Triangle Inequality

Given y = (x+y)+(-x),
I have to prove, using triangle inequality that ||x|-|y|| <= |x+y|

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This is what I did:

|y| = |(x+y)+(-x)| <= |x+y|+|-x|

so, |y| <= |x+y|+|-x|

=> |y|-|x| <= |x+y|

How do I get to the conclusion?

so . Reversing the roles of we get . So we have

and

which means...

For this we need to notice that:
if and only if