easy inequality to prove Prove that $\log_2(x)+\frac{1-x}{x} > 0$ 
I think the answer is easy but I've no clue how to go about it.

Let $f(x) = x\ln x + 1 - x$, then $f'(x) = \ln x + 1 - 1 = \ln x < 0$ since $x < 1$. So $f$ decreases, and $f(x) > f(1) = 0$, so $$x\ln x + 1 - x > 0\implies \ln x + \frac{1 - x}x > 0.$$
