Solve the inequality $LaTeX: \displaystyle \frac{7}{x^{2} - 25}<\frac{8}{x^{2} - 4 x - 5}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle \frac{7}{\left(x - 5\right) \left(x + 5\right)} - \frac{8}{\left(x - 5\right) \left(x + 1\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 5\right) \left(x + 1\right) \left(x + 5\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{7 x + 7 - (8 x + 40)}{\left(x - 5\right) \left(x + 1\right) \left(x + 5\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{- x - 33}{\left(x - 5\right) \left(x + 1\right) \left(x + 5\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{-33\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-5, -1, 5\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -33\right) \cup \left(-5, -1\right) \cup \left(5, \infty\right)$