This document provides examples of solving inequalities by multiplying or dividing each side by positive and negative numbers. It begins with 6 examples of solving inequalities such as x > 18, g ≤ -23, and t ≤ 19. Then it discusses the properties of inequalities when multiplying or dividing each side by positive and negative numbers. Specifically, it states that the inequality sign remains the same when multiplying by a positive number but must be reversed when multiplying by a negative number. Finally, it provides step-by-step examples of solving inequalities by multiplying and dividing each side by positive and negative numbers. This includes writing the inequality, simplifying by multiplying or dividing, and determining the solution set.

1. Solve each inequality.
1. x – 15 > 3
2. g + 3 ≤ –20
3. 11 ≥ t – 8
Solve each inequality. Graph the solution set on a number line.
4. 34 < 6 + d
5. 5 – f ≥ 10
6. Jonas has $75 to spend on new clothes. The pair of jeans he
wants costs $48.50. What is the most Jonas can spend on shirts?
Course 2, Lesson 6-7

2. ANSWERS
1. x > 18
2. g ≤ –23
3. t ≤ 19
4. d > 28;
5. f ≤ –5;
6. $26.50
Course 2, Lesson 6-7

3. WHAT does it mean to say
two quantities are equal?
Expressions and Equations
Course 2, Lesson 6-7

4. • 7.EE.4
Use variables to represent quantities in a real-world or mathematical
problem, and construct simple equations and inequalities to solve
problems by reasoning about the quantities.
• 7.EE.4b
Solve word problems leading to inequalities of the form px + q > r
or px + q < r, where p, q, and r are specific rational numbers. Graph
the solution set of the inequality and interpret it in the context of
the problem.
Course 2, Lesson 6-7 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and
Council of Chief State School Officers. All rights reserved.
Expressions and Equations

5. Mathematical Practices
1 Make sense of problems and persevere in solving them.
2 Reason abstractly and quantitatively.
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.
7 Look for and make use of structure.
Course 2, Lesson 6-7 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and
Council of Chief State School Officers. All rights reserved.
Expressions and Equations

6. • To solve inequalities when multiplying
or dividing by a positive number
• To solve inequalities when multiplying
or dividing by a negative number
Course 2, Lesson 6-7
Expressions and Equations

7. Course 2, Lesson 6-7
Expressions and Equations
• Multiplication Property of Inequality
• Division Property of Inequality

8. Course 2, Lesson 6-7
Expressions and Equations
Words The and the
state that an inequality
remains true when you multiply or divide each side of an
inequality by a positive number.
Symbols For all numbers a, b, and c, where c > 0,
1. if a > b, then ac > bc and
2. if a < b, then ac < bc and
These properties are also true for a ≥ b and a ≤ b.
Multiplication Property of Inequality
Division Property of Inequality
.
a b
c c

.
a b
c c


9. 1
Need Another Example?
2
3
4
Step-by-Step Example
1. Solve 8x ≤ 40.
Write the inequality.
The solution is x ≤ 5. You can check this solution by
substituting 5 or a number less than 5 into the inequality.
Simplify.
8x ≤ 40
Divide each side by 8.
x ≤ 5

10. Answer
Need Another Example?
Solve 5x > 30.
x > 6

11. Need Another Example?
Step-by-Step Example
2. Solve > 7.
1
2
3
4
Write the inequality.
The solution is d > 14. You can check this solution by
substituting a number greater than 14 into the inequality.
Simplify.
> 7
Multiply each side by 2.
d > 14
2 > 2(7)

12. Answer
Need Another Example?
Solve 3 ≥ .
12 ≥ h or h ≤ 12

13. Course 2, Lesson 6-7
Expressions and Equations
Words When you multiply or divide each side of an inequality by a
negative number, the inequality symbol must be reversed
for the inequality to remain true.
Symbols For all numbers a, b, and c, where c < 0,
1. if a > b, then ac < bc and
2. if a < b, then ac > bc and
Examples 7 > 1 –4 < 16
–2(7) < –2(1) Reverse the symbols >
–14 < –2 1 > –4
These properties are also true for a ≥ b and a ≤ b.
.
a b
c c

.
a b
c c

4
4

16
4

14. 1
Need Another Example?
2
3
4
Step-by-Step Example
3. Solve −2g < 10. Graph the solution set on a number line.
Write the inequality.
Simplify.
– 2g < 10
Divide each side by –2
and reverse the symbol.
g > –5

15. Answer
Need Another Example?
Solve –4x ≤ 4. Graph the solution set on a
number line.
x ≥ –1

16. 1
Need Another Example?
2
3
4
Step-by-Step Example
4. Solve ≤ 4. Graph the solution set on a number line.
Write the inequality.
Simplify.
≤ 4
Multiply each side by –3
and reverse the symbol.
x ≥ –12

17. Answer
Need Another Example?
Solve > – 5. Graph the solution set on a number line.
x < 20

18. 1
Need Another Example?
2
3
4
5
Step-by-Step Example
5. Ling earns $8 per hour working at the zoo. Write and solve an
inequality that can be used to find how many hours she must
work in a week to earn at least $120. Interpret the solution.
Write the inequality.
Simplify.
8x ≥ 120
Divide each side by 8.
x ≥ 15
Word
Variable
Inequality
Amount earned times number is at amount earned
per hour of hours least each week.
Let x represent the number of hours.
8 x ≥ 120
So, Ling must work at least 15 hours.

19. Answer
Need Another Example?
A plate weighs pound. A shelf can hold at most 20
pounds. Write and solve an inequality to find how
many plates the shelf can hold. Interpret the solution.
x ≤ 20; x ≤ 80; The shelf can hold at
most 80 plates.

20. How did what you learned
today help you answer the
WHAT does it mean to say
two quantities are equal?
Course 2 Lesson 6-7
Expressions and Equations

21. How did what you learned
today help you answer the
WHAT does it mean to say
two quantities are equal?
Course 2 Lesson 6-7
Expressions and Equations
Sample answers:
• To keep the same inequality sign when multiplying or
dividing each side of an inequality by a positive
number
• To reverse the inequality sign when multiplying or
dividing each side of an inequality by a negative
number

22. How did the previous lesson
on solving inequalities by
adding or subtracting help
you with today’s lesson?
Ratios and Proportional RelationshipsExpressions and Equations
Course 2 Lesson 6-7