Solve the inequality $LaTeX: \displaystyle \frac{4}{x^{2} - 16}<\frac{7}{x^{2} + x - 20}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{7}{\left(x - 4\right) \left(x + 5\right)} + \frac{4}{\left(x - 4\right) \left(x + 4\right)}< 0$ . 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 $LaTeX: \displaystyle \frac{4 x + 20 - (7 x + 28)}{\left(x - 4\right) \left(x + 4\right) \left(x + 5\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{- 3 x - 8}{\left(x - 4\right) \left(x + 4\right) \left(x + 5\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{- \frac{8}{3}\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-5, -4, 4\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -5\right) \cup \left(-4, - \frac{8}{3}\right) \cup \left(4, \infty\right)$