How do I find supremum and infimum of the set : $\{x\in\Bbb R\mid \exp(-x^2) < 1/2\}$ without using graph?

Question

I tried this approach- $\exp(-x^2) < 1/2$ Taking ln both sides $$-x^2 < \ln(1/2) \iff x^2 > -\ln(1/2)$$ This implies that x is bounded below. But according to graph this is unbounded from above and below. How can I prove this without graph?

Answer 1

Hint:

First note this means $\; x^2>\ln 2\enspace (>0)$.

Next, use a basic result on inequalities:

$$\text{If }\;A\ge 0,\;\text{ then }\quad \begin{cases} x^2<A\iff -\sqrt A<x<\sqrt A,\\[1ex] x^2>A\iff x<-\sqrt A\enspace\text{ or }\enspace x>\sqrt A. \end{cases} $$