Inequalities: Solve Using Multiplication and Division
Tool Box:
Equivalent Inequalities
Properties of Inequalities
Summary:
Question:
Solve an Inequality by Multiplying
𝟒 < 𝟗
(2)4 < 9(2)
8 < 18
𝟖 > 𝟓
(2)8 > 5(2)
16 > 10
Inequality
Multiply each side of the inequality by 2
Multiplying each side of an inequality by a
positive number, the inequality remains
true
𝟒 < 𝟗
𝟖 > 𝟓
(−2)4 < 9(−2)
(−2)8 > 5(−2)
−8 ≮ −18
−16 ≯ −10
*In order to make the inequality true, you must
flip the inequality symbol (the symbol must be
reversed)
−8 > −18
−16 > −10
Inequality
Multiply each side of the inequality by -2
Multiplying each side of an equality by a
negative number, the inequality becomes
false.
To make it true you must flip the inequality
symbol
Multiply by a Positive Number
𝒃
EXAMPLE
≥ 𝟐𝟓
𝟕
Solve for “b”
𝒃
𝟕
𝒃
𝟕
𝒃
(𝟕)
𝟕
1. Write the inequality
≥ 𝟐𝟓
2. Train track
3. Isolate the variable “b”
≥ 𝟐𝟓
4. Multiplication Property of Inequality
(multiply both sides of the inequality by 7)
5. Simplify
≥ 𝟐𝟓(𝟕)
𝑏 ≥ 175
Since the inequality was multiplied by a positive number, the inequality symbol
stays the same
Set Notation
𝑏 | 𝑏 ≥ 175
140
150
160
170 175 180
Multiply by a Negative Number
𝟐
EXAMPLE
− 𝟓 𝒑 < −𝟏𝟒
𝟐
−𝟓𝒑
< −𝟏𝟒
190
200
210
220
Solve for “p”
1. Write the inequality
230
2. Draw railroad tracks
3. Isolate the variable “p”
𝟐
− 𝒑 < −𝟏𝟒
𝟓
𝟓
𝟐
𝟓
(− ) − 𝒑 < −𝟏𝟒 (− )
𝟐
𝟓
𝟐
𝑝
>
35
Since the inequality was multiplied by a
negative number, the inequality symbol had to
be flipped
0
5
10
15 20
25 30
35
40
45
4. Multiplication Property of Inequality
𝟓
2
(multiply − 𝟐 , the reciprocal of − 5, to
both sides of the inequality)
5. Simplify
** Flip the inequality symbol**
The solution set is {𝑝| 𝑝 > 35}
50 55
60
65 70
75 80
85
90
TRY IT!!!!
Solve and Graph the Inequalities
𝑛
1
−5
𝑤
2
6
≥ −2
< −0.8
Solve an Inequality by Dividing
6 < 15
6
15
<
3
3
𝟐 < 𝟓
3 > −27
3
−27
>
3
3
𝟏 >−𝟗
Inequality
Divide each side of the inequality by 3
6 < 15
15
<
−3
−3
−𝟐 ≮ −𝟓
3 > −27
3
−27
>
−3
−3
−𝟏 ≯ 𝟗
Inequality
Divide each side of the inequality by -3
6
*In order to make the inequality true, you must
flip the inequality symbol (the symbol must be
reversed)
−𝟐 > −𝟓
−𝟏 < 𝟗
Dividing each side of an inequality by a
positive number, the inequality remains
true
Dividing each side of an equality by a
negative number, the inequality becomes
false.
To make it true you must flip the inequality
symbol
Divide by a Positive Number
EXAMPLE:
𝟏𝟒𝒉 < 𝟗𝟏
𝟏𝟒𝒉 < 𝟗𝟏
𝟏𝟒𝒉 < 𝟗𝟏
Solve for “h”
1. Write the inequality
2. Draw railroad tracks
2. Isolate the variable “h”
𝟏𝟒𝒉
𝟗𝟏
<
𝟏𝟒
3. Division Property of Inequality (Divide
both sides of the inequality by 14)
4. Combine Like Terms/Simplify
Set Builder Notation
𝟏𝟒
ℎ < 6.5
{𝒉| 𝒉 < 𝟔. 𝟓 }
2
2.5
3
3.5
4
4.5
5
6
6.5
7
8
8.5
9
9.5
10
Divide by a Negative Number
Solve for “t”
EXAMPLE:
−5𝑡
≥
275
−5𝑡
−5𝑡
−5
≥
275
275
−5
𝑡
≤
≥
1 Write the inequality
2 Draw railroad tracks
3. Isolate the variable.
4. Division Property of Inequality
(Divide both sides of the inequality by -5)
−55
5. Simplify
** Flip the inequality symbol**
Since the inequality was divided by a negative
number, the inequality symbol had to be
flipped
{𝒕 | 𝒕 ≤ −𝟓𝟓 }
Set Builder Notation
-90 -85 -80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5
0
Real World Problem
Pizza Party: You have a budget of $45 to buy pizza for a student council meeting. Pizzas cost
$7.50 each. Write and solve an inequality to find the possible numbers of pizzas that you can
buy
Variable: Let p = possible number of pizzas
Inequality: Price per pizza
$7.50
●
●
Number of Pizzas
p
≤
≤
Budget Amount
$45
Solve the Inequality
7.5𝑝 ≤
45
7.5𝑝 ≤ 45
7.5𝑝
45
≤
7.5
7.5
𝒑 ≤ 𝟔
𝑝 ≤ 6
1. Write the inequality
2. Train Tracks
3. Isolate the variable “p”
4. Division Property of Inequality (divide
7.5 from each side of the inequality)
5. Combine Like Terms/Simplify
{𝑝 | 𝑝 ≤ 6}
You can buy at the most 6 pizzas
Try It!!!! Solve and Graph the Inequality
1. −9𝑘 < 36
2. 10𝑛 ≥ 140