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

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: \(|x-4| = 5\)

\(x-4 = 5\)

\(x = 9\)

\(x = 9\)

and

\(x-4 = -5\)

\(x = -1\)

\(x = -1\)

Plotted on the number line:

- The two solutions are both 5 units away from 4

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 \(x > \frac{c - b}{a}\)

**Example:** \(|x + 12| > 2\)

\(x + 12 < -2\)

\(x < -14\)

or

\(x + 12 > 2\)

\(x > -10\)

\(x > -10\)

**Example:** \(|4x + 5| ≤ 13\)

\(4x + 5 ≥ -13\)

\(4x ≥ -18\)

\(x ≥ - \frac{9}{2}\)

\(4x ≥ -18\)

\(x ≥ - \frac{9}{2}\)

and

\(4x + 5 ≤ 13\)

\(4x ≤ 8\)

\(x ≤ 2\)

\(4x ≤ 8\)

\(x ≤ 2\)