Negative of Absolute Value/Corollary 2
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Corollary to Negative of Absolute Value
Let $x, y \in \R$ be a real numbers.
Let $\size x$ be the absolute value of $x$.
Then:
- $\size x \le y \iff -y \le x \le y$
that is:
- $\size x \le y \iff \begin {cases} x & \le y \\ -x & \le y \end {cases}$
Proof
Necessary Condition
Let $\size x \le y$.
If $\size x < y$ then from Corollary 1:
- $-y < x < y$
Thus:
- $-y \le x \le y$
Otherwise, if $\size x = y$ then either $x = y$ or $-x = y$.
Hence the result.
$\Box$
Sufficient Condition
Let $-y \le x \le y$.
If $-y < x < y$ then from Corollary 1:
- $\size x < y$
Hence:
- $\size x \le y$
Otherwise, if either $-y = x$ or $x = y$ then:
- $\size x = y$
Hence the result.
$\blacksquare$