Absolute Value

For real numbers $a,b$ and positive real number $k$ :

$\left\lvert a \right\rvert \equiv\begin{cases} a, & \text{if $a\ge 0$} \\ -a, & \text{if $a<0$} \\ \end{cases}$   Alternatively,$\qquad\left\lvert a \right\rvert \equiv\sqrt{a^2}$

$$\left\lvert a \right\rvert \ge 0 \left\lvert-a \right\rvert =\left\lvert a \right\rvert \left\lvert a \right\rvert \cdot\left\lvert b \right\rvert =\left\lvert ab \right\rvert$$

$$\bigg\vert\frac{a}{b}\bigg\vert=\frac{\left\lvert a \right\rvert }{\left\lvert b \right\rvert },\;\; b\ne 0 \left\lvert a+b \right\rvert \le\left\lvert a \right\rvert +\left\lvert b \right\rvert \left\lvert a^n \right\rvert =\left\lvert a \right\rvert ^n$$

$$-\left\lvert a \right\rvert \le a\le \left\lvert a \right\rvert \underbrace{\left\lvert a \right\rvert \le k\iff -k\le a\le k}_{\text{also true if}\;\le\;\text{is replaced by}\;<} \underbrace{k\le\left\lvert a \right\rvert \iff k\le a \;\;\text{or}\;\; a\le -k}_{\text{also true if}\;\le\;\text{is replaced by}\;<}$$

$$\left\lvert a \right\rvert =b \iff a=b\;\;\textrm{or}\;\;a=-b$$