Question

How do you solve `|x + 3| = abs(x-2)`?

Answer 1

`x=-1/2`

Explanation:

`|x+3| = |x-2|`

`=> x+3 = |x-2|`
or
`-(x+3) = |x-2|`

`=> x+3 = x-2`
or
`x+3 = -(x-2)`
or
`-(x+3)=x-2`
or
`-(x+3)=-(x-2)`

The first and fourth equations have no solution as you can cancel `x` (or `-x`) from both sides to obtain `2=-3`, which is a contradiction.

The second and third equations have the same solution, as one can be obtained from the other by multiplying both sides by `-1`. Then, we may simply solve

`x+3 = -(x-2)`

`=> x+3 = -x+2`

`=>2x=-1`

`:.x = -1/2`