[br][color=#000000]An absolute value of [i]x[/i] means positive number describing the distance from zero[/color][color=#000000]. [/color][br][br]  [math]\Large\textcolor{blue}{|x|=}\begin{cases}\textcolor{blue}{x,}& \textcolor{blue}{\text{ if } x\geq 0}\\[br]\textcolor{blue}{-x,} &\textcolor{blue}{\text{ if } x<0}\end{cases}[/math][br][br] [br][br][color=#0000ff]Examples:[/color][br][br]1. [math] |-3|=3[/math][br]2. [math] |\pi-3|=\pi-3[/math], as [math] \pi >3.[/math][br]3. [math] |\sqrt 3-3|=-(\sqrt 3-3)=3-\sqrt 3,[/math] as [math] \sqrt 3 < 3.[/math][br] [br][br][color=#0000ff]Example 4. [/color][br][br][math]\begin{eqnarray} |x-5|&=\begin{cases}x-5,& \text{if } x\geq 5\\-(x-5),&\text{if } x<5\end{cases}\\[br]&=\begin{cases}x-5,&\text{if } x\geq 5\\5-x,&\text{if } x<5\end{cases}\end{eqnarray}[/math][br][br] [br][br][color=#0000ff]Example 5. [color=#000000]Simplify [math] |x-5|+|x+5|,[/math] [/color][/color] if [math] -5\leq x\leq 5.[/math] [br][br]The first term of the expression is always negative on a given interval:[br][br]  [math] |x-5| =-(x-5)=5-x.[/math][br] [br]The second term is always positive, so[br][br]   [math] |x+5| =x+5.[/math][br] [br]Addition of expressions give us[br][br]  [math] |x-5|+|x+5| =(5-x)+(x+5)=5-x+x+5=10.[/math][br] [br][br]