Trying to show $|a|\geq |b|-|b-a|$ for $a,b \in \mathbb{R}$

Question

I'm currently solving a question that boils down to showing this inequality. The only tool I've got in my belt to show this the triangle inequality essentially.

My first step is $|a|=|a-b+b|$, so that I can apply the triangle inequality. I've tried many manipulations on $|a-b+b|$ to prove my inequality but I cannot seem to land on the right step.

Any help would be greatly appreciated!

Answer 1

By the triangle inequality, $|a|+|b-a|\ge|a+b-a|=|b|$.

Now subtract $|b-a|$ from both sides.

Answer 2

|a|-|b|≥-|b-a| since |b| always greater or equal to zero, there are only two cases: case 1: |a|-|b| = |a-b| this happens if a ≥ b case 2: |a|-|b| = |b-a| this happens if a ≤ b furthermore, |a-b|≥ 0 and |b-a| ≥ 0 therefore in both cases |a|-|b| ≥ -|b-a| rearrange again to prove |a|≥|b|-|b-a|