Question

Solve the equation. StartFraction x Over x squared minus 1 EndFraction minus StartFraction x plus 6 Over x squared minus x EndFraction equalsStartFraction negative 6 Over x squared plus x EndFraction

Select the correct choice below and fill in any answer boxes in your choice.

A. The solution set is StartSet nothing EndSet . ​(Simplify your​ answer.)
B. There is no solution.

Answer 1

`x=-12`.

Explanation:

We have been given an equation
`(x)/(x^2-1)-(x+6)/(x^2-x)=-(6)/(x^2+x)`. We are asked to solve the given equation.

Upon rewriting the equation, we will get:

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

Multiply both sides of equation by
`x(x+1)(x-1)`:

`(x)/((x+1)(x-1))*x(x+1)(x-1)-(x+6)/(x(x-1))*x(x+1)(x-1)=-(6)/(x(x+1))*x(x+1)(x-1)`

`x*x-(x+6)(x+1)=-6(x-1)`

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

`x^2-x^2-x-6x-6=-6x+6`

`-7x-6=-6x+6`

`-7x-6+6=-6x+6+6`

`-7x=-6x+12`

`-7x+6x=-6x+6x+12`

`-x=12`

`(-x)/(-1)=(12)/(-1)`

`x=-12`

Therefore,
`x=-12` is the solution of our given equation.