From $$x^2\gt 2$$ we can conclude that $$\sqrt{x^2}\gt \sqrt{2}.$$ However, the important point to remember is that $\sqrt{x^2}$ is not equal to $x$, it is equal to $|x|$, the absolute value of $x$. That is, we have $$|x|\gt \sqrt{2}.$$
And, by the definition of the absolute value, $|x|\gt\sqrt{2}$ if and only if $x\gt 0$ and $x\gt \sqrt{2}$, or $x\lt 0$ and $-x\gt\sqrt{2}$, which is equivalent to $x\lt-\sqrt{2}$; so $$|x|\gt\sqrt{2}\text{ is equivalent to }x\gt\sqrt{2}\text{ or }x\lt-\sqrt{2}.$$ You either write it as two inequalities, with an "or" connective, or as the single inequality using the absolute value.