For any real number $a$, the absolute value (also called the modulus) of $a$, denoted by $\left|a\right|$, is $a$ itself if $a\ge 0$, and $-a$ if $a\le 0$. Thus $\left|a\right|$ is positive except when $a=0$. The following properties hold:

1. (i) $\left|ab\right|=\left|a\right|\left|b\right|.$

2. (ii) $\left|a+b\right|\le \left|a\right|+\left|b\right|.$

3. (iii) $\left|a-b\right|\ge \left|\left|a\right|-\left|b\right|\right|.$

4. (iv) For $a>0$, $\left|x\right|\le a$ if and only if $-a\le x\le a$.

(ii) is the triangle inequality and (iii) is sometimes known as the reverse triangle inequality.

Graph of absolute value