Question

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

Answer 1

You have to use the definition of absolute value:

`|a|=a` if `a>=0`
`|a|=-a` if `a<0`

Explanation:

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

Let's first consider `x>=0`. In this case `|x|=x`, and the inequality reduces to `x<3`. So, the solution are `all` the values `x` that satisfy both `x>=0` and `x<3`, that is `0<=x<3`

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

Putting together both solutions we have `-3 < x < 3`, that is, the interval `(-3, 3)`