Studying probability theory using Grimmett and Stirzaker's book and their solution for this question was (predictably) sparse, so I'm hoping someone else can explain it:
For $X$ a random variable with moment generating function $M_{X}$, show that:
$P(X\geq x) \leq \inf_{t \geq 0} \left \{e^{-tx}M_{X}(t) \ \right \} $
They used Chernoff's bound. Which itself is derived from Markov's bound.
Markov's bound is (for non-negative random variable $X$) $$\mathbb{P}(X>x)\le \frac {\mathbb{E}(X)}{x}$$ Then
$$\mathbb{P}(e^{tX}>e^{tx})\le \frac {\mathbb{E}(e^{tX})}{e^{tx}}= M_X(t)e^{-tx} $$ Since this is true for all $t > 0 $ (since when $t<0$ the inequality inside the probability gets reversed), you can choose $t$ such that the RHS is minimized.