Solve the inequality $LaTeX: \displaystyle \frac{6}{x^{2} - 16}<\frac{9}{x^{2} + 9 x + 20}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{9}{\left(x + 4\right) \left(x + 5\right)} + \frac{6}{\left(x - 4\right) \left(x + 4\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 4\right) \left(x + 4\right) \left(x + 5\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{6 x + 30 - (9 x - 36)}{\left(x - 4\right) \left(x + 4\right) \left(x + 5\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{66 - 3 x}{\left(x - 4\right) \left(x + 4\right) \left(x + 5\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{22\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-5, -4, 4\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -5\right) \cup \left(-4, 4\right) \cup \left(22, \infty\right)$