Prove $||a| - |b||$ is less than or equal to $|a-b|$

Question

I was given the hint to split it into two cases ($|a| - |b|$ being positive and negative) and then use the triangle inequality. However, since the triangle inequality says that $|a+b|$ is less than or equal to $|a| + |b|$, so I don't see how I can use that to help since I'm dealing with subtraction. I've tried using the definition of absolute value, and I was able to find that $||a| - |b||$ = $|a| - |b|$ in my first case, but I couldn't do much with that.

Answer 1

You may write $$ x=x-y+y $$ giving, by the triangle inequality, $$ |x|\leq |x-y|+|y| $$ or

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

then do the same starting this time with $$y=y-x+x.$$

Answer 2

You may prove $(|a|-|b|)^2 \leq (a-b)^2$, since $2|a||b| \geq 2ab$. By taking positive absolute values, you obtain $||a|-|b|| \leq |a-b|$.