Suppose $g$ is a function from $\mathbb{R}\to\mathbb{R}$ with the following properties:
1) $g$ is continuous and increasing
2) $g(x)\leq 1$ for all $x>0$
3) $g(0)=0$
Suppose $X$ is a random variable. Show that $P(|x| > b) \geq E[g(x)] - g(b)$ for all $b\geq0$
I've thought about the Chebychev inequality, but that doesn't help since the inequality sign is in the wrong direction over there. I also tried changing the probability term on the left hand side to Expectations of appropriate indicator functions. Doesn't seem to be going anywhere. Any ideas?