1. Solving Equations and Inequalities with Absolute Value
Section 3.5
Solving Equations and Inequalities with Absolute Value
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2. Objectives Solve equations with absolute value.
Solve inequalities with absolute value.

3. Equations with Absolute Value
For a > 0 and an algebraic expression X: | X | = a is equivalent to X = a or X = a.

4. Example Solve: The solutions are –5 and 5.
To check, note that –5 and 5 are both 5 units from 0 on the number line.

5. Example
Solve:
First, add one to both sides to get the expression in the form | X | = a.
Let’s check the possible solutions –2 and 8.

6. Example (continued) The possible solutions are –2 and 8. Check x = –2:
TRUE
TRUE
The solutions are –2 and 8.

7. More About Absolute Value Equations
When a = 0, | X | = a is equivalent to X = 0. Note that for a < 0, | X | = a has no solution, because the absolute value of an expression is never negative. The solution is the empty set, denoted

8. Inequalities with Absolute Value
Inequalities sometimes contain absolute-value notation. The following properties are used to solve them. For a > 0 and an algebraic expression X: | X | < a is equivalent to a < X < a. | X | > a is equivalent to X < a or X > a. Similar statements hold for | X |  a and | X |  a.

9. Inequalities with Absolute Value
For example, | x | < 3 is equivalent to 3 < x < 3 | y | ≥ 1 is equivalent to y ≤ 1 or y ≥ 1 | 2x + 3 | ≤ 4 is equivalent to 4 < 2x + 3 < 4

10. Example
Solve and graph the solution set:
The solution set is

11. Example
Solve and graph the solution set:
The solution set is