# Negative of Absolute Value/Corollary 3

## Theorem

Let $x \in \R$.

Let $y \in \R_{\ge 0}$.

Let $z \in \R$.

Then:

$\left\vert{x - z}\right\vert < y \iff z - y < x < z + y$

## Proof

 $\displaystyle \left\vert{x - z}\right\vert$ $<$ $\, \displaystyle y \,$ $\displaystyle$ $\displaystyle \iff \ \$ $\displaystyle -y$ $<$ $\, \displaystyle x - z \,$ $\, \displaystyle <\,$ $\displaystyle y$ Negative of Absolute Value: Corollary 1 $\displaystyle \iff \ \$ $\displaystyle z - y$ $<$ $\, \displaystyle x \,$ $\, \displaystyle <\,$ $\displaystyle z + y$ Real Number Ordering is Compatible with Addition

$\blacksquare$