Solution Set Inequality Continuous Function, Probability Theory
Solution 1:
If $f:\mathcal{Y}\subseteq \mathbb{R}\to\mathbb{R}$ is a strictly increasing function, $$ y>c \Leftrightarrow f(y)>f(c). $$ Thus, for a random variable $Y$, $$ \{\omega: Y(\omega)>c\} =\{\omega: f(Y(\omega))>f(c)\}. $$ In your case, $Y=|X-\mathsf{E}X|$ and $f(y)=y^2$ which is strictly increasing on $\mathcal{Y}=\mathbb{R}_{>0}$.