absolute value

Absolute Value

The absolute value is a function defined as:

$$|a|:=\{\begin{array}{l}a\text{if }\ge 0\\ -a\text{if }<0\end{array}$$

$|a|\ge \pm a$
$\sqrt{x^2}=|x|$

$|a|<b⟺-b<a<b$

Determine the set of all real numbers $x\in \mathbb{R}$ with $||x-2|-2|<1$

$$\begin{array}{rl}& ||x-2|-2|<1\\ & ⟺-1<|x-2|-2<1\\ & ⟺1<|x-2|<3\\ & ⟺-3<x-2<-1\text{or}1<x-2<3\\ & ⟺-1<x<1\text{or}3<x<5\\ & x\in (-1,1)\cup (3,5)\end{array}$$