# Absolute Value – Definition and in inequality

The mark of absolute value is two straight lines on either side of a mathematical expression. For example, the expression:

$$|x|$$

is called x in absolute value.

Once there is an absolute value, then the expression within the absolute value becomes positive. That is, if it is positive, nothing happens to it, but if it is negative, the absolute value makes it positive. For example,

$$|2|=2$$

$$|-2|=2$$

Absolute Value Definition

$$|x| = \begin{cases} x, &\quad x\geq 0\\ -x, &\quad x < 0\\ \end{cases}$$

When we want to remove an absolute value, we use this definition – if the expression is positive, we simply remove the absolute value, and if the expression is negative, we remove the absolute value and multiply the expression by minus one.

Hence, another definition arises:

$$|x|=\sqrt{x^2}$$

Absolute Value in Inequality

These inequalities are mathematically equivalent:

$$|x|

$$|x|>a \Longleftrightarrow x<-a \text{ or } x>a$$

It is worth remembering them and using them to solve inequalities with absolute value.

Press here for exercises and solutions using absolute value

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