2. CHAPTER 3: Quadratic Functions and Equations; Inequalities
3.1 The Complex Numbers
3.2 Quadratic Equations, Functions, Zeros, and Models
3.3 Analyzing Graphs of Quadratic Functions
3.4 Solving Rational Equations and Radical Equations
3.5 Solving Equations and Inequalities with Absolute Value
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3. 3.5 Solving Equations and Inequalities with Absolute Value
Solve equations with absolute value.
Solve inequalities with absolute value.
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4. Equations with Absolute Value
For a > 0 and an algebraic expression X:
| X | = a
is equivalent to
X = a or X = a.
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5. Copyright © 2009 Pearson Education, Inc.
Example
Solve:
Solution:
The solutions are –5 and 5.
To check, note that –5 and 5 are both 5 units from 0 on the number line.
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6. Copyright © 2009 Pearson Education, Inc.
Example
Solve:
Solution: First, add one to both sides to get the expression in the form | X | = a.
Let’s check the possible solutions –2 and 8.
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7. Copyright © 2009 Pearson Education, Inc.
Example (continued)
The possible solutions are –2 and 8.
Check x = –2:
Check x = 8:
TRUE
TRUE
The solutions are –2 and 8.
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8. 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
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9. 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.
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10. 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
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11. Copyright © 2009 Pearson Education, Inc.
Example
Solve: Solve and graph the solution set:
Solution:
The solution set is
Copyright © 2009 Pearson Education, Inc.

12. Copyright © 2009 Pearson Education, Inc.
Example
Solve: Solve and graph the solution set:
Solution:
The solution set is
Copyright © 2009 Pearson Education, Inc.