Let $X$ and $Y$ be two independent identically distributed random variables with finite expectation $\Bbb{E}(X) = \Bbb{E}(Y) < \infty$. Prove that
$$\Bbb{E}(|X-Y|) \le \Bbb{E}(|X+Y|)$$
I think that this inequality may follow somehow from Jensen's inequality, but I failed to use it here. Or maybe it is worth considering an expression $|x+y|-|x-y|$ and making use of some of its properties?
I am interested to see a proof of this fact or some favorable ideas that may help here. Any suggestions would be greatly appreciated.