Solve the inequality $LaTeX: \displaystyle \frac{8}{x^{2} - 25}<\frac{7}{x^{2} - x - 30}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle \frac{8}{\left(x - 5\right) \left(x + 5\right)} - \frac{7}{\left(x - 6\right) \left(x + 5\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 6\right) \left(x - 5\right) \left(x + 5\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{8 x - 48 - (7 x - 35)}{\left(x - 6\right) \left(x - 5\right) \left(x + 5\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{x - 13}{\left(x - 6\right) \left(x - 5\right) \left(x + 5\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{13\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-5, 5, 6\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-5, 5\right) \cup \left(6, 13\right)$