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

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{2}{\left(x - 1\right) \left(x + 8\right)} + \frac{9}{\left(x - 1\right) \left(x + 1\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 1\right) \left(x + 1\right) \left(x + 8\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{9 x + 72 - (2 x + 2)}{\left(x - 1\right) \left(x + 1\right) \left(x + 8\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{7 x + 70}{\left(x - 1\right) \left(x + 1\right) \left(x + 8\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{-10\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-8, -1, 1\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-10, -8\right) \cup \left(-1, 1\right)$