# Math Help - Triangle Inequality

1. ## 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?

2. $|y|=|(x+y)-x| \leq |y+x|+|x|$ so $|y|-|x|\leq |x+y|$. Reversing the roles of $x,y$ we get $|x|-|y| \leq |x+y|$. So we have

$|y|-|x| \leq |x+y|$

and

$-(|y|-|x|) \leq |x+y|$

which means...

3. For this we need to notice that:
$|a|\le |b|$ if and only if $-|b|\le a \le |b|$