inequality. Graph and check the solution.
My Notes
A -4x > 52
STEP 1
Solve the inequality.
-4x > 52
Divide both sides by -4.
Reverse the inequality symbol.
-4x
52
____
< ___
-4
-4
x < -13
STEP 2
Graph the solution.
STEP 3
Check your answer using substitution.
?
Substitute -15 for x in -4x > 52.
-4(-15) > 52
-8
The statement is true.
y
B - _3 < -5
STEP 1
Solve the inequality.
y
- _3 < -5
y
-3(- _3 ) > -3(-5)
Multiply both sides by -3.
Reverse the inequality symbol.
y > 15
STEP 2
Graph the solution.
10 11 12 13 14 15 16 17 18 19 20
STEP 3
Check your answer using substitution.
?
18 <
- ___
-5
3
-6 < -5
368
Unit 4
y
Substitute 18 for y in - __ < -5.
3
The inequality is true.
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60 > 52
-15 -14 -13 -12 -11 -10 -9
YOUR TURN
Solve each inequality. Graph and check the solution.
1. -10y < 60
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
2. 7 ≥ - __t
6
0 1
-47 -46 -45 -44 -43 -42 -41 -40
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Solving a Real-World Problem
Although elevations below sea level are represented by negative numbers, we
often use absolute value to describe these elevations. For example, -50 feet
relative to sea level might be described as 50 feet below sea level.
EXAMPL 2
EXAMPLE
Problem
Solving
Math On the Spot
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6.10.A
A marine submersible descends more than 40 feet below sea level. As it
descends from sea level, the change in elevation is -5 feet per second. For
how many seconds does it descend?
Analyze Information
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Rotman/Peter Arnold Inc/Getty Images
Rewrite the question as a statement.
• Find the number of seconds that the submersible decends below
sea level.
List the important information:
• The final elevation is greater than 40 feet below sea level or < -40 feet.
• The rate of descent is -5 feet per second.
Formulate a Plan
Write and solve an inequality. Use this fact:
Rate of change in elevation × Time in seconds = Total change in elevation
Justify and Evaluate
Solve
-5t < -40
-5t > ____
-40
____
-5
-5
t> 8
Rate of change × Time < Maximum elevation
Divide both sides by -5. Reverse the inequality symbol.
The submersible descends for more than 8 seconds.
Justify and Evaluate
Check your answer by substituting a value greater than 8 seconds in the
original inequality.
?
Substitute 9 for t in the inequality -5t < -40.
-5(9) < -40
-45 < -40
The statement is true.
Lesson 13.4
369
YOUR TURN
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3. Every month, $35 is withdrawn from Tom’s savings account to pay for
his gym membership. He has enough savings to withdraw no more than
$315. For how many months can Tony pay for his gym membership?
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Guided Practice
Solve each inequality. Graph and check the solution.
(Explore Activity and Example 1)
1. -7z ≥ 21
2. -__t > 5
4
-10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
-50 -40 -30 -20 -10
3. 11x < -66
t >5
4. -___
10
0
10
20
30
-8 -7 - 6 - 5 - 4 - 3 - 2 - 1
40
0
50
0 1 2
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10
0
5. For a scientific experiment, a physicist must make sure that the
temperature of a metal does not get colder than -80 °C. The metal
begins the experiment at 0 °C and is cooled at a steady rate of -4 °C per
hour. How long can the experiment run? (Example 2)
b. Solve the inequality in part a. How long will it take the physicist
to change the temperature of the metal?
c. The physicist has to repeat the experiment if the metal gets
cooler than -80 °C. How many hours would the physicist have
to cool the metal for this to happen?
?
?
ESSENTIAL QUESTION CHECK-IN
6. Suppose you are solving an inequality. Under what circumstances do
you reverse the inequality symbol?
370
Unit 4
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a. Let t represent time in hours. Write an inequality. Use the
fact that the rate of change in temperature times the
number of seconds equals the final temperature.
Name
Class
Date
13.4 Independent Practice
6.9.B, 6.10.A, 6.10.B
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Solve each inequality. Graph and check your
solution.
q
7. - __ ≥ -1
7
0 1 2 3 4 5 6 7 8 9 10
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14. A veterinarian tells Max that his cat
should lose no more than 30 ounces. The
veterinarian suggests that the cat should
lose 7 ounces or less per week. What is the
shortest time in weeks and days it would
take Max’s cat to lose the 30 ounces?
8. -12x < 60
-10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0
y
9. 0.5 ≤ __
8
0 1 2 3 4 5 6 7 8 9 10
10. 36 < -6r
-10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0
15. The elevation of an underwater cave is
-120 feet relative to sea level. A submarine
descends to the cave. The submarine’s rate
of change in elevation is no greater than
-12 feet per second. How long will it take
to reach the cave?
