Solve the inequality $LaTeX: \displaystyle \frac{5}{x^{2} - 9}<\frac{3}{x^{2} + 4 x - 21}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{3}{\left(x - 3\right) \left(x + 7\right)} + \frac{5}{\left(x - 3\right) \left(x + 3\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 3\right) \left(x + 3\right) \left(x + 7\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{5 x + 35 - (3 x + 9)}{\left(x - 3\right) \left(x + 3\right) \left(x + 7\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{2 x + 26}{\left(x - 3\right) \left(x + 3\right) \left(x + 7\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{-13\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-7, -3, 3\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-13, -7\right) \cup \left(-3, 3\right)$