For a maximal subgroup $H$ and $x \not\in H$, prove that $G = \langle H, x\rangle$

Question

Let G be a group and H a maximal subgroup of G,

I want to prove that if $x \not\in H$, then $G = \langle H, x\rangle$. I could easily answer in the case G finite, but in general I don't really know what to do ....

Answer 1

Questions:

1. Is $\langle H,x\rangle$ a subgroup of $G$?
2. Is $H\subsetneq \langle H, x\rangle$?

Answer 2

So with the definition you just gave $\langle H,x\rangle$ is a subgroup of $G$ such that $H\subsetneq\langle H,x\rangle$ and because $H$ is maximal we have $\langle H,x\rangle=G$