Solve the inequality LaTeX: \displaystyle \frac{5}{x^{2} - 1}<\frac{8}{x^{2} - 8 x - 9}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{5}{\left(x - 1\right) \left(x + 1\right)} - \frac{8}{\left(x - 9\right) \left(x + 1\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 9\right) \left(x - 1\right) \left(x + 1\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{5 x - 45 - (8 x - 8)}{\left(x - 9\right) \left(x - 1\right) \left(x + 1\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{- 3 x - 37}{\left(x - 9\right) \left(x - 1\right) \left(x + 1\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{- \frac{37}{3}\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-1, 1, 9\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, - \frac{37}{3}\right) \cup \left(-1, 1\right) \cup \left(9, \infty\right)