# proving an inequality

Let G be a non-abelian finite group, Z(G) be the center of G and $C_{G}(x)$ be the centralizer of the element x in G. Let $x \in G \setminus Z(G)$, so we have $|G| \geq 2|C_{G}(x)|$. Now how can I prove that $|G \setminus C_{G}(x)|>(|G|-|Z(G)|)/2$ ?