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

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{2}{\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 - (2 x - 4)}{\left(x - 2\right) \left(x + 1\right) \left(x + 2\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{5 x + 11}{\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\{- \frac{11}{5}\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(- \frac{11}{5}, -2\right) \cup \left(-1, 2\right)