if $(|a\rangle - |b\rangle)\langle b| + |b\rangle(\langle a| - \langle b|) = 0$ then $|a\rangle = |b\rangle$, where $|a\rangle, |b\rangle$ unit

Question

Essentially the problem above. I've tried approaching it by arguing that $|b\rangle(\langle a| - \langle b|)$ is the adjoint of $(|a\rangle - |b\rangle)\langle b|$, and since $|b\rangle$ nonzero, since unitary, then it must be such that $|a\rangle-|b\rangle = 0$, except, I can't quite formalise that argument, since I cannot factorise it out.

What steps am I missing?

Alternatively, how can it be shown?