1. Warm Up Solve each equation. 1. 2x – 5 = –17 2. –6 14
Solve each inequality and graph the
solutions.
3. 5 < t + 9
t > –4 4.
a ≤ –8

2. Objective
Solve inequalities that contain more than one operation.

3. Inequalities that contain more than one operation require more than one step to solve. Use inverse operations to undo the operations in the inequality one at a time.

4. Example 1A: Solving Multi-Step Inequalities
Solve the inequality and graph the solutions.
45 + 2b > 61
Since 45 is added to 2b, subtract 45 from both sides to undo the addition.
45 + 2b > 61
– –45
2b > 16
Since b is multiplied by 2, divide both sides by 2 to undo the multiplication.
b > 8 2 4 6 8 10 12 14 16 18 20

5. Example 1B: Solving Multi-Step Inequalities
Solve the inequality and graph the solutions.
8 – 3y ≥ 29
Since 8 is added to –3y, subtract 8 from both sides to undo the addition.
8 – 3y ≥ 29
– –8
–3y ≥ 21
Since y is multiplied by –3, divide both sides by –3 to undo the multiplication. Change ≥ to ≤.
y ≤ –7
–10 –8 –6 –4 –2 2 4 6 8 10 –7

6. Solve the inequality and graph the solutions.
Check It Out! Example 1a
Solve the inequality and graph the solutions.
–12 ≥ 3x + 6
Since 6 is added to 3x, subtract 6 from both sides to undo the addition.
–12 ≥ 3x + 6
– – 6
–18 ≥ 3x
Since x is multiplied by 3, divide both sides by 3 to undo the multiplication.
–6 ≥ x
–10 –8 –6 –4 –2 2 4 6 8 10

7. Solve the inequality and graph the solutions.
Check It Out! Example 1b
Solve the inequality and graph the solutions.
Since x is divided by –2, multiply both sides by –2 to undo the division. Change > to <.
–5 –5
x + 5 < –6
Since 5 is added to x, subtract 5 from both sides to undo the addition.
x < –11
–20
–12 –8 –4
–16
–11

8. Solve the inequality and graph the solutions.
Check It Out! Example 1c
Solve the inequality and graph the solutions.
Since 1 – 2n is divided by 3, multiply both sides by 3 to undo the division.
1 – 2n ≥ 21
Since 1 is added to –2n, subtract 1 from both sides to undo the addition.
– –1
–2n ≥ 20
Since n is multiplied by –2, divide both sides by –2 to undo the multiplication. Change ≥ to ≤.
n ≤ –10
–10
–20
–12 –8 –4
–16

9. To solve more complicated inequalities, you may first need to simplify the expressions on one or both sides by using the order of operations, combining like terms, or using the Distributive Property.

10. Example 2A: Simplifying Before Solving Inequalities
Solve the inequality and graph the solutions.
2 – (–10) > –4t
12 > –4t
Combine like terms.
Since t is multiplied by –4, divide both sides by –4 to undo the multiplication. Change > to <.
–3 < t
(or t > –3) –3
–10 –8 –6 –4 –2 2 4 6 8 10

11. Example 2B: Simplifying Before Solving Inequalities
Solve the inequality and graph the solutions.
–4(2 – x) ≤ 8
–4(2 – x) ≤ 8
Distribute –4 on the left side.
–4(2) – 4(–x) ≤ 8
Since –8 is added to 4x, add 8 to both sides.
–8 + 4x ≤ 8
4x ≤ 16
Since x is multiplied by 4, divide both sides by 4 to undo the multiplication.
x ≤ 4
–10 –8 –6 –4 –2 2 4 6 8 10

12. Example 2C: Simplifying Before Solving Inequalities
Solve the inequality and graph the solutions.
Multiply both sides by 6, the LCD of the fractions.
Distribute 6 on the left side.
4f + 3 > 2
Since 3 is added to 4f, subtract 3 from both sides to undo the addition.
–3 –3
4f > –1

13. Example 2C Continued
4f > –1
Since f is multiplied by 4, divide both sides by 4 to undo the multiplication.

14. Solve the inequality and graph the solutions.
Check It Out! Example 2a
Solve the inequality and graph the solutions.
2m + 5 > 52
Simplify 52.
2m + 5 > 25
– 5 > – 5
Since 5 is added to 2m, subtract 5 from both sides to undo the addition.
2m > 20
m > 10
Since m is multiplied by 2, divide both sides by 2 to undo the multiplication. 2 4 6 8 10 12 14 16 18 20

15. Solve the inequality and graph the solutions.
Check It Out! Example 2b
Solve the inequality and graph the solutions.
3 + 2(x + 4) > 3
Distribute 2 on the left side.
3 + 2(x + 4) > 3
3 + 2x + 8 > 3
Combine like terms.
2x + 11 > 3
Since 11 is added to 2x, subtract 11 from both sides to undo the addition.
– 11 – 11
2x > –8
Since x is multiplied by 2, divide both sides by 2 to undo the multiplication.
x > –4
–10 –8 –6 –4 –2 2 4 6 8 10

16. Check It Out! Example 2c
Solve the inequality and graph the solutions.
Multiply both sides by 8, the LCD of the fractions.
Distribute 8 on the right side.
5 < 3x – 2
Since 2 is subtracted from 3x, add 2 to both sides to undo the subtraction.
7 < 3x

17. Check It Out! Example 2c Continued
Solve the inequality and graph the solutions.
7 < 3x
Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. 4 6 8 2 10

18. daily cost at We Got Wheels
Example 3: Application
To rent a certain vehicle, Rent-A-Ride charges $55.00 per day with unlimited miles. The cost of renting a similar vehicle at We Got Wheels is $38.00 per day plus $0.20 per mile. For what number of miles is the cost at Rent-A-Ride less than the cost at We Got Wheels?
Let m represent the number of miles. The cost for Rent-A-Ride should be less than that of We Got Wheels.
Cost at Rent-A-Ride
must be less than
daily cost at We Got Wheels
plus
$0.20
per mile
times
# of miles. 55
< 38 +
0.20  m

19. Example 3 Continued
55 < m
Since 38 is added to 0.20m, subtract 38 from both sides to undo the addition.
–38 –38
55 < m
17 < 0.20m
Since m is multiplied by 0.20, divide both sides by 0.20 to undo the multiplication.
85 < m
Rent-A-Ride costs less when the number of miles is more than 85.

20.   Example 3 Continued Check Check the endpoint, 85. 55 38 + 17 55 55 
55 = m
(85)
Check a number greater than 85.
55 <
55 < 56 
55 < (90)
55 < m

21. is greater than or equal to
Check It Out! Example 3
The average of Jim’s two test scores must be at least 90 to make an A in the class. Jim got a 95 on his first test. What grades can Jim get on his second test to make an A in the class?
Let x represent the test score needed. The average score is the sum of each score divided by 2.
First test score
plus
second test score
divided by
number of scores
is greater than or equal to
total score
(95 + x)  2 ≥ 90

22. Check It Out! Example 3 Continued
Since 95 + x is divided by 2, multiply both sides by 2 to undo the division.
95 + x ≥ 180
Since 95 is added to x, subtract 95 from both sides to undo the addition.
– –95
x ≥ 85
The score on the second test must be 85 or higher.

23. Check It Out! Example 3 Continued
Check the end point, 85.
Check a number greater than 85. 90 
90.5 ≥ 90 