Absolute Value

Distance From Zero

Absolute value answers one question: how far is this number from zero?

We write it with vertical bars: $|x|$

Examples

Absolute value strips away the sign. It tells you how far, not which direction.

Definition

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

The second line might look strange. But if $x$ is negative, then $-x$ is positive.

Example: $|-7| = -(-7) = 7$

Key Properties

• $|x| \geq 0$ — always non-negative
• $|x| = 0$ only when $x = 0$
• $|xy| = |x| \cdot |y|$ — absolute value of a product
• $|x + y| \leq |x| + |y|$ — the triangle inequality