Solve the inequality $LaTeX: \displaystyle \frac{6}{x^{2} - 25}<\frac{3}{x^{2} - 4 x - 5}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle \frac{6}{\left(x - 5\right) \left(x + 5\right)} - \frac{3}{\left(x - 5\right) \left(x + 1\right)}< 0$ . 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 $LaTeX: \displaystyle \frac{6 x + 6 - (3 x + 15)}{\left(x - 5\right) \left(x + 1\right) \left(x + 5\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{3 x - 9}{\left(x - 5\right) \left(x + 1\right) \left(x + 5\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{3\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-5, -1, 5\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-5, -1\right) \cup \left(3, 5\right)$