ThmDex – An index of mathematical definitions, results, and conjectures.

Absolute value strictly less than real number iff in symmetric open interval

Formulation 0

Let $t \mapsto |t|$ be the D412: Absolute value function.
Let $x, a \in \mathbb{R}$ each be a D993: Real number.

Then $|x| < a$ if and only if \begin{equation} - a < x < a \end{equation}

Formulation 1

Let $t \mapsto |t|$ be the D412: Absolute value function.
Let $x, a \in \mathbb{R}$ each be a D993: Real number.

Then $|x| < a$ if and only if \begin{equation} x > -a \quad \text{ and } \quad x < a \end{equation}

Formulation 2

Let $t \mapsto |t|$ be the D412: Absolute value function.
Let $x, a \in \mathbb{R}$ each be a D993: Real number.

Then $|x| < a$ if and only if \begin{equation} x \in (-a, a) \end{equation}