Negative of Absolute Value/Corollary 3

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

$\blacksquare$