Solve the inequality $LaTeX: \displaystyle \frac{7}{x^{2} - 25}<\frac{5}{x^{2} - 13 x + 40}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle \frac{7}{\left(x - 5\right) \left(x + 5\right)} - \frac{5}{\left(x - 8\right) \left(x - 5\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 8\right) \left(x - 5\right) \left(x + 5\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{7 x - 56 - (5 x + 25)}{\left(x - 8\right) \left(x - 5\right) \left(x + 5\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{2 x - 81}{\left(x - 8\right) \left(x - 5\right) \left(x + 5\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{\frac{81}{2}\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-5, 5, 8\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-5, 5\right) \cup \left(8, \frac{81}{2}\right)$