Negative of Absolute Value/Corollary 2

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

Let $\size x$ be the absolute value of $x$.

Then:
 * $\size x \le y \iff -y \le x \le y$

that is:
 * $\size x \le y \iff \begin {cases} x & \le y \\ -x & \le y \end {cases}$

Necessary Condition
Let $\size x \le y$.

If $\size x < y$ then from Corollary 1:
 * $-y < x < y$

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

Otherwise, if $\size x = y$ 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:
 * $\size x < y$

Hence:
 * $\size x \le y$

Otherwise, if either $-y = x$ or $x = y$ then:
 * $\size x = y$

Hence the result.