Solve the inequality LaTeX: \displaystyle \frac{9}{x^{2} - 25}<\frac{5}{x^{2} - 14 x + 45}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{9}{\left(x - 5\right) \left(x + 5\right)} - \frac{5}{\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{9 x - 81 - (5 x + 25)}{\left(x - 9\right) \left(x - 5\right) \left(x + 5\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{4 x - 106}{\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\{\frac{53}{2}\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(-5, 5\right) \cup \left(9, \frac{53}{2}\right)