Question

What is the solution set for `abs(2x-4)<6`?

Answer 1

`-1 < x < 5`

Explanation:

Consider the 2 possibilities `(2x-4 < 0)` and `(2x-4) >= 0`

Possibility 1
If `2x-4 < 0`
`rArr x < 2`
and
`abs(2x-4) = 4 -2x`
so
`color(white)("XXXX")4 -2x < 6`
`color(white)("XXXX")rArr x > -1`

So with Possibility 1: `-1 < x < 2`

Possibility 2
If `2x-4 >= 0`
`rArr x>=2`
and
`abs(2x-4) = 2x-4`
so
`color(white)("XXXX")2x-4 < 6`
`color(white)("XXXX")rArr x < 5`

So with Possibility 2: `2 <= x < 5`

Combining
Valid possibilities are:
`color(white)("XXXX")-1 < x < 2`
or
`color(white)("XXXX")2 <= x < 5`

`rArr -1 < x < 5`