Solve the inequality $LaTeX: \displaystyle \frac{7}{x^{2} - 9}<\frac{8}{x^{2} + 7 x + 12}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{8}{\left(x + 3\right) \left(x + 4\right)} + \frac{7}{\left(x - 3\right) \left(x + 3\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 3\right) \left(x + 3\right) \left(x + 4\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{7 x + 28 - (8 x - 24)}{\left(x - 3\right) \left(x + 3\right) \left(x + 4\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{52 - x}{\left(x - 3\right) \left(x + 3\right) \left(x + 4\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{52\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-4, -3, 3\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -4\right) \cup \left(-3, 3\right) \cup \left(52, \infty\right)$