Solve the inequality $LaTeX: \displaystyle \frac{2}{x^{2} - 25}<\frac{6}{x^{2} + 14 x + 45}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{6}{\left(x + 5\right) \left(x + 9\right)} + \frac{2}{\left(x - 5\right) \left(x + 5\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 5\right) \left(x + 5\right) \left(x + 9\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{2 x + 18 - (6 x - 30)}{\left(x - 5\right) \left(x + 5\right) \left(x + 9\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{48 - 4 x}{\left(x - 5\right) \left(x + 5\right) \left(x + 9\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{12\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-9, -5, 5\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -9\right) \cup \left(-5, 5\right) \cup \left(12, \infty\right)$