Solve the inequality $LaTeX: \displaystyle \frac{3}{x^{2} - 4}<\frac{7}{x^{2} + 5 x - 14}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{7}{\left(x - 2\right) \left(x + 7\right)} + \frac{3}{\left(x - 2\right) \left(x + 2\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 2\right) \left(x + 2\right) \left(x + 7\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{3 x + 21 - (7 x + 14)}{\left(x - 2\right) \left(x + 2\right) \left(x + 7\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{7 - 4 x}{\left(x - 2\right) \left(x + 2\right) \left(x + 7\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{\frac{7}{4}\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-7, -2, 2\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -7\right) \cup \left(-2, \frac{7}{4}\right) \cup \left(2, \infty\right)$