Absolute Value

Definition

Definition

We define the absolute value of a real $x$ by $|x| = \begin{cases} x \quad & \text{if } x \geqslant 0, \\ -x \quad & \text{if } x \leqslant 0. \end{cases}$

Results

Proposition

Let $a \in \mathbb R$ and $h \in \mathbb R_+$. For all real number $x \in \mathbb R$, we have $|x - a| \leqslant h$ if, and only if, $a-h \leqslant x \leqslant a+h$.

Proposition : Triangle inequalities

If $x$ and $y$ are any two real numbers, then we have :

$$|x + y| \leqslant |x| + |y| \qquad \text{and} \qquad ||x| - |y|| \leqslant |x - y|.$$