Warm Up
Solve each equation.
1. –5a = 30 –6
2.
3.
4.
Graph each inequality.
5. x ≥ –10
6. x < –3
–10
Lesson 3.3
Solving Inequalities
by Multiplying or Dividing
Less than or equal to
Less than
≤
<
greater than
>
greater than or equal to
≥
California
Standards
Preparation for
5.0
Students solve multistep problems, including
word problems, involving linear equations and
linear inequalities in one variable and provide
justification for each step.
Inequalities and their Graphs
Terms you see and need to know
to graph inequalities correctly
≤ is less than or equal to
≥ is greater than or equal to
Notice colored in circles
Inequalities and their Graphs
Terms you see and need to know
to graph inequalities correctly
< is less than
> is greater than
Notice
open
circles
Solving Inequalities Using Multiplication
x  -2
3
Translation: x divided by 3 is
greater than or equal to
negative 2.
x  -2
• 3
3 • 3
x  -6
The opposite of divided by 3
is Multiplying by 3.
Graph
-6
-4
-2
0
4
What you do on one side
you do on the other!
Solving Inequalities Using Division
Translation: 3times x is less than
or equal to 12.
3x ≤ 12
The opposite of multiplying by 3 is
divided by 3.
3x ≤ 12
3
3
1
What you do on one side you do on
the other!
x ≤ 4
Graph
-4
-2
0
2
4
Solve the inequality. Then
graph the solution.
1)  6 
r
3
2 )  10  2 k
Multiplying and Dividing Inequalities by a
Negative Number
Solve the inequality.
Step1: solve for m
12 > -3m
-divide by -3
12 <> -3m
-3
-3
Step 2: reverse the
inequality symbol
4 < m
When multiplying or dividing by a negative
number reverse the inequality symbol.
Multiplying and Dividing Inequalities by a
Negative Number
Solve the inequality.
Step1: solve for k
k
<
-3
-2.
.-2
-Multiply by -2
-2
k >>6
Step 2: reverse the
inequality symbol
When multiplying or dividing by a negative
number reverse the inequality symbol.
Tell whether the operation reverses
the sign of the inequality.
1)
2)
Multiply each side by -1
1
Multiply each side b 4
3) Divide each side by 4.

4) Divide each side by -5.
You try: Solve the Inequalities
1)  2 x  16
2)
x
 10
 5 
3)  4 k  12
4)7 
m
4
Additional Example: Application
Jill has a $20 gift card to an art supply store
where 4 oz tubes of paint are $4.30 each after
tax. What are the possible numbers of tubes
that Jill can buy?
Let p represent the number of tubes of paint that Jill
can buy.
$4.30
times
4.30
•
number of tubes
is at most
$20.00.
p
≤
20.00
Additional Example Continued
Jill has a $20 gift card to an art supply store
where 4 oz tubes of paint are $4.30 each after
tax. What are the possible numbers of tubes
that Jill can buy?
4.30p ≤ 20.00
Since p is multiplied by 4.30,
divide both sides by 4.30. The
symbol does not change.
p ≤ 4.65…
Since Jill can buy only whole numbers of tubes,
she can buy 0, 1, 2, 3, or 4 tubes of paint.
Lesson Quiz
Solve each inequality and graph the solutions.
1. 8x < –24 x < –3
2. –5x ≥ 30
x ≤ –6
3.
4.
x≥6
x > 20
5. A soccer coach plans to order more shirts for
her team. Each shirt costs $9.85. She has $77
left in her uniform budget. What are the
possible numbers of shirts she can buy?
0, 1, 2, 3, 4, 5, 6, or 7 shirts