Question

How do you solve `|x| <3`?

Answer 1

You have to use the definition of `|x|`:
`|x|=x`, if `x>=0`
`|x|=-x`, if `x<0`

Explanation:

So we first consider `x>=0`. In this case `|x|=x`, and the inequality becomes `x<3`. Hence, all `x` such that `x>=0` and `x<3` satisfy the inequality. That is all `x`, `0<=x<3`

Now consider `x<0`; is this case `|x|=-x`, and the inequality becomes `-x<3`. This is the same as `-3 < x`, so all the `x` such that `-3 < x < 0` satisfy the inequality.

Putting both together, the solution are all `x` such that `-3 < x <3`