1. & Multiplying or Dividing
Modules 4-2 & 4-3
Objectives
Solve one-step inequalities by using addition.
Solve one-step inequalities by using subtraction.
Solve one-step inequalities by using multiplication.
Solve one-step inequalities by using division.

2. Solving one-step inequalities is much like solving one-step equations
Solving one-step inequalities is much like solving one-step equations. To solve an inequality, you need to isolate the variable using the properties of inequality and inverse operations.

4. Example 1A: Using Addition and Subtraction to Solve
Inequalities
Solve the inequality and graph the solutions.
x + 12 < 20
x + 12 < 20
Since 12 is added to x, subtract 12 from both sides to undo the addition.
–12 –12
x + 0 < 8
x < 8
Draw an empty circle at 8.
–10 –8 –6 –4 –2 2 4 6 8 10
Shade all numbers less than 8 and draw an arrow pointing to the left.

5. Example 1B: Using Addition and Subtraction to Solve
Inequalities
Solve the inequality and graph the solutions.
d – 5 > –7
d + 0 > –2
d > –2
d – 5 > –7
Since 5 is subtracted from d, add 5 to both sides to undo the subtraction.
Draw an empty circle at –2.
–10 –8 –6 –4 –2 2 4 6 8 10
Shade all numbers greater than –2 and draw an arrow pointing to the right.

6. Example 1C: Using Addition and Subtraction to Solve
Inequalities
Solve the inequality and graph the solutions.
0.9 ≥ n – 0.3
0.9 ≥ n – 0.3
Since 0.3 is subtracted from n, add 0.3 to both sides to undo the subtraction.
1.2 ≥ n – 0
1.2 ≥ n 
1.2
Draw a solid circle at 1.2. 1 2
Shade all numbers less than 1.2 and draw an arrow pointing to the left.

7. Example 2: Problem-Solving Application
Sami has a gift card. She has already used $14 of the total value, which was $30. Write, solve, and graph an inequality to show how much more she can spend.
Understand the problem 1
The answer will be an inequality and a graph that show all the possible amounts of money that Sami can spend.
List important information:
• Sami can spend up to, or at most $30.
• Sami has already spent $14.

8. Example 2 Continued 2
Make a Plan
Write an inequality.
Let g represent the remaining amount of money Sami can spend.
Amount remaining
plus
$30.
is at most
amount used g + 14 ≤ 30
g + 14 ≤ 30

9. The amount spent cannot be negative.
Example 2 Continued
Solve 3
g + 14 ≤ 30
Since 14 is added to g, subtract 14 from both sides to undo the addition.
– 14 – 14
g + 0 ≤ 16
g ≤ 16
Draw a solid circle at 0 and16. 2 4 6 8 10 12 14 16 18
Shade all numbers greater than 0 and less than 16.
The amount spent cannot be negative.

10.   Example 2 Continued Look Back 4 Check Check a number less than 16.
g ≤ 30
≤ 30
20 ≤ 30 
Check the endpoint, 16.
g + 14 = 30
30 30
Sami can spend from $0 to $16.

11. Example 3: Application
Mrs. Lawrence wants to buy an antique bracelet at an auction. She is willing to bid no more than $550. So far, the highest bid is $475. Write and solve an inequality to determine the amount Mrs. Lawrence can add to the bid. Check your answer.
Let x represent the amount Mrs. Lawrence can add to the bid.
$475
plus
amount can add
is at most
$550. x +
475 ≤
550
475 + x ≤ 550

12. Example 3 Continued
475 + x ≤ 550
– – 475
x ≤ 75
0 + x ≤ 75
Since 475 is added to x, subtract 475 from both sides to undo the addition.
Check the endpoint, 75.
Check a number less than 75.
x = 550 
x ≤ 550
≤ 550
525 ≤ 550 
Mrs. Lawrence is willing to add $75 or less to the bid.

13. Remember, solving inequalities is similar to solving equations
Remember, solving inequalities is similar to solving equations. To solve an inequality that contains multiplication or division, undo the operation by dividing or multiplying both sides of the inequality by the same number.
The following rules show the properties of inequality for multiplying or dividing by a positive number. The rules for multiplying or dividing by a negative number appear later in this lesson.

15. Solve the inequality and graph the solutions.
Example 1A: Multiplying or Dividing by a Positive Number
Solve the inequality and graph the solutions.
7x > –42
7x > –42
>
Since x is multiplied by 7, divide both sides by 7 to undo the multiplication.
1x > –6
x > –6
–10 –8 –6 –4 –2 2 4 6 8 10

16. Solve the inequality and graph the solutions.
Example 1B: Multiplying or Dividing by a Positive Number
Solve the inequality and graph the solutions.
Since m is divided by 3, multiply both sides by 3 to undo the division.
3(2.4) ≤ 3
7.2 ≤ m
(or m ≥ 7.2) 2 4 6 8 10 12 14 16 18 20

17. Solve the inequality and graph the solutions.
Example 1C: Multiplying or Dividing by a Positive Number
Solve the inequality and graph the solutions.
Since r is multiplied by , multiply both sides by the reciprocal of .
r < 16 2 4 6 8 10 12 14 16 18 20

18. If you multiply or divide both sides of an inequality by a negative number, the resulting inequality is not a true statement. You need to reverse the inequality symbol to make the statement true.
This means there is another set of properties of inequality for multiplying or dividing by a negative number.

20. Caution!
Do not change the direction of the inequality symbol just because you see a negative sign. For example, you do not change the symbol when solving 4x < –24.

21. Solve the inequality and graph the solutions.
Example 2A: Multiplying or Dividing by a Negative Number
Solve the inequality and graph the solutions.
–12x > 84
Since x is multiplied by –12, divide both sides by –12. Change > to <.
x < –7
–10 –8 –6 –4 –2 2 4 6
–12
–14 –7

22. Solve the inequality and graph the solutions.
Example 2B: Multiplying or Dividing by a Negative Number
Solve the inequality and graph the solutions.
Since x is divided by –3, multiply both sides by –3. Change to .
24  x
(or x  24) 16 18 20 22 24 10 14 26 28 30 12

23. Let p represent the number of tubes of paint that Jill can buy.
Example 3: 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

24. Example 3 Continued
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.

25. Tonight’s HW:
p #11-15, #26-29 all
p. 101 #51-54 all
p. 104 #15-25 all