1. 1.8 Solving Absolute-Value Equations and Inequalities
Objective
Write, solve, and graph absolute-value equations and inequalities in mathematical and real-world situations.

2. Glossary Term absolute value
1.8 Solving Absolute-Value Equations and Inequalities
Glossary Term
absolute value

3. Rules and Properties Absolute Value Let x be any real number.
1.8 Solving Absolute-Value Equations and Inequalities
Rules and Properties
Absolute Value
Let x be any real number.
Algebraic definition:
If x  0, thenx= x.
If x  0, thenx= x.
Geometric definition:
The absolute value of x is the distance from
x to 0 on the number line.

4. Rules and Properties Absolute-Value Equations
1.8 Solving Absolute-Value Equations and Inequalities
Rules and Properties
Absolute-Value Equations
If a > 0 and |x| = a, then x = a or x = –a.
Absolute-Value Inequalities
If a > 0 and |x| < a, then x > –a and x < a.
If a > 0 and |x| > a, then x < –a and x > a.
Similar statements are true for |x|  a and |x|  a.

5. Key Skills Solve and graph absolute-value equations. |x + 1| = 7
1.8 Solving Absolute-Value Equations and Inequalities
Key Skills
Solve and graph absolute-value equations.
|x + 1| = 7
x + 1 = 7 or x + 1 = –7
x = 6 or x = –8

6. Key Skills Solve and graph absolute-value equations. 4|x| = 8
1.8 Solving Absolute-Value Equations and Inequalities
Key Skills
Solve and graph absolute-value equations.
4|x| = 8
4x = 8 or 4x = –8
x = 2 or x = –2

7. Key Skills Solve and graph absolute-value inequalities. |x + 52|  76
1.8 Solving Absolute-Value Equations and Inequalities
Key Skills
Solve and graph absolute-value inequalities.
|x + 52|  76
x + 52  76 or –(x + 52)  76
x  24 or x  –128

8. Key Skills Solve and graph absolute-value inequalities. |x + 52|  76
1.8 Solving Absolute-Value Equations and Inequalities
Key Skills
Solve and graph absolute-value inequalities.
|x + 52|  76
x + 52  76 and –(x + 52)  76
x  24 and x  –128
TOC