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

Jun 19, 2016

$x = \frac{1}{2}$

#### Explanation:

Cross multiply the terms,

${\left(x - 2\right)}^{2} = {\left(x + 1\right)}^{2}$

$| x - 2 | = | x + 1 |$

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,

$2 - x = x + 1$

$2 x = 1$

$x = \frac{1}{2}$

Jun 19, 2016

$x = \frac{1}{2}$

#### Explanation:

Given:

$\frac{x - 2}{x + 1} = \frac{x + 1}{x - 2}$

Multiply both sides by $\left(x + 1\right) \left(x - 2\right)$ to get:

${\left(x - 2\right)}^{2} = {\left(x + 1\right)}^{2}$

Expanding both sides this becomes:

${x}^{2} - 4 x + 4 = {x}^{2} + 2 x + 1$

Subtract ${x}^{2}$ from both sides to get:

$- 4 x + 4 = 2 x + 1$

Add $4 x$ to both sides to get:

$4 = 6 x + 1$

Subtract $1$ from both sides to get:

$3 = 6 x$

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

$x = \frac{1}{2}$