# Absolute Value

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# Definition

 Absolute Value Is the value for a real number, disregarding the sign. Written $\left|x\right|$. Mathematically defined as: $\left|x\right|=x\text { if }x\geqq 0$ $\left|x\right|=-x\text { if }x<0$ Geometrically, $\left|x\right|$ is the distance of $x \text { to } 0\,$, and from $-x \text { to } 0\,$.

## Examples

$\left|+3\right| = 3$

$\left|-45\right| = 45$

$\left|0\right| = 0$

$\left|+5\right| =\left|-5\right| = 5$

### Absolute value properties

$\text{1) if }\left|x\right| = a\qquad\text{then}\quad x=a \qquad\text{or}\quad x=-a$

$\text{2) if }\left|x\right| < a\qquad\text{then}\quad -a

$\text{3) if }\left|x\right| > a\qquad\text{then}\quad x>a \qquad\text{or}\quad x<-a$

$\text{4) }\left|x\right|^2 = x ^ 2$

$\text{5) }\left|x\right| = sqr(x^2)$