Check that $|x|=\max\{-x,x\}=\max\{-x,x,0\}$

Question

Let $x \in \mathbb{R}$, $|x|=\max\{-x,x\}$, is correct also $|x|=\max\{-x,0,x\}$?

Answer 1

$|x|\geq 0$, because $x\geq 0$ or, when $x<0$, $-x>0$. Thus, $0\leq \max\{x,-x\}$, so the extra $0$ is ok.

Answer 2

Yes, and $|x|=\max\{-x,-42,x\}$.