Solve the inequality $LaTeX: \displaystyle \frac{3}{x^{2} - 9}<\frac{9}{x^{2} - 3 x - 18}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle \frac{3}{\left(x - 3\right) \left(x + 3\right)} - \frac{9}{\left(x - 6\right) \left(x + 3\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 6\right) \left(x - 3\right) \left(x + 3\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{3 x - 18 - (9 x - 27)}{\left(x - 6\right) \left(x - 3\right) \left(x + 3\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{9 - 6 x}{\left(x - 6\right) \left(x - 3\right) \left(x + 3\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{\frac{3}{2}\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-3, 3, 6\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -3\right) \cup \left(\frac{3}{2}, 3\right) \cup \left(6, \infty\right)$