Variation of the triangle inequality on arbitrary normed spaces

Bipolarity
The following inequality can easily be proved on ##ℝ## :

## ||x|-|y|| \leq |x-y| ##

I was wondering if it extends to arbitrary normed linear spaces, since I can't seem to prove it using the axioms for linear spaces. (I can however, prove it using the definition of the norm on ##ℝ## by using casework).

Suggestions? Hints?

BiP

$$\begin{split} & \| x\| = \| (x - y) + y \|, \\ & \| y\| = \| (y - x) + x \|. \end{split}$$