Prove one case of the Reverse Triangle Inequality $|x-y|≥|x|-|y|$ for all reals $x$ and $y$

Question

Prove this inequality for all reals $x$ and $y$:

$$|x-y|≥|x|-|y|$$

Answer 1

By the triangle inequality,

$$\begin{align}|x| &= |(x-y) + y|\\ &\le |x-y| + |y|\end{align}$$

Rearrange the terms to get $|x-y| \ge |x| - |y|$.