1. Warm Up
Lesson Presentation
Lesson Quiz

2. Warm Up Solve each inequality and graph the solution. 1. x + 7 < 42.
x < –3 –5 –4 –3 –2 –1 1 2 3 4 5
14x ≥ 28
x ≥ 2 –5 –4 –3 –2 –1 1 2 3 4 5
x > 1
x > –2 –5 –4 –3 –2 –1 1 2 3 4 5

3. Sunshine State Standards
MA.912.A.3.6 Solve and graph the solutions of absolute value…inequalities with one variable.

4. Objective
Solve inequalities in one variable involving absolute-value expressions.

5. When an inequality contains an absolute-value expression, it can be written as a compound inequality. The inequality |x| < 5 describes all real numbers whose distance from 0 is less than 5 units. The solutions are all numbers between –5 and 5, so |x| < 5 can be rewritten as –5 < x < 5, or as x > –5 AND x < 5.

7. Write as a compound inequality.
Additional Example 1A: Solving Absolute-Value Inequalities Involving <
Solve the inequality and graph the solutions.
|x|– 3 < –1
Since 3 is subtracted from |x|, add 3 to both sides to undo the subtraction.
|x| < 2
|x|– 3 < –1
x > –2 AND x < 2
Write as a compound inequality. –2 –1 1 2
2 units

8. Write as a compound inequality. +1 +1 +1 +1 Solve each inequality.
Additional Example 1B: Solving Absolute-Value Inequalities Involving <
Solve the inequality and graph the solutions.
|x – 1| ≤ 2
x – 1 ≥ –2 AND x – 1 ≤ 2
Write as a compound inequality.
+1 +1
Solve each inequality.
x ≥ –1
x ≤ 3
AND
Write as a compound inequality. –2 –1 1 2 3 –3

9. Just as you do when solving absolute-value equations, you first isolate the absolute-value expression when solving absolute-value inequalities.
Helpful Hint

10. Write as a compound inequality.
Check It Out! Example 1a
Solve the inequality and graph the solutions.
2|x| ≤ 6
2|x| ≤ 6
Since x is multiplied by 2, divide both sides by 2 to undo the multiplication.
|x| ≤ 3
x ≥ –3 AND x ≤ 3
Write as a compound inequality. –2 –1 1 2
3 units –3 3

11. Isolate the absolute value function. |x + 3|– 4.5 ≤ 7.5
Check It Out! Example 1b
Solve each inequality and graph the solutions.
Isolate the absolute value function.
|x + 3|– 4.5 ≤ 7.5
|x + 3| ≤ 12
|x + 3|– 4.5 ≤ 7.5
Since 4.5 is subtracted from |x + 3|, add 4.5 to both sides to undo the subtraction.
x + 3 ≥ –12 AND x + 3 ≤ 12
–3 –3
–3 –3
x ≥ –15 AND x ≤ 9
Write as a compound inequality.
–20
–15
–10 –5 5 10 15

12. The inequality |x| > 5 describes all real numbers whose distance from 0 is greater than 5 units. The solutions are all numbers less than –5 or greater than 5. The inequality |x| > 5 can be rewritten as the compound inequality x < –5 OR x > 5.

14. Solve the inequality and graph the solutions.
Additional Example 2A: Solving Absolute-Value Inequalities Involving >
Solve the inequality and graph the solutions.
|x| + 14 ≥ 19
Isolate the absolute value function.
– 14 –14
|x| + 14 ≥ 19
Since 14 is added to |x|, subtract 14 from both sides to undo the addition.
|x| ≥ 5
x ≤ –5 OR x ≥ 5
Write as a compound inequality. The solution set is {x: x ≤ –5 OR x ≥ 5}.
5 units
–10 –8 –6 –4 –2 2 4 6 8 10

15. Solve the inequality and graph the solutions.
Additional Example 2B: Solving Absolute-Value Inequalities Involving >
Solve the inequality and graph the solutions.
Isolate the absolute value function.
3 + |x + 2| > 5
|x + 2| > 2
– – 3
3 + |x + 2| > 5
Since 3 is added to |x + 2|, subtract 3 from both sides to undo the addition.
x + 2 < –2 OR x + 2 > 2
–2 –2
–2 –2
x < –4 OR x > 0
Write as a compound inequality. Solve each inequality.
Write as a compound inequality.
–10 –8 –6 –4 –2 2 4 6 8 10

16. Solve each inequality and graph the solutions.
Check It Out! Example 2a
Solve each inequality and graph the solutions.
|x| + 10 ≥ 12
Isolate the absolute value function.
|x| + 10 ≥ 12
Since 10 is added to |x|, subtract 10 from both sides to undo the addition.
– 10 –10
|x| ≥ 2
x ≤ –2 OR x ≥ 2
Write as a compound inequality.
2 units –5 –4 –3 –2 –1 1 2 3 4 5

