Solve the inequality $LaTeX: \displaystyle \frac{3}{x^{2} - 4}<\frac{5}{x^{2} - x - 2}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle \frac{3}{\left(x - 2\right) \left(x + 2\right)} - \frac{5}{\left(x - 2\right) \left(x + 1\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 2\right) \left(x + 1\right) \left(x + 2\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{3 x + 3 - (5 x + 10)}{\left(x - 2\right) \left(x + 1\right) \left(x + 2\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{- 2 x - 7}{\left(x - 2\right) \left(x + 1\right) \left(x + 2\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{- \frac{7}{2}\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-2, -1, 2\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, - \frac{7}{2}\right) \cup \left(-2, -1\right) \cup \left(2, \infty\right)$