Solve the inequality $LaTeX: \displaystyle \frac{7}{x^{2} - 4}<\frac{4}{x^{2} + 2 x - 8}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{4}{\left(x - 2\right) \left(x + 4\right)} + \frac{7}{\left(x - 2\right) \left(x + 2\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 2\right) \left(x + 2\right) \left(x + 4\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{7 x + 28 - (4 x + 8)}{\left(x - 2\right) \left(x + 2\right) \left(x + 4\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{3 x + 20}{\left(x - 2\right) \left(x + 2\right) \left(x + 4\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{- \frac{20}{3}\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-4, -2, 2\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(- \frac{20}{3}, -4\right) \cup \left(-2, 2\right)$