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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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 in 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

12. 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.

13. Lesson Quiz: Part I
Solve each inequality and graph the solutions.
– 2x ≥ 21
x ≤ –4
2. – < 3p
p > –3
3. 23 < –2(3 – t)
t > 7 4.

14. Lesson Quiz: Part II
5. A video store has two movie rental plans. Plan A includes a $25 membership fee plus $1.25 for each movie rental. Plan B costs $40 for unlimited movie rentals. For what number of movie rentals is plan B less than plan A?
more than 12 movies