Solve the inequality $LaTeX: \displaystyle \frac{2}{x^{2} - 1}<\frac{9}{x^{2} - 6 x - 7}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle \frac{2}{\left(x - 1\right) \left(x + 1\right)} - \frac{9}{\left(x - 7\right) \left(x + 1\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 7\right) \left(x - 1\right) \left(x + 1\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{2 x - 14 - (9 x - 9)}{\left(x - 7\right) \left(x - 1\right) \left(x + 1\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{- 7 x - 5}{\left(x - 7\right) \left(x - 1\right) \left(x + 1\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{- \frac{5}{7}\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-1, 1, 7\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -1\right) \cup \left(- \frac{5}{7}, 1\right) \cup \left(7, \infty\right)$