Question

How do you solve the rational equation `1 / (x+1) = (x-1)/ x + 5/x`?

Answer 1

`x=-2`

Explanation:

`1/(x+1)=(x-1)/x+5/x`

First, recognize that the fractions on the right hand side can be added since they have the same denominator.

`1/(x+1)=(x-1+5)/x`

`1/(x+1)=(x+4)/x`

Now, cross multiply.

`x*1=(x+4)*(x+1)`

You will have to FOIL on the right hand side.

`x=x^2+x+4x+4`

`0=x^2+4x+4`

From here, you could find the roots by using the quadratic formula, completing the square, or simply by factoring and recognizing this is a perfect square trinomial.

`0=(x+2)^2`

`0=x+2`

`x=-2`

When working with rational functions, always check that this won't cause any domain errors (making a denominator equal `0`). In this case, that won't happen.