Solve the inequality $LaTeX: \displaystyle \frac{7}{x^{2} - 4}<\frac{8}{x^{2} + 3 x + 2}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{8}{\left(x + 1\right) \left(x + 2\right)} + \frac{7}{\left(x - 2\right) \left(x + 2\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 2\right) \left(x + 1\right) \left(x + 2\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{7 x + 7 - (8 x - 16)}{\left(x - 2\right) \left(x + 1\right) \left(x + 2\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{23 - x}{\left(x - 2\right) \left(x + 1\right) \left(x + 2\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{23\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-2, -1, 2\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -2\right) \cup \left(-1, 2\right) \cup \left(23, \infty\right)$