Solve the inequality LaTeX: \displaystyle \frac{8}{x^{2} - 25}<\frac{9}{x^{2} - 4 x - 45}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{8}{\left(x - 5\right) \left(x + 5\right)} - \frac{9}{\left(x - 9\right) \left(x + 5\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 9\right) \left(x - 5\right) \left(x + 5\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{8 x - 72 - (9 x - 45)}{\left(x - 9\right) \left(x - 5\right) \left(x + 5\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{- x - 27}{\left(x - 9\right) \left(x - 5\right) \left(x + 5\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\{-5, 5, 9\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -27\right) \cup \left(-5, 5\right) \cup \left(9, \infty\right)