Solve the inequality LaTeX: \displaystyle \frac{2}{x^{2} - 25}<\frac{3}{x^{2} - 9 x + 20}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{2}{\left(x - 5\right) \left(x + 5\right)} - \frac{3}{\left(x - 5\right) \left(x - 4\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 5\right) \left(x - 4\right) \left(x + 5\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{2 x - 8 - (3 x + 15)}{\left(x - 5\right) \left(x - 4\right) \left(x + 5\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{- x - 23}{\left(x - 5\right) \left(x - 4\right) \left(x + 5\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{-23\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-5, 4, 5\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -23\right) \cup \left(-5, 4\right) \cup \left(5, \infty\right)