How do I turn an inequality involving only |a| and |b| into an inequality involving |b + a| and |b - a|?

Question

I'm working on a proof which involves showing that if $|a| < 2|b|$, then either $|b + a| < |b|$ or $|b - a| < |b|$. I've tried using contradiction (assuming that both $|b + a|,|b - a| \geq |b|$ and then various techniques (in particular, the reverse triangle inequality), but I've had no luck on this; most of my arguments just end up showing that an absolute value is nonnegative. Could anyone give a hint as to the direction I should be pursuing?

Answer 1

Try using the following:

If $$sign(a) = sign(b)$$ then: $$|a| + |b| = |a + b|$$ otherwise: $$|a| + |b| = |a - b|$$