$||x|-|y||\leq|x-y|\Rightarrow |x-y|\geq|x|-|y|$ and $|x+y|\geq|x|-|y|$.

Question

I think I can prove the inequality but in order to do so I Need to understand whether if

$|a|>|b|$ then $|a|> b$ and $|a| > - b (*)$

My proof would be then

$||x|-|y||\leq|x-y|\Rightarrow |x|-|y|\leq|x-y|$

And then one can choose for $y$ its negative value and would get

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

If my idea is Right please help me to prove $(*)$

Otherwise I would like a hint so I can find it out myself

Answer 1

We know that $|b|=\max\{b, -b\}$, that is $|b| \ge b$ and $|b| \ge -b$.

Hence $|a| > |b| \ge b$, that is we have $|a| > b$.

Similarly for $-b$.

Answer 2

Because by the triangle inequality

$$|x-y|+|y|\geq|x-y+y|\geq|x|,$$ which gives $$|x-y|\geq|x|-|y|$$ and by the triangle inequality again: $$|x+y|+|y|=|x+y|+|-y|\geq|x+y-y|=|x|,$$ which gives $$|x+y|\geq|x|-|y|.$$