11. -12 > 2x
-8 -7 - 6 - 5 - 4 - 3 - 2 - 1
0 1 2
x ≤ -0.5
12. - __
6
© Houghton Mifflin Harcourt Publishing Company
-5 -4 - 3 - 2 - 1
0 1 2 3 4 5
13. Multistep Parav is playing a game in
which he flips a counter that can land on
either a -6 or a 6. He adds the point values
of all the flips to find his total score. To win,
he needs to get a score less than -48.
a. Assuming Parav only gets -6s when
he flips the counter, how many times
does he have to flip the counter?
16. The temperature of a freezer is never
greater than -2 °C. Yesterday the
temperature was -10 °C, but it increased
at a steady rate of 1.5 °C per hour. How
long in hours and minutes did the
temperature increase inside the freezer?
17. Explain the Error A student's solution to
the inequality -6x > 42 was x > -7. What
error did the student make in the solution?
What is the correct answer?
b. Suppose Parav flips the counter and
gets five 6s and twelve -6s when he
plays the game. Does he win? Explain.
Lesson 13.4
371
Solve each inequality.
18. 18 ≤ -2x
x < - __
1
20. - __
8
2
1
22. 4x < __
5
x ≤ -23
19. - __
7
21. 0.4 < -x
x ≤ -30
23. - ___
0.8
24. Use the order of operations to simplify the left side of the
inequality below. What values of x make the inequality a true
statement? - _12 (32 + 7)x > 32
FOCUS ON HIGHER ORDER THINKING
Work Area
26. Communicate Mathematical Thinking Van thinks that the answer
to -3x < 12 is x < -4. How would you convince him that his answer
is incorrect?
372
Unit 4
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25. Counterexamples John says that if one side of an inequality is 0, you
don’t have to reverse the inequality symbol when you multiply or divide
both sides by a negative number. Find an inequality that you can use
to disprove John’s statement. Explain your thinking.
MODULE QUIZ
Ready
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13.1 Writing Inequalities
Online Assessment
and Intervention
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Write an inequality to represent each situation, then graph the solutions.
1. There are fewer than 8 gallons of gas in the tank.
0 1 2 3 4 5 6 7 8 9 10
2. There are at least 3 pieces of gum left in the pack.
0 1 2 3 4 5 6 7 8 9 10
3. The valley was at least 4 feet below sea level.
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0 1 2 3 4 5 6 7 8 9 10
13.2 Addition and Subtraction Inequalities
Solve each inequality. Graph the solution.
4. c - 28 > -32
-10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
5.
0
v + 17 ≤ 20
0 1 2 3 4 5 6 7 8 9 10
6. Today’s high temperature of 80 °F is at least 16 ° warmer than yesterday’s high
temperature. What was yesterday’s high temperature?
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13.3, 13.4 Multiplication and Division Inequalities
Solve each inequality. Graph the solution.
8. __a2 < 4
7. 7f ≤ 35
0 1 2 3 4 5 6 7 8 9 10
k
<3
10. ___
-3
9. -25g ≥ 150
-10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0 1 2 3 4 5 6 7 8 9 10
0
-10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0
Module 13
373
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MODULE 13 MIXED REVIEW
Texas Test Prep
Selected Response
1. Em saves at least 20% of what she earns
each week. If she earns $140 each week
for 4 weeks, which inequality describes the
total amount she saves?
A t > 112
B t ≥ 112
C
t < 28
D t ≤ 28
2. Which number line represents the
inequality r > 6?
A
0 1 2 3 4 5 6 7 8 9 10
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5. The number line below represents the
solution to which inequality?
0 1 2 3 4 5 6 7 8 9 10
m
A __ > 2.2
4
C
B 2m < 17.6
D 5m > 40
m
__
> 2.5
3
6. Which number line shows the solution to
w - 2 ≤ 8?
A
0 1 2 3 4 5 6 7 8 9 10
B
0 1 2 3 4 5 6 7 8 9 10
C
0 1 2 3 4 5 6 7 8 9 10
B
0 1 2 3 4 5 6 7 8 9 10
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D
0 1 2 3 4 5 6 7 8 9 10
C
0 1 2 3 4 5 6 7 8 9 10
Gridded Response
0 1 2 3 4 5 6 7 8 9 10
3. For which inequality below is z = 3
a solution?
A z+5≥9
7. Hank needs to save at least $150 to ride
the bus to his grandparent’s home. If he
saves $12 a week, what is the least number
of weeks he needs to save?
B z+5>9
z+5≤8
0
0
0
0
0
0
D z+5<8
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
C
4. What is the solution to the inequality
−6x < −18?
A x>3
B x<3
C
x≥3
D x≤3
374
.
Unit 4
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D