How to break apart $\operatorname*P(X^2 < 0.5)$?

Question

I get confused easily whenever I see expression like this:

$\operatorname*P(X^2 < 0.5)$ to $\operatorname*P(X > -\sqrt{0.5})$ and $\operatorname*P(X < \sqrt{0.5})$.

Is there a rule or process to break it down? Currently I am using trial and error, thanks! Any tips would be greatly appreciated!

Answer 1

We have this:

$$x^2<0.5$$

Thus, taking the square root of both sides:

$$|x|=\sqrt{x^2}<\sqrt{0.5}$$

$$|x|<\sqrt{0.5}$$

So, either $x$ is positive or $x$ is negative:

$$\begin{cases}+x<\sqrt{0.5}\\-x<\sqrt{0.5}\end{cases}\implies\begin{cases}x<+\sqrt{0.5}\\x>-\sqrt{0.5}\end{cases}$$

Recalling that signs flip when $x$ is negative and we remove the absolute value bars.