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

Jan 9, 2016

$x = - 2$

#### Explanation:

$\frac{1}{x + 1} = \frac{x - 1}{x} + \frac{5}{x}$

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

$\frac{1}{x + 1} = \frac{x - 1 + 5}{x}$

$\frac{1}{x + 1} = \frac{x + 4}{x}$

Now, cross multiply.

$x \cdot 1 = \left(x + 4\right) \cdot \left(x + 1\right)$

You will have to FOIL on the right hand side.

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

$0 = {x}^{2} + 4 x + 4$

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.

$0 = {\left(x + 2\right)}^{2}$

$0 = x + 2$

$x = - 2$

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.