Although it's simply stated, this is neither a homework problem or trivial (I think, but I'd be happy to be proven wrong :) ).

Let $A,B$ be matrices and $x$ be a vector. Is it true that $$ \|P_{A+B} x\| \geq \|P_A x\| - \|P_B x\|, $$ where $P_A$ is the projection onto the range space of $A$? (or is it true if you square the norms?)

I'm having difficulty even figuring out how to attack this: every attempt I've made falters on the facts that the range space of $A + B$ is not simply related to those of $A$ and $B$ and that the projection is nonlinear. Random instances haven't yet provided counterexamples to the inequality.