Thomas' Calculus 13th Edition

$x\in(-\sqrt{2},\sqrt{2})$
The square root function is defined for nonnegative real numbers, and is an increasing function. This means that if a and b are nonnegative, $a \lt b\Rightarrow\sqrt{a} \lt \sqrt{b}$ An alternate definition of the absolute value is $|x|=\sqrt{x^{2}}$. Both sides of the inequality sign are nonnegative, so we may take the square root of both sides, with the direction of inequality being unchanged: $\sqrt{x^{2}} \lt \sqrt{2}$ $|x| \lt \sqrt{2}$ by property 6 from the table 'Absolute Values and Intervals", $-\sqrt{2} \lt x \lt \sqrt{2}$ In interval form, we can write this result as $x\in(-\sqrt{2},\sqrt{2})$