Definition and Properties

Symmetry. For any $x\in R$, $|x|=|-x|$ (both measure the same distance from $0$). Consequently $|x|=|y|$ exactly when $x=y$ or $x=-y$.

Equation form. $|x-a|=r$ if and only if $r\ge0$ and ($x-a=r$ or $x-a=-r$) -- i.e. $x=a+r$ or $x=a-r$. If $r<0$, the equation has no solution at all, since an absolute value can never equal a negative number. …