1. Solving Compound Inequalities (5-4)
Objective: Solve compound inequalities containing the word and and graph their solution sets. Solve compound inequalities containing the word or and graph their solution sets.

2. Inequalities Containing and
When considered together, two inequalities such as h ≥ 52 and h ≤ 72 form a compound inequality.
A compound inequality containing and is only true if both inequalities are true.
Its graph is where the graphs of the two inequalities overlap.
This is called the intersection of the two graphs.
The intersection can be found by graphing each inequality and then determining where the graphs intersect.

3. Inequalities Containing and
x ≥ 3
x < 7
x ≥ 3 and x < ≤ x < 7
The statement 3 ≤ x < 7 can be read as “x is greater than or equal to 3 and less than 7” or “x is between 3 and 7 including 3”.

4. Example 1 Solve 7 < z + 2 ≤ 11. Then graph the solution set.
5 < z ≤ 9
{z|5 < z ≤ 9}

5. Check Your Progress Choose the best answer for the following.
Solve -3 < x – 2 < 5. Then graph the solution set.
{x|-1 < x < 7}
{x|-5 < x < 3}
{x|x < 7}
{x|-1 < x < 3}
-3 < x – 2 < 5

6. Inequalities Containing or
Another type of compound inequality contains the word or.
A compound inequality containing or is true if at least one of the inequalities is true.
Its graph is the union of the graphs of two inequalities.

7. Inequalities Containing and
x > 2
x ≤ -1
x > 2 or x ≤ -1
When solving problems involving inequalities, within is meant to be inclusive, so use ≥ or ≤.
Between is meant to be exclusive, so use < or >.

8. Example 2
A ski resort has several types of hotel rooms and cabins. The hotel rooms cost at most $89 per night, and the cabins cost at least $109 per night. Write and graph a compound inequality that describes the amount a guest would pay per night at the resort.
c ≤ 89 or c ≥ 109
{c|c ≤ 89 or c ≥ 109}

9. Check Your Progress Choose the best answer for the following.
A professional hockey arena has seats available in the Lower Bowl level that cost at most $65 per seat. The arena also has seats available at the Club Level and above that cost at least $80 per seat. Write and graph a compound inequality that describes the amount a spectator would pay for a seat at the hockey game.
c ≤ 65 or c ≥ 80
c ≥ 65 or c ≤ 80
c ≥ 65 or c ≥ 80
c ≤ 65 or c ≤ 80

10. Example 3 Solve 4k – 7 ≤ 25 or 12 – 9k ≥ 30. Then graph the solution.
4k ≤ 32
-9k ≥ 18
k ≤ 8
k ≤ -2
{k|k ≤ 8}

11. Check Your Progress 5x + 15 > 20 -15 -15 5x > 5 5 5
5x > 5
Choose the best answer for the following.
Solve -2x + 5 < 15 or 5x + 15 > 20. Then graph the solution set.
{x|x > 1}
{x|x < -5}
{x|x > -5}
{x|x < 1}
x > 1
-2x + 5 < 15
-2x < 10
x > -5