Solve the inequality $LaTeX: \displaystyle \frac{2}{x^{2} - 9}<\frac{3}{x^{2} + 12 x + 27}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{3}{\left(x + 3\right) \left(x + 9\right)} + \frac{2}{\left(x - 3\right) \left(x + 3\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 3\right) \left(x + 3\right) \left(x + 9\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{2 x + 18 - (3 x - 9)}{\left(x - 3\right) \left(x + 3\right) \left(x + 9\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{27 - x}{\left(x - 3\right) \left(x + 3\right) \left(x + 9\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{27\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-9, -3, 3\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -9\right) \cup \left(-3, 3\right) \cup \left(27, \infty\right)$