Solve the inequality LaTeX: \displaystyle \frac{5}{x^{2} - 25}<\frac{6}{x^{2} + 4 x - 45}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{6}{\left(x - 5\right) \left(x + 9\right)} + \frac{5}{\left(x - 5\right) \left(x + 5\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 5\right) \left(x + 5\right) \left(x + 9\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{5 x + 45 - (6 x + 30)}{\left(x - 5\right) \left(x + 5\right) \left(x + 9\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{15 - x}{\left(x - 5\right) \left(x + 5\right) \left(x + 9\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{15\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-9, -5, 5\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -9\right) \cup \left(-5, 5\right) \cup \left(15, \infty\right)