Negative of Absolute Value/Corollary 3

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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

$\blacksquare$