Negative of Absolute Value/Corollary 1

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

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

Then from Negative of Absolute Value:
 * $x \le \left|{x}\right|$

and:
 * $\left|{x}\right| \ge -x$

So $x < y$ and $-x < y$, and so $x > -y$ from Ordering of Inverses in Ordered Monoid.

It follows that $-y < x < y$.

Sufficient Condition
Let $-y < x < y$.

Then $x < y$ and $-x < y$.

For all $x$:
 * $\left|{x}\right| = x$

or:
 * $\left|{x}\right| = -x$

Thus it follows that $\left|{x}\right| < y$.