# How do you solve and write the following in interval notation: |x| <3?

Jul 24, 2016

You have to use the definition of absolute value:

$| a | = a$ if $a \ge 0$
$| a | = - a$ if $a < 0$

#### Explanation:

For example $| 5 | = 5$, $| - 5 | = 5$ and $| 0 | = 0$

Let's first consider $x \ge 0$. In this case $| x | = x$, and the inequality reduces to $x < 3$. So, the solution are $a l l$ the values $x$ that satisfy both $x \ge 0$ and $x < 3$, that is $0 \le x < 3$

If $x < 0$, then $| x | = - x$, and the inequality reduces to $- x < 3$, and so $- 3 < x$. Thus the solution are $a l l$ the values $x$ that satisfy $- 3 < x < 0$

Putting together both solutions we have $- 3 < x < 3$, that is, the interval $\left(- 3 , 3\right)$