Negative of Absolute Value/Corollary 2

Corollary to Negative of Absolute Value
Let $x, y \in \R$ be a real numbers.

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

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

Necessary Condition
Let $\left|{x}\right| \le y$.

If $\left|{x}\right| < y$ then from Corollary 1 $-y < x < y$.

Thus:
 * $-y \le x \le y$

Otherwise, if $\left|{x}\right| = y$ then either $x = y$ or $-x = y$.

Hence the result.

Sufficient Condition
Let $-y \le x \le y$.

If $-y < x < y$ then from Corollary 1 $\left|{x}\right| < y$.

Hence:
 * $\left|{x}\right| \le y$

Otherwise, if either $-y = x$ or $x = y$ then:
 * $\left|{x}\right| = y$

Hence the result.