Solve the inequality $LaTeX: \displaystyle \frac{2}{x^{2} - 9}<\frac{7}{x^{2} + 11 x + 24}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{7}{\left(x + 3\right) \left(x + 8\right)} + \frac{2}{\left(x - 3\right) \left(x + 3\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 3\right) \left(x + 3\right) \left(x + 8\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{2 x + 16 - (7 x - 21)}{\left(x - 3\right) \left(x + 3\right) \left(x + 8\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{37 - 5 x}{\left(x - 3\right) \left(x + 3\right) \left(x + 8\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{\frac{37}{5}\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-8, -3, 3\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -8\right) \cup \left(-3, 3\right) \cup \left(\frac{37}{5}, \infty\right)$