1. Solve Compound Inequalities
Warm Up
Lesson Presentation
Lesson Quiz

2. Warm-Up
Solve the inequality. 1.
8 > x + 10
ANSWER
all real numbers less than –2
6  2x – 4 2.
ANSWER
all real numbers greater than or equal to 5 3.
You estimate you can read at least 8 history text
pages per day. What are the possible numbers of
day it will take you to read at most 118 pages?
ANSWER
at most 15 days

3. Example 1
Translate the verbal phrase into an inequality. Then
graph the inequality. a.
All real numbers that are greater than –2 and less
than 3
Inequality:
–2 < x < 3
Graph: b.
All real numbers that are less than 0 or greater than
or equal to 2
Inequality:
x < 0 or x  2
Graph:

4. Guided Practice
Translate the verbal phrase into an inequality. Then
graph the inequality.
All real numbers that are less than –1 or greater than
or equal to 4 1.
Inequality:
x < –1 or x  4 2.
All real numbers that are greater than or equal
To –3 and less than 5
Inequality:
x  –3 and x < 5
= –3  x < 5

5. Example 2
CAMERA CARS
A crane sits on top of a camera car and faces toward the front. The crane’s maximum height and minimum height above the ground are shown. Write and graph a compound inequality that describes the possible heights of the crane.

6. Example 2
SOLUTION
Let h represent the height (in feet) of the crane. All possible heights are greater than or equal to 4 feet and less than or equal to 18 feet. So, the inequality is 4  h  18.

7. Guided Practice
Investing
An investor buys shares of a stock and will sell them if the change c in value from the purchase price of a share is less than –$3.00 or greater than $4.50. Write and graph a compound inequality that describes the changes in value for which the shares will be sold. 3.
c < –3 or c > 4.5
ANSWER

8. Solve 2 < x + 5 < 9. Graph your solution.
Example 3
Solve 2 < x + 5 < 9. Graph your solution.
SOLUTION
Separate the compound inequality into two inequalities. Then solve each inequality separately.
2 < x + 5
x + 5 < 9
and
Write two inequalities.
2 – 5 < x + 5 – 5
x + 5 – 5 < 9 – 5
and
Subtract 5 from each side.
–3 < x
x < 4
and
Simplify.
The compound inequality can be written as –3 < x < 4.

9. Example 3
ANSWER
The solutions are all real numbers greater than –3 and
less than 4.
Graph:

10. Solve the inequality. Graph your solution. –7 < x – 5 < 4 4.
Guided Practice
Solve the inequality. Graph your solution.
–7 < x – 5 < 4 4.
–2 < x < 9
ANSWER
– – – 9
Graph:
10  2y + 4  24 5.
3  y  10
ANSWER
Graph: 3
–7 < –z – 1 < 3 6.
–4 < z < 6
ANSWER

11. Solve –5  –x – 3  2. Graph your solution.
Example 4
Solve –5  –x – 3  2. Graph your solution.
–5  –x – 3  2
Write original inequality.
–5 + 3  –x –  2 + 3
Add 3 to each expression.
–2  –x  5
Simplify.
–1(–2)  –1(–x)  –1(5)
Multiply each expression by –1 and reverse both inequality symbols.
2  x  –5
Simplify.
–5  x  2
Rewrite in the form
a  x  b.

12. Example 4
ANSWER
The solutions are all real numbers greater than or equal to –5 and less than or equal to 2.

13. Guided Practice
Solve the inequality. Graph your solution.
7. –14 < x – 8 < –1
–6 < x < 7
ANSWER
8. –1  –5t + 2  4
–  t  2 5 3
ANSWER

14. Solve 2x + 3 < 9 or 3x – 6 > 12. Graph your solution.
Example 5
Solve 2x + 3 < 9 or 3x – 6 > 12. Graph your solution.
SOLUTION
Solve the two inequalities separately.
2x + 3 < 9 or
3x – 6 > 12
Write original
inequality.
2x + 3 – 3 < 9 – 3 or
3x – >
Addition or
Subtraction
property of
inequality
2x < 6 or
3x > 18
Simplify.

15. The solutions are all real numbers less than 3 or greater than 6.
Example 5 2x 2 6
< or 3x 3 18
>
Division property of inequality
x < 3 or
x > 6
Simplify.
ANSWER
The solutions are all real numbers less than 3 or greater
than 6.

16. Guided Practice
Solve the inequality. Graph your solution.
h + 1< – 5 or
2h – 5 > 7
h < –2 or h > 6
ANSWER
c + 1  –3 or
5c – 3 > 17
c  –1 or c > 4
ANSWER

17. Example 6
Astronomy
The Mars Exploration Rovers Opportunity and Spirit are robots that were sent to Mars in 2003 in order to gather geological data about the planet. The temperature at the landing sites of the robots can range from 100°C to 0°C.
• Write a compound inequality that describes the possible temperatures (in degrees Fahrenheit) at a landing site.
• Solve the inequality. Then graph your solution.
• Identify three possible temperatures (in degrees Fahrenheit) at a landing site.

18. Celsius. Use the formula 5 9 C = (F – 32).
Example 6
SOLUTION
Let F represent the temperature in degrees Fahrenheit, and let C represent the temperature in degrees
Celsius. Use the formula 5 9
C = (F – 32).
STEP 1
Write a compound inequality. Because the temperature at a landing site ranges from –100°C to 0°C, the lowest possible temperature is –100°C, and the highest possible temperature is 0°C.
–100  C  0
Write inequality using C.
(F – 32)
–100   0 5 9
Substitute (F – 32) for C. 9 5

19. Solve the inequality. Then graph your solution.
Example 6
STEP 2
Solve the inequality. Then graph your solution.
(F – 32)
–100   0 5 9
Write inequality from Step 1.
Multiply each expression by 5 9
–180  (F – 32 )  0
–148  F  32
Add 32 to each expression.

20. Example 6
STEP 3
Identify three possible temperatures.
The temperature at a landing site is greater than or equal to –148°F and less than or equal to 32°F. Three possible temperatures are –115°F, 15°F, and 32°F.

21. Guided Practice
Mars has a maximum temperature of 27°C at the equator and a minimum temperature of –133°C at the winter pole.
11.
• Write and solve a compound inequality that describes the possible temperatures (in degree Fahrenheit) on Mars.
ANSWER
–133 ≤ (F– 32) ≤ 27; 207.4 ≤ F ≤ 80.6 5 9
• Graph your solution. Then identify three possible temperatures (in degrees Fahrenheit) on Mars.
ANSWER
Sample answer: 100°F, 0°F, 25°F

22. Lesson Quiz
1. Solve x + 4 < 7 or 2x – 5 > 3. Graph your solution.
ANSWER
all real numbers less than 3 or greater than 4 2.
Solve 3x + 2 > –7 and 4x – 1 < –5. Graph your solution.
ANSWER
all real numbers greater than –3 and less than –1

23. Lesson Quiz
The smallest praying mantis is 0.4 inch in length. The largest is 6 inches. Write and graph a compound inequality that describes the possible lengths L of a praying mantis. 3.
ANSWER
L 6
< –