Solve the inequality LaTeX: \displaystyle \frac{3}{x^{2} - 25}<\frac{9}{x^{2} - 2 x - 15}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{3}{\left(x - 5\right) \left(x + 5\right)} - \frac{9}{\left(x - 5\right) \left(x + 3\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 5\right) \left(x + 3\right) \left(x + 5\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{3 x + 9 - (9 x + 45)}{\left(x - 5\right) \left(x + 3\right) \left(x + 5\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{- 6 x - 36}{\left(x - 5\right) \left(x + 3\right) \left(x + 5\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{-6\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-5, -3, 5\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -6\right) \cup \left(-5, -3\right) \cup \left(5, \infty\right)