Prove that if $\sigma(x) \le 2x$ and $n|x$ where $n\neq x$, then $\sigma(n)<2n$.

Question

Prove that if $\sigma(x) \le 2x$ and $n|x$ where $n\neq x$, then $\sigma(n)<2n$.

So far I tried that if $n|x$, then $\sigma(n)<\sigma(x)\le 2x$, but $2n<2x$, so this doesn't work. Any ideas?