Chapter 1 - Equations and Inequalities - 1.8 Absolute Value Equations and Inequalities - 1.8 Exercises - Page 167: 42

See explanations.

Step 1. If $x\ge0$, we have $LHS=|x|=x$ and $RHS=\sqrt {x^2}=x=LHS$ Step 2. If $x\lt0$, we have $LHS=|x|=-x$ and $RHS=\sqrt {x^2}=\sqrt {(-x)^2}=-x=LHS$ Step 3. Thus for any $x\in(-\infty,\infty)$, $LHS=RHS$