Distance to the complementary of a convex set.

Question

Let $E$ be a subset of $\mathbb{R}^n$ such that $\mathbb{R}^n \setminus E$ is convex.

Let $x,y$ be in $\mathbb{R}^n$. Is it true that for $t\in [0,1]$, we have: $$d(tx+(1-t)y,E) \geq td(x,E) - (1-t)d(y,\mathbb{R}^n \setminus E)?$$ If yes, do you know why?