17. Solve the inequality and graph the solutions.
Check It Out! Example 2b
Solve the inequality and graph the solutions.
Since is added to |x |, subtract from both sides to undo the addition.
Write as a compound inequality. Solve each inequality. OR
x ≤ –6
x ≥ 1
Write as a compound inequality. –7 –6 –5 –4 –3 1 2 3 –2 –1

18. Additional Example 3: Application
A pediatrician recommends that a baby’s bath water be 95°F, but it is acceptable for the temperature to vary from this amount by as much as 3°F. Write and solve an absolute-value inequality to find the range of acceptable temperatures. Graph the solutions.
Let t represent the actual water temperature.
The difference between t and the ideal temperature is at most 3°F.
t – ≤ 3

19. Additional Example 3 Continued
t – 95 ≥ –3 AND t – 95 ≤ 3
Solve the two inequalities.
t ≥ 92 AND t ≤ 98 98
100 96 94 92 90
The range of acceptable temperature is 92 ≤ t ≤ 98.

20. Check It Out! Example 3
A dry-chemical fire extinguisher should be pressurized to 125 psi, but it is acceptable for the pressure to differ from this value by at most 75 psi. Write and solve an absolute-value inequality to find the range of acceptable pressures. Graph the solutions.
Let p represent the desired pressure.
The difference between p and the ideal pressure is at most 75 psi.
p – ≤

21. Check It Out! Example 3 Continued
p – 125 ≥ –75 AND p – 125 ≤ 75
Solve the two inequalities.
p ≥ AND p ≤ 200
200
225
175
150
125
100 75 50 25
The range of pressure is 50 ≤ p ≤ 200.

22. When solving an absolute-value inequality, you may get a statement that is true for all values of the variable. In this case, all real numbers are solutions of the original inequality. If you get a false statement when solving an absolute-value inequality, the original inequality has no solutions.

23. Additional Example 4A: Special Cases of Absolute-Value Inequalities
Solve the inequality.
|x + 4|– 5 > – 8
|x + 4|– 5 > – 8
|x + 4| > –3
Add 5 to both sides.
Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers.
The solution set is all real numbers.

24. Additional Example 4B: Special Cases of Absolute-Value Inequalities
Solve the inequality.
|x – 2| + 9 < 7
|x – 2| + 9 < 7
– 9 – 9
|x – 2| < –2
Subtract 9 from both sides.
Absolute-value expressions are always nonnegative. Therefore, the statement is false for all values of x.
The inequality has no solutions.

25. An absolute value represents a distance, and distance cannot be less than 0.
Remember!

26. Check It Out! Example 4a
Solve the inequality.
|x| – 9 ≥ –11
|x| – 9 ≥ –11
+9 ≥ +9
|x| ≥ –2
Add 9 to both sides.
Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers.
The solution set is all real numbers.

27. Check It Out! Example 4b
Solve the inequality.
4|x – 3.5| ≤ –8
4|x – 3.5| ≤ –8 4
|x – 3.5| ≤ –2
Divide both sides by 4.
Absolute-value expressions are always nonnegative. Therefore, the statement is false for all values of x.
The inequality has no solutions.

28. Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems

29. Solve each inequality and graph the solutions.
Lesson Quiz: Part I
Solve each inequality and graph the solutions.
1. 3|x| > 15
x < –5 OR x > 5 –5
–10 5 10
2. |x + 3| + 1 < 3
–5 < x < –1 –2 –1 –3 –4 –5 –6
3. A number, n, is no more than 7 units away from 5. Write and solve an inequality to show the range of possible values for n.
|n– 5| ≤ 7; –2 ≤ n ≤ 12

30. Lesson Quiz: Part II
Solve each inequality.
4. |3x| + 1 < 1
no solutions
5. |x + 2| – 3 ≥ – 6
all real numbers

31. Lesson Quiz for Student Response Systems
1. Solve the inequality.
2|x| ≤ 10
A. x < –5 OR x > 5
C. x < 5
B. x < 5 OR x > –5
D. no solution

32. Lesson Quiz for Student Response Systems
2. Solve the inequality.
2|x| ≥ 10
A. x < –5 OR x > 5
C. x < 5
B. x < 5 OR x > –5
D. no solution

33. Lesson Quiz for Student Response Systems
3. Solve the inequality.
2|x|+ 3 ≤ 9
A. x < –3 OR x > 3
C. x < –6 OR x > 6
B. x < 3 OR x > –3
D. x < 6 OR x > –6

34. Lesson Quiz for Student Response Systems
4. Graph the inequality.
6|x| ≤ 24 A.
–10 –8 –6 –4 –2 2 4 6 8 10 B.
–10 –8 –6 –4 –2 2 4 6 8 10 C.
–10 –8 –6 –4 –2 2 4 6 8 10 D.
–10 –8 –6 –4 –2 2 4 6 8 10