A general method for solving inequations with absolute values

Question

I've been asked to find which $b$ satisfy $|a + b| = |a| + |b|$ for $a \geq 0$. I'm familiar with the method described here and I tried to apply it but I'm confused about what I should do with the solutions of each equation. Any leads?

Answer 1

Hint: As both sides are non-negative,

$$|a + b| = |a| + |b| \ \ \text{ if and only if } \ \ |a+b|^2 = (|a| + |b|)^2$$

The right-hand side in turn is equivalent to $$a^2 + 2ab + b^2 = |a|^2 + 2|a||b| + |b|^2$$ Simplify that expression and you'll find a condition on $b$. Note that there are two cases: $a = 0$ and $a > 0$.

When you've found those conditions, think about what they mean geometrically on the number line.