Solve the inequality $LaTeX: \displaystyle \frac{9}{x^{2} - 25}<\frac{9}{x^{2} + 7 x + 10}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{9}{\left(x + 2\right) \left(x + 5\right)} + \frac{9}{\left(x - 5\right) \left(x + 5\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 5\right) \left(x + 2\right) \left(x + 5\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{9 x + 18 - (9 x - 45)}{\left(x - 5\right) \left(x + 2\right) \left(x + 5\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{63}{\left(x - 5\right) \left(x + 2\right) \left(x + 5\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \emptyset$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-5, -2, 5\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -5\right) \cup \left(-2, 5\right)$