Question

How do you solve `(x - 2)/( x + 1) =( x + 1)/(x - 2)`?

Answer 1

`x=1/2`

Explanation:

Cross multiply the terms,

`(x-2)^2 = (x+1)^2`

`|x-2| = |x+1|`

If both x-2 and x+1 are positive or negative, then this has no solution.

If x+1 is negative, then x-2 is also negative. (`x+1<0` means `x<-1` which means maximum value of `x-2` is less than `-1-2=-3` which is also negative)

So our only option is if x-2 is negative and x+1 is positive.

In this case, `|x-2| = 2-x` and `|x+1| = x+1`

We have,

`2-x = x+1`

`2x=1`

`x=1/2`

Answer 2

`x = 1/2`

Explanation:

Given:

`(x-2)/(x+1) = (x+1)/(x-2)`

Multiply both sides by `(x+1)(x-2)` to get:

`(x-2)^2 = (x+1)^2`

Expanding both sides this becomes:

`x^2-4x+4 = x^2+2x+1`

Subtract `x^2` from both sides to get:

`-4x+4 = 2x+1`

Add `4x` to both sides to get:

`4 = 6x+1`

Subtract `1` from both sides to get:

`3 = 6x`

Divide both sides by `6` and transpose to get:

`x = 1/2`