Absolute Value Equations and Inequalities Study Guide

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Absolute value equations and inequalities are very similar to linear equations and inequalities. In both cases, the goal is to solve for a variable. However, unlike linear equations and linear inequalities, the variable is not a specific number. Instead, the variable represents a specific distance from zero. When solving absolute value equations and inequalities, two options need to be considered: when the expression inside the absolute value is not negative and when the expression inside the absolute value is negative.

Absolute Value Equations

An absolute value equation is an equation that contains an absolute value expression

Example: $| a x + b | = c, where c \geq 0$

To solve an absolute value equation, split it into two equations and solve individually.

Example: $| a x + b | = c$ , where $c \geq 0$

Split into two equations: $a x + b = c$ and $a x + b = - c$
Solve: $x = \frac{c - b}{a}$ and $x = \frac{c + b}{a}$ are both solutions

Do not start to solve until the absolute value equation is splitinto two equations.

Example: $| x - 4 | = 5$

x - 4 = 5

x = 9

and

x - 4 = - 5

x = - 1

Plotted on the number line:

The two solutions are both 5 units away from 4

Absolute Value Inequalities

An absolute value inequality is an inequality that contains an absolute value expression

To solve an absolute value inequality, split into two inequalities, and solve individually.

Example: $| a x + b | < c$ , where $c \geq 0$

Split into two inequalities: $a x + b > - c$ and $a x + b < c$ (can be rewritten as $- c < a x + b < c$ )
Absolute value inequalities can be rewritten as compound inequalities with “and”
Solve: $x > - \frac{c + b}{a}$ and $x < \frac{c - b}{a}$ (can be rewritten as $- \frac{c + b}{a} < x < \frac{c - b}{a}$ )

Example: $| a x + b | > c$ , where $c \geq 0$

Split into two inequalities: $a x + b < - c or a x + b > c$
Absolute value inequalities can be rewritten as compound inequalities with “or”
Solve: $x < - \frac{c + b}{a}$ or $x > \frac{c - b}{a}$

If the sign is less thAN, then it’s a compound inequality with AND. If the sign is greatER than, then it’s a compound equality with OR. Remember: less thAND, greatOR.

Example: $| x + 12 | > 2$

x + 12 < - 2

x < - 14

x + 12 > 2

x > - 10

Example: $| 4 x + 5 | \leq 13$

4 x + 5 \geq - 13

4 x \geq - 18

x \geq - \frac{9}{2}

and

4 x + 5 \leq 13

4 x \leq 8

x \leq 2

Algebra I

Absolute Value: Equations & Inequalities

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Key Terms

Absolute Value Equations

Absolute Value Inequalities

Notes