Question

How do you solve `abs(x - 7)<10`?

Answer 1

We have two situations
(1) `x-7 < 0`
and
(2) `x-7 >= 0`

Case (1)
If `x-7 < 0` which implies `x < 7`
then
`abs(x-7) < 10`
is equivalent to
`7-x < 10`
`-x < 3`
`x > -3`
so the inequality holds if `-3 < x < 7`

Case (2)
If `x-7 >= 0` which implies `x >= 7`
`abs(x-7) < 10`
is equivalent to
`x-7 <10`
`x<17`
so the inequality holds if `7 <= x < 17`

Solution
The solution to the given inequality includes all values in the 2 ranges
`-3 < x < 7`
and
`7 <= x < 17`
which can be combined into a single range
`-3 < x < 17`