1. Objective The student will be able to:
solve inequalities using multiplication and division.

2. Solving Inequalities Using Multiplication
x  -2 3
x  -6
Translation: x divided by 3 is greater than or equal to negative 2.
The opposite of divided by 3 is Multiplying by 3.
What you do on one side you do on the other!
• 3
3 •
Graph -6 -4 -2 4

3. Solving Inequalities Using Division
Translation: 3times x is less than or equal to 12.
The opposite of multiplying by 3 is divided by 3.
What you do on one side you do on the other!
3x ≤ 12 3 1 3
x ≤ 4
Graph -4 -2 2 4

4. Solve the inequality. Then graph the solution.

5. Multiplying and Dividing Inequalities by a Negative Number
Solve the inequality.
12 > -3m
4 < m
Step1: solve for m
-divide by -3
Step 2: reverse the inequality symbol
12 > -3m
< -3 -3
When multiplying or dividing by a negative number reverse the inequality symbol.

6. 1) Solve > 13 k < -52 o -52 -51 -53 Draw “the river”
Clear the fraction - Multiply both sides by -4
NEW STEP!! When multiplying BOTH sides by a NEGATIVE number, SWITCH the inequality!
Simplify
Check your answer
Graph the solution
k < -52 o
-52
-51
-53

7. 1) Which graph represents the correct answer to > 1-4 -3 -5 o -4 -3 -5 ● -4 -3 -5 ● -4 -3 -5
Answer Now

8. 2) When solving > -10 will the inequality switch?
Yes!
No!
I still don’t know!
Answer Now

9. 2) Solve < -10 x < -30 o -30 -29 -31 Draw “the river”
Clear the fraction - Multiply both sides by 3. Do you switch the inequality?
No - Both sides are being multiplied by a positive number
Simplify
Check your answer
Graph the solution
x < -30 o
-30
-29
-31

10. 3) When solving will the inequality switch?
Yes!
No!
I still don’t know!
Answer Now

11. 3) Solve a > -24 o -24 -23 -25 Draw “the river”
Clear the fraction - Multiply both sides by -4. Do you switch the inequality?
Yes - Both sides are being multiplied by a negative number
Simplify
Check your answer
Graph the solution
a > -24 o
-24
-23
-25

12. 4) Solve -8p ≥ -96
p ≥ 12
p ≥ -12
p ≤ 12
p ≤ -12
Answer Now

13. 4) Solve -8p ≥ -96 p ≤ 12 -8(12) = -96 ● 12 13 11 Draw “the river”
Divide both sides by -8. Do you switch the inequality?
Yes - Both sides are being divided by a negative number
Simplify
Check your answer
Graph the solution
p ≤ 12
-8(12) = -96 ● 12 13 11

14. ● ● 5) Solve 7v < -105 o o -15 -14 -16 -15 -14 -16 -15 -14 -16 -15
Answer Now

15. You try: Solve the Inequalities

16. 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
number of tubes
is at most
$20.00.
4.30 • p ≤
20.00

17. 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 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.