I know that for $x,y \in \mathbb{R}$ we have that $$|x-y| \ge ||x|-|y||$$
which can be proven by writing $$|x|=|x+y+(-y)|$$ and $$|y|=|y+x+(-x)|$$ and applying triangle inequality.
But I am wondering if it could also be proven somehow by the following
Noting that $$|x|= max(-x,x)$$ for all $x \in \mathbb{R}$
and doing something like
$$|x+(-y)| \le |x|+|y| \ge |x|-|y|$$
and $$|y+(-x)| \le |y|+|x| \ge |y|-|x|$$
is there any wayI could do something like this for an alternate way of showing it Thanks