1. Absolute Value Equations and Inequalities
Section 3-7 Part 2

2. Goals Goal Rubric To solve inequalities involving absolute value.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
None

4. Absolute Value Inequalities
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.

5. “Less Than” Absolute Value Inequalities

6. Example: “Less Than” Absolute Value Inequalities
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 < x < 2}.
Write as a compound inequality.
The solution set is
x > –2 AND x < 2 –2 –1 1 2
2 units

7. Example: “Less Than” Absolute Value Inequalities
Solve the inequality and graph the solutions.
|x – 1| ≤ 2
x – 1 ≥ –2 AND x – 1 ≤ 2
Write as a compound inequality.
Solve each inequality.
x ≥ –1
x ≤ 3
AND
{x: –1 ≤ x ≤ 3}.
Write as a compound inequality.
The solution set is –2 –1 1 2 3 –3

8. Procedure Solving “Less Than” Inequalities
Solving Absolute Inequalities:
Isolate the absolute value.
Break it into 2 inequalities (and statement) – one positive and the other negative reversing the sign.
Solve both inequalities.
Check both answers.

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. Your Turn: Solve the inequality and graph the solutions. 2|x| ≤ 6
Since x is multiplied by 2, divide both sides by 2 to undo the multiplication.
|x| ≤ 3
{x: –3 ≤ x ≤ 3}.
Write as a compound inequality.
The solution set is
x ≥ –3 AND x ≤ 3 –2 –1 1 2
3 units –3 3

11. Your Turn: Solve each inequality and graph the solutions.
|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
Write as a compound inequality.
–3 –3
–3 –3
x ≥ –15 AND x ≤ 9
{x: –15 ≤ x ≤ 9}.
The solution set is
–20
–15
–10 –5 5 10 15

12. “Greater Than” Absolute Value Inequalities
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.

13. “Greater Than” Absolute Value Inequalities

14. Example: “Greater Than” Absolute Value Inequalities
Solve the inequality and graph the solutions.
|x| + 14 ≥ 19
– 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. Example: “Greater Than” Absolute Value Inequalities
Solve the inequality and graph the solutions.
3 + |x + 2| > 5
Since 3 is added to |x + 2|, subtract 3 from both sides to undo the addition.
|x + 2| > 2
– – 3
3 + |x + 2| > 5
Write as a compound inequality. Solve each inequality.
x + 2 < –2 OR x + 2 > 2
–2 –2
–2 –2
x < –4 OR x > 0
Write as a compound inequality.
The solution set is
{x: x < –4 or x > 0}.
–10 –8 –6 –4 –2 2 4 6 8 10

16. Procedure Solving “Greater Than” Inequalities
Solving Absolute Inequalities:
Isolate the absolute value.
Break it into 2 inequalities (or statement) – one positive and the other negative reversing the sign.
Solve both inequalities.
Check both answers.

17. Your Turn: Solve each inequality and graph the solutions.
|x| + 10 ≥ 12
|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. The solution set is {x: x ≤ –2 or x ≥ 2}.
2 units –5 –4 –3 –2 –1 1 2 3 4 5

18. Your Turn: 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
Write as a compound inequality. The solution set is {x: x ≤ –6 or x ≥ 1}
x ≤ –6
x ≥ 1 –7 –6 –5 –4 –3 1 2 3 –2 –1

19. Example: 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

20. Example: Continued t – 95 ≤ 3 |t – 95| ≤ 3 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.

21. Your Turn:
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 solution.
Let p represent the desired pressure.
The difference between p and the ideal pressure is at most 75 psi.
p – ≤

22. Your Turn: Continued p – 125 ≤ 75 |p – 125| ≤ 75
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.

23. Solutions to Absolute Value Inequalities
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. Its solution set is ø.

24. Example: 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.

25. Example: 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. The solution set is ø.

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

27. Your Turn: Solve the inequality. |x| – 9 ≥ –11 |x| – 9 ≥ –11 +9 ≥ +9
+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.

28. Your Turn: Solve the inequality. 4|x – 3.5| ≤ –8 4|x – 3.5| ≤ –8 4
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. The solution set is ø.

29. Summary

30. Absolute Value Inequalities (< , )
Inequalities of the Form < or  Involving Absolute Value
If a is a positive real number and if u is an algebraic expression, then
|u| < a is equivalent to u < a and u > -a
|u|  a is equivalent to u  a and u > -a
Note: If a = 0, |u| < 0 has no real solution, |u|  0 is equivalent to u = 0. If a < 0, the inequality has no real solution.

31. Absolute Value Inequalities (> , )
Inequalities of the Form > or  Involving Absolute Value
If a is a positive real number and u is an algebraic expression, then
|u| > a is equivalent to u < – a or u > a
|u|  a is equivalent to u  – a or u  a.

32. Procedure Solving Absolute Inequalities: Isolate the absolute value.
Break it into 2 inequalites (“< , ≤ and statement” – “> , ≥ or statement”) – one positive and the other negative reversing the sign.
Solve both inequalities.
Check both answers.

33. Solving an Absolute-Value Equation
Recall that | x | is the distance between x and 0. If | x |  8, then any number between 8 and 8 is a solution of the inequality.
Recall that x is the distance between x and 0. If x  8, then any number between 8 and 8 is a solution of the inequality.
Recall that | x | is the distance between x and 0. If | x |  8, then any number between 8 and 8 is a solution of the inequality.
 8  7  6  5  4  3  2 
You can use the following properties to solve absolute-value inequalities and equations.

34. SOLVING ABSOLUTE-VALUE INEQUALITIES
means
| a x  b |  c
means
a x  b  c and a x  b   c.
When an absolute value is less than a number, the
means
| a x  b |  c
means
a x  b  c and a x  b   c.
inequalities are connected by and. When an absolute
value is greater than a number, the inequalities are
connected by or.
means
| a x  b |  c
means
a x  b  c or a x  b   c.
means
| a x  b |  c
means
a x  b  c or a x  b   c.

35. Solving an Absolute-Value Inequality
Solve | x  4 | < 3
x  4 IS POSITIVE
x  4 IS NEGATIVE
| x  4 |  3
| x  4 |  3
x  4  3
x  4  3 
x  7
x  1
Reverse inequality symbol.
The solution is all real numbers greater than 1 and less than 7.
This can be written as 1  x  7.
–10 –8 –6 –4 –2 2 4 6 8 10

36. Solve | 2x  1 | 3  6 and graph the solution.
Solving an Absolute-Value Inequality
| 2x  1 |  3  6
| 2x  1 |  9
2x  1  +9
x  4
2x  8
| 2x  1 | 3  6
2x  1  9
2x  10
x  5
2x + 1 IS POSITIVE
2x + 1 IS NEGATIVE
Solve | 2x  1 | 3  6 and graph the solution.
2x + 1 IS POSITIVE
2x + 1 IS NEGATIVE
| 2x  1 |  3  6
| 2x  1 | 3  6
| 2x  1 |  9
| 2x  1 |  9
2x  1  +9
2x  1  9 
2x  8
2x  10
The solution is all real numbers greater than or equal to 4 or less than or equal to  5. This can be written as the compound inequality x   5 or x  4.
x  4
x  5
Reverse inequality symbol.
 5 4.
 6  5  4  3  2 

37. Joke Time What's the difference between chopped beef and pea soup?
Everyone can chop beef, but not everyone can pea soup!
What do you get when you cross an elephant and a rhino?
el-if-i-no
What’s a quick way to double your money?
You fold it!

38. Assignment
3.7 pt 2 Exercises Pg. 228 – 229: #6 – 42 even