Absolute Value

The absolute value of a real number $x$, written $|x|$, is its distance from $0$ on the number line — always non-negative. Formal definition:

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

Common rules:

• $|ab| = |a||b|$
• $|a/b| = |a|/|b|$ (with $b \neq 0$)
• $|a + b| \leq |a| + |b|$ — the triangle inequality.

Solving $|x - 3| = 5$ requires considering both cases: $x - 3 = 5$ or $x - 3 = -5$, giving $x = 8$ or $x = -2$.

Generalisations: in the complex plane, $|z|$ is the distance from $0$ in 2D. In vector spaces, $|\vec{v}|$ becomes the norm. The absolute value generalises to any structure where "size" or "distance" makes sense.