Inequality involving modulus

Question

If $\vert x\vert\leqslant a$ and $\vert y\vert\leqslant b$ can we create some inequality that contains $\vert\vert x\vert-\vert y\vert\vert$?

Answer 1

You can write

$$ ||x|-|y|| \le \frac{|a+b|+|a-b|}{2} $$

Answer 2

For example: $\vert\vert x\vert-\vert y\vert\vert \leq max(a,b)$

Since we can assume that one variable is equal to 0 and the second one equals $max(a,b)$, there's nothing better that we can get.