Let f : $\mathbb{R} \to \mathbb{R}$ be differentiable function and suppose that $|f'(x)|\le 0.49$ for all $x \in \mathbb{R}$. Prove that the equation $f(x) =\frac{2x+\sin(x)}{2}$ has a unique solution in $\mathbb{R}$.
I started by defining a function $g$ where $g(x)= f(x)-\frac{2x+\sin(x)}{2}$ and applying the mean value theorem. I don't know how to move on from there, how should I proceed?