1. Solving Two-Step Inequalities
Use the same process when solving two-step inequalities that you used to solve two-step equations.

2. Solving Two-Step Inequalities
Draw an open circle when the inequality includes the < or > symbol.
Draw a closed circle when the inequality includes the < or > symbol.
Remember!

3. Example 1: Solving Two-Step Inequalities
Solve. Then graph the solution on a number line.
4y – 5 < 11
4y – 5 < 11
Add 5 to both sides.
4y < 16
4y < 16
Divide both sides by 4. 4 4
y < 4 –6 –4 –2 2 4 6

4. Solving Two-Step Inequalities
Example 2
Solve. Then graph the solution on a number line.
2y – 4 > –12
2y – 4 > –12
Add 4 to both sides.
2y > –8
2y > –8
Divide both sides by 2. 2 2
y > –4 –6 –4 –2 2 4 6

5. Solving Two-Step Inequalities
Example 3
Solve. Then graph the solution set on a number line. h 7
+ 1 > –1 h 7
+ 1 > –1
– – 1
Subtract 1 from both sides. h 7
> –2 h 7
> (7)(–2)
(7)
Multiply both sides by 7.
h > –14 7 14 21 –

6. Solving Inqualities by Multiplying or Dividing
Notes:
When you multiply or divide both sides by a negative number, you need to reverse the direction of the inequality symbol.

7. Solving Two-Step Inequalities
Example 4
Solve. Then graph the solution set on a number line.
D. –9x + 4 ≤ 31
–9x + 4 ≤ 31
– 4 –4
Subtract 4 from both sides.
–9x ≤ 27
Divide both sides by –9, and reverse the inequality symbol.
–9x ≥ 27
– –9
x ≥ –3 3 6 9 –3 –6 –9

8. Example 5: Solving Two-Step Inequalities
Solve. Then graph the solution set on a number line. m –3
+ 8 m –3
5 ≥
– –8
Subtract 8 from both sides. m –3
-3 ≥ m –3
(–3) ≤
(–3)
Multiply both sides by –3, and
reverse the inequality symbol.
m ≥ 9 • –

9. Example 6: Solving Two-Step Inequalities
Solve. Then graph the solution set on a number line. y 2
– 6 > 1 y 2
– 6 > 1
Add 6 to both sides. y 2
> 7 y 2
> (2)7
(2)
Multiply both sides by 2.
y > 14 7 14 21 –

10. Example 7: Solving Two-Step Inequalities
Solve. Then graph the solution set on a number line.
–4 ≥ –3x + 5
–4 ≥ –3x + 5
– –5
Subtract 5 from both sides.
–9 ≤ –3x
Divide both sides by –3, and reverse the inequality symbol.
–3 –3
x ≥ 3 3 6 9 –3 –6 –9

11. Solving Two-Step Inequalities
Example 8
Solve. Then graph the solution set on a number line. m –2
+ 1 ≥ 7 m –2
+ 1 ≥ 7
– 1 –1
Subtract 1 from both sides. m –2
≥ 6 m –2
≤ (6)
(–2)
Multiply both sides by –2, and
reverse the inequality symbol.
m ≤ –12 •
–18 –12 –

12. Solving Two-Step Inequalities
Application
Sun-Li has $30 to spend at the carnival. Admission is $5, and each ride costs $2. What is the greatest number of rides she can ride?
Let r represent the number of rides Sun-Li can ride.
5 + 2r ≤ 30
– –5
Subtract 5 from both sides.
2r ≤ 25
Divide both sides by 2.
2r ≤ 25
r ≤ 12
252
, or 12
Sun-Li can ride only a whole number of rides, so the most she can ride is 12.

13. Solving Two-Step Inequalities
Application 2
Brice has $30 to take his brother and his friends to the movies. If each ticket costs $4.00, and he must buy tickets for himself and his brother, what is the greatest number of friends he can invite?
Let t represent the number of tickets.
8 + 4t ≤ 30
– –8
Subtract 8 from both sides.
4t ≤ 22
Divide both sides by 4.
4t ≤ 22
t ≤ 5.5
Brice can only buy a whole number of tickets, so the most people he can invite is 5.

14. Solving Two-Step Inequalities
Lesson Quiz: Part I
Solve. Then graph each solution set on a number line. –4 –2 2 –6 –8
–10
s > –7
1. 7s + 14 > –35
–62
–60
–58
–56
–64
–66
–68
y < –64 y –8
> 20 2 3 4 5 1 –1
n ≤ 3
3. 18n – 22 ≤ 32

15. Solving Two-Step Inequalities
Lesson Quiz: Part II
4. A cyclist has $7.00. At the first stop on the tour, energy bars are $1.15 each, and a sports drink is $1.75. What is the greatest number of energy bars the cyclist can buy if he buys one sports drink? 4