Answer
Domain: $(-\infty,-1)\cup(5,\infty)$
Work Step by Step
Logarithmic functions are defined for positive arguments only.
$ f(x)=\displaystyle \ln(\frac{x+1}{x-5})$ is defined when $\displaystyle \frac{x+1}{x-5}>0.$
The boundary points are $-1$ and $5.$
Testing each interval $:$
$\left[\begin{array}{ccccccc}
& -\infty & & -1 & & 5 & & \infty\\
\text{test point} & & -10 & & 0 & & 10 & \\
f(x) & & \frac{-9}{-15} & & \frac{1}{-5} & & \frac{11}{5} & \\
\text{sign} & & + & & - & & + &
\end{array}\right]$
Domain: $(-\infty,-1)\cup(5,\infty)$ .
