1. 2.5 Solving Compound Inequalities
Objective 1
Write, solve and graph compound inequalities.
Objective 2
Model a real-life situation with a compound inequality.
EQ: How do you solve a compound inequality?
VOCABULARY
A compound inequality consists of two inequalities connected by the word and or the word or.

2. EX: Graph x > –2 and x < 3
NOTE: Solutions are all numbers that are greater than or equal to –2 and less than 3.
Since X is between –2 and 3 the inequality is usually written as
–2 < X < 3
EX: Graph X < 3 or X > 6
Graph both equations on the same graph

3. Write an inequality that describes each condition.
Example 2
Compound Inequalities in Real Life
Write an inequality that describes each condition.
a. Water is a non-liquid when the temperature is 32°F or below or is at least 212°F.
T ≤ 32 or T ≥ 212
b. A refrigerator is designed to work on an electric line carrying from 115 volts to 120 volts.
115 ≤ V ≤ 120

4. Solving a Compound Inequality with And
Example 3
Solving a Compound Inequality with And
Solve -5 ≤ 2x + 3 < 7. Then graph the solution.
Isolate the variable between the inequality symbols.
–5 ≤ 2x + 3 < 7
Subtract 3 from all three sides
–8 ≤ 2x < 4
Divide each side by 2.
–4 ≤ x < 2
The solution is all real numbers that are greater than or equal to -4 and less than 2.

5. Solving a Compound Inequality with Or
Solve x + 5 < or 3x > 12
Solve each part separately. or
3x >12
x + 5 ≤ –6
x ≤ –11 or
x > 4

6. Solve –3 < –1 – 2x ≤ 5. Then graph the solution.
Example 5
Reversing the sign
Solve –3 < –1 – 2x ≤ 5. Then graph the solution.
–2 < – 2x ≤ 6
Reverse the inequalities when you divide by a negative
– – –2
1 > x ≥ –3

7. Try these Solve and graph the inequality. 1. 2. 3. 4. 2. 3. 4.

8. Modeling with a Compound Inequality
The park is 2 miles from your house. The mall is 3 miles from your house.
a. Find the minimum distance between the park and the mall.
b. Find the maximum distance between the park and the mall.
c. Write an inequality that describes the possible distances d between the park and the mall.
Solution – DRAW A DIAGRAM
The shortest distance is 1 mile.
The mall is somewhere on this circle
b. The longest distance is 5 miles. 3 2
c. The values of d can be described by the inequality 1 ≤ d ≤ 5
House
The park is somewhere on this circle