Solve the inequality $LaTeX: \displaystyle \frac{6}{x^{2} - 25}<\frac{7}{x^{2} - 8 x + 15}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle \frac{6}{\left(x - 5\right) \left(x + 5\right)} - \frac{7}{\left(x - 5\right) \left(x - 3\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 5\right) \left(x - 3\right) \left(x + 5\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{6 x - 18 - (7 x + 35)}{\left(x - 5\right) \left(x - 3\right) \left(x + 5\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{- x - 53}{\left(x - 5\right) \left(x - 3\right) \left(x + 5\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{-53\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-5, 3, 5\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -53\right) \cup \left(-5, 3\right) \cup \left(5, \infty\right)$