Absolute Value

The absolute value of a real number x is a nonnegative real number denoted by $| x |$ , and defined as follows:

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

Quick Reference

Concept	Formula	Notes
Definition	$\| x \| = x$ if $x \geq 0$ , $\| x \| = - x$ if $x < 0$	Always non-negative
Bounded Inequality	$\| x \| \leq r ⟺ - r \leq x \leq r$	Valid for $r > 0$
Exterior Inequality	$r \leq \| x \| ⟺ x \leq - r or r \leq x$	Valid for $r > 0$
Triangle Inequality	$\| a + b \| \leq \| a \| + \| b \|$	Fundamental upper bound for sums

For arbitrary numbers a and b:

$| a | \geq 0$
$| a | = 0$ if and only if $a = 0$
$\sqrt{a^{2}} = | a |$
$| - a | = | a |$
$| a b | = | a | | b |$
$| \frac{a}{b} | = \frac{| a |}{| b |}$ (provided $b \neq 0$ )
If $r > 0$ :
\begin{aligned} |x| \le r &\quad\text{is equivalent to}\quad -r \le x \le r &&\text{(i)} \\ r \le |x| &\quad\text{is equivalent to}\quad x \le -r \quad\text{or}\quad r \le x && \text{(ii)} \\ |x| = r &\quad\text{is equivalent to}\quad x = r \quad\text{or}\quad x = -r && \text{(iii)} \end{aligned}
$| a + b | \leq | a | + | b |$ (known as the triangle inequality)