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

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{7}{\left(x + 2\right) \left(x + 5\right)} + \frac{2}{\left(x - 5\right) \left(x + 5\right)}< 0$ . 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 $LaTeX: \displaystyle \frac{2 x + 4 - (7 x - 35)}{\left(x - 5\right) \left(x + 2\right) \left(x + 5\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{39 - 5 x}{\left(x - 5\right) \left(x + 2\right) \left(x + 5\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{\frac{39}{5}\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-5, -2, 5\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -5\right) \cup \left(-2, 5\right) \cup \left(\frac{39}{5}, \infty\right)$