Question

How do you solve `\frac { x - 6} { x + 2} + 1= \frac { x - 2} { x + 3}`?

Answer 1

`-8.899` and `0.899`

Explanation:

The first step is to get rid of all the fractions. You can easily do this by every term by the product of every denominator. So you have two denominators: `x+2` and `x+3`.

The product of the denominators is: `(x+2)(x+3)`

Multiplying every single term by this product gives,

`(x+2)(x+3) (x-6)/(x+2)+(x+2)(x+3)(1)=(x+2)(x+3) (x-2)/(x+3)`

Cancel out terms that are in both numerator and denominator:

`cancel((x+2))(x+3) (x-6)/cancel((x+2))+(x+2)(x+3)(1)=(x+2)cancel((x+3)) (x-2)/cancel((x+3))`

This simplifies to

`(x+3) (x-6)+(x+2)(x+3)(1)=(x+2) (x-2)`

On the left hand side, the term `(x+3)` factors out of both terms.

`(x+3)(x-6+x+2)=(x+2)(x-2)`

Simplifying the left even more gives

`(x+3)(2x-4)=(x+2)(x-2)`

Factoring out the left and right gives

`2x^2+8x-12=x^2-4`

`x^2+8x-12=-4`

`x^2+8x-8=0`

This doesn't factor easily, so using the quadratic equation gives

`x=(-8+-sqrt(64-4(1)(-8)))/(2(1))=-4+-sqrt(96)/(2)~~-8.899` and `0.899`