# How do you solve (n+5)/(n+8)=1+6/(n+1) and check for extraneous solutions?

Dec 2, 2016

$n = - \frac{17}{3}$

#### Explanation:

Put everything on the least common denominator of $\left(n + 8\right) \left(n + 1\right)$.

$\frac{\left(n + 5\right) \left(n + 1\right)}{\left(n + 8\right) \left(n + 1\right)} = \frac{1 \left(n + 8\right) \left(n + 1\right)}{\left(n + 8\right) \left(n + 1\right)} + \frac{6 \left(n + 8\right)}{\left(n + 1\right) \left(n + 8\right)}$

We can now solve without the denominators, considering everything is equivalent.

${n}^{2} + 5 n + n + 5 = {n}^{2} + 8 n + n + 8 + 6 n + 48$

$6 n - 8 n - n - 6 n = 56 - 5$

$- 9 n = 51$

$n = - \frac{17}{3}$

Our original restrictions on the variable are $n \ne - 8 , - 1$, and since this solution includes neither of those numbers, the solution is valid.

Hopefully this helps!