Solve the inequality $LaTeX: \displaystyle \frac{3}{x^{2} - 4}<\frac{7}{x^{2} + 11 x + 18}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{7}{\left(x + 2\right) \left(x + 9\right)} + \frac{3}{\left(x - 2\right) \left(x + 2\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 2\right) \left(x + 2\right) \left(x + 9\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{3 x + 27 - (7 x - 14)}{\left(x - 2\right) \left(x + 2\right) \left(x + 9\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{41 - 4 x}{\left(x - 2\right) \left(x + 2\right) \left(x + 9\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{\frac{41}{4}\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-9, -2, 2\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -9\right) \cup \left(-2, 2\right) \cup \left(\frac{41}{4}, \infty\right)$