Supremum and infimum of an inequality

Question

I feel like this is elementary but I can't seem to determine the supremum/infimum (if they exist) of the following example:

Let $S = \{x \in \mathbb{R} : x^2 > 2x + 8\}$

I rewrote it as $$x^2 -2x - 8 > 0$$ $$(x-4)(x+2) > 0 $$

So it is clear that there is no supremum.

But how do I determine the infimum?

Thanks in advance.

Answer 1

$S = (-\infty,-2)\cup (4,\infty)$, thus $\inf(S) = -\infty$, and $\sup(S) = \infty$.