Solve the inequality LaTeX: \displaystyle \frac{4}{x^{2} - 9}<\frac{6}{x^{2} - 5 x - 24}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{4}{\left(x - 3\right) \left(x + 3\right)} - \frac{6}{\left(x - 8\right) \left(x + 3\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 8\right) \left(x - 3\right) \left(x + 3\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{4 x - 32 - (6 x - 18)}{\left(x - 8\right) \left(x - 3\right) \left(x + 3\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{- 2 x - 14}{\left(x - 8\right) \left(x - 3\right) \left(x + 3\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{-7\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-3, 3, 8\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -7\right) \cup \left(-3, 3\right) \cup \left(8, \infty\right)