Solve the inequality $LaTeX: \displaystyle \frac{7}{x^{2} - 1}<\frac{2}{x^{2} + 8 x - 9}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{2}{\left(x - 1\right) \left(x + 9\right)} + \frac{7}{\left(x - 1\right) \left(x + 1\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 1\right) \left(x + 1\right) \left(x + 9\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{7 x + 63 - (2 x + 2)}{\left(x - 1\right) \left(x + 1\right) \left(x + 9\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{5 x + 61}{\left(x - 1\right) \left(x + 1\right) \left(x + 9\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{- \frac{61}{5}\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-9, -1, 1\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(- \frac{61}{5}, -9\right) \cup \left(-1, 1\right)$