3.5
Solving Inequalities Using
Multiplication or Division
Goal: Solve inequalities using multiplication or division.
Multiplication Property of Inequality
Words Multiplying each side of an inequality by a positive
number produces an equivalent inequality.
Multiplying each side of an inequality by a negative number and
reversing the direction of the inequality symbol produces an
equivalent inequality.
The multiplication
properties of
inequality are also
true for inequalities
involving >, ≤, and ≥.
Algebra If a < b and c > 0, then ac
bc.
If a < b and c < 0, then ac
bc.
Solving an Inequality Using Multiplication
Example 1
m
4
Solve > 2.
m
> 2
4
Write original inequality.
m
4
Multiply each side by
.
Reverse inequality symbol.
p2
Simplify.
m
Checkpoint Solve the inequality. Graph your solution.
t
5
b
8
1. < 3
9
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10
2. ≤ 1
11
12
13
14
15
16
9 8 7 6 5 4 3 2
Chapter 3 • Pre-Algebra Notetaking Guide
57
3.5
Solving Inequalities Using
Multiplication or Division
Goal: Solve inequalities using multiplication or division.
Multiplication Property of Inequality
Words Multiplying each side of an inequality by a positive
number produces an equivalent inequality.
Multiplying each side of an inequality by a negative number and
reversing the direction of the inequality symbol produces an
equivalent inequality.
The multiplication
properties of
inequality are also
true for inequalities
involving >, ≤, and ≥.
Algebra If a < b and c > 0, then ac < bc.
If a < b and c < 0, then ac > bc.
Solving an Inequality Using Multiplication
Example 1
m
4
Solve > 2.
m
> 2
4
4
m
4
<
m <
Write original inequality.
4 p 2
Multiply each side by 4 .
Reverse inequality symbol.
8
Simplify.
Checkpoint Solve the inequality. Graph your solution.
t
5
b
8
1. < 3
2. ≤ 1
b ≥ 8
t < 15
9
Copyright © Holt McDougal. All rights reserved.
10
11
12
13
14
15
16
9 8 7 6 5 4 3 2
Chapter 3 • Pre-Algebra Notetaking Guide
57
Division Property of Inequality
Words Dividing each side of an inequality by a positive number
produces an equivalent inequality.
Dividing each side of an inequality by a negative number and
reversing the direction of the inequality symbol produces an
equivalent inequality.
The division
properties of
inequality are also
true for inequalities
involving >, ≤, and ≥.
a
c
a
If a < b and c < 0, then c
Algebra If a < b and c > 0, then b
.
c
b
.
c
Solving an Inequality Using Division
Example 2
Solve 11t ≥ 33.
11t ≥ 33
11t
Write original inequality.
33
Divide each side by
.
Reverse inequality symbol.
Simplify.
t
Checkpoint Solve the inequality. Graph your solution.
4. 3x > 12
3. 4y ≤ 36
3
58
4
5
Chapter 3 • Pre-Algebra Notetaking Guide
6
7
8
9
10
9 8 7 6 5 4 3 2
Copyright © Holt McDougal. All rights reserved.
Division Property of Inequality
Words Dividing each side of an inequality by a positive number
produces an equivalent inequality.
Dividing each side of an inequality by a negative number and
reversing the direction of the inequality symbol produces an
equivalent inequality.
The division
properties of
inequality are also
true for inequalities
involving >, ≤, and ≥.
a
b
c
c
a
b
If a < b and c < 0, then > .
c
c
Algebra If a < b and c > 0, then < .
Solving an Inequality Using Division
Example 2
Solve 11t ≥ 33.
11t ≥ 33
11t
11
Write original inequality.
33
≤ 11
3
t ≤
Divide each side by 11 .
Reverse inequality symbol.
Simplify.
Checkpoint Solve the inequality. Graph your solution.
4. 3x > 12
3. 4y ≤ 36
x < 4
y≤9
3
58
4
5
Chapter 3 • Pre-Algebra Notetaking Guide
6
7
8
9
10
9 8 7 6 5 4 3 2
Copyright © Holt McDougal. All rights reserved.