Question

How do you solve `abs(x+3)=abs(7-x)`?

Answer 1

One way(Method 1) would be to replace each of the absolute value expressions with 2 possible expressions (one positive and one negative). For the given example there are two absolute value expression and all 4 possible combinations would need to be considered.

An alternative (Method 2) would be to square both sides and solve.

Method 1
`-(x+3) = -(7-x)`
`rarr -2x = -4`
`rarr x = 2`

`-(x+3) = +(7-x)`
`rarr -3 = 7` impossible; extraneous result; ignore

`+(x+3) = -(7-x)`
`rarr 3=-7` Impossible; extraneous result; ignore

`+(x+3) = +(7-x)`
`rarr 2x = 4`
`rarr x = 2` (a duplicate of the second result)

The only solution is `x=2`

Method 2
`(abs(x+3))^2 = (abs(7-x))^2`

`cancel(x^2) +6x + 9 = 49 -14x +cancel(x^2)`
`rarr 20x = 40`
`x=2`