# Absolute Value: Equations & Inequalities

## Big Picture

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.

## Key Terms

Absolute Value: The absolute value of a number is the distance of that number from 0.

## Absolute Value Equations

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

Example: $$|ax+b| = c, \text{ where } c ≥ 0$$

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

Example: $$|ax+b| = c$$, where $$c ≥ 0$$

• Split into two equations: $$ax+b = c$$ and $$ax+b = -c$$
• 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: $$|ax + b| < c$$, where $$c ≥ 0$$

• Split into two inequalities: $$ax + b > -c$$ and $$ax + b < c$$ (can be rewritten as $$-c < ax + b < c$$)
• 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: $$|ax + b| > c$$, where $$c ≥ 0$$

• Split into two inequalities: $$ax + b < -c \text{ or } ax + b > c$$
• 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: $$|x + 12| > 2$$

$$x + 12 < -2$$
$$x < -14$$
or
$$x + 12 > 2$$
$$x > -10$$

Example: $$|4x + 5| ≤ 13$$

$$4x + 5 ≥ -13$$
$$4x ≥ -18$$
$$x ≥ - \frac{9}{2}$$
and
$$4x + 5 ≤ 13$$
$$4x ≤ 8$$
$$x ≤ 2$$