Negative of Absolute Value

Theorem
Let $x \in \R$ be a real number.

Let $\left|{x}\right|$ be the absolute value of $x$.

Then:
 * $- \left|{x}\right| \le x \le \left|{x}\right|$

Proof
Either $x \ge 0$ or $x < 0$.


 * If $x \ge 0$, then $- \left|{x}\right| \le 0 \le x = \left|{x}\right|$.


 * If $x < 0$, then $- \left|{x}\right| = x < 0 < \left|{x}\right|$.