Absolute Value Equations

What is Absolute Value?

The absolute value of a number is its distance from zero on the number line.

$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

Examples: $|5| = 5$, $|-3| = 3$, $|0| = 0$

Solving Absolute Value Equations

For $|x| = a$ where $a \geq 0$: $x = a \quad \text{or} \quad x = -a$

Example: Solve $|x| = 7$ $x = 7 \quad \text{or} \quad x = -7$

Multi-Step Absolute Value Equations

For $|ax + b| = c$:

1. Isolate the absolute value
2. Split into two equations: $ax + b = c$ or $ax + b = -c$
3. Solve both equations

Example: Solve $|2x - 1| = 5$

Case 1: $2x - 1 = 5$ → $x = 3$ Case 2: $2x - 1 = -5$ → $x = -2$

Solutions: $x = 3$ or $x = -2$