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:
- $\size {x - z} < y \iff z - y < x < z + y$
Proof
\(\ds \size {x - z}\) | \(<\) | \(\, \ds y \, \) | \(\ds \) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds -y\) | \(<\) | \(\, \ds x - z \, \) | \(\, \ds < \, \) | \(\ds y\) | Negative of Absolute Value: Corollary 1 | ||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds z - y\) | \(<\) | \(\, \ds x \, \) | \(\, \ds < \, \) | \(\ds z + y\) | Real Number Ordering is Compatible with Addition |
$\blacksquare$