Solve the inequality LaTeX: \displaystyle \frac{9}{x^{2} - 25}<\frac{9}{x^{2} + 2 x - 35}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{9}{\left(x - 5\right) \left(x + 7\right)} + \frac{9}{\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 + 7\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{9 x + 63 - (9 x + 45)}{\left(x - 5\right) \left(x + 5\right) \left(x + 7\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{18}{\left(x - 5\right) \left(x + 5\right) \left(x + 7\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \emptyset , or the zeros of the denominator LaTeX: \displaystyle \left\{-7, -5, 5\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -7\right) \cup \left(-5, 5\right)