Solve the inequality LaTeX: \displaystyle \frac{5}{x^{2} - 4}<\frac{2}{x^{2} + 8 x + 12}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{2}{\left(x + 2\right) \left(x + 6\right)} + \frac{5}{\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 + 6\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{5 x + 30 - (2 x - 4)}{\left(x - 2\right) \left(x + 2\right) \left(x + 6\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{3 x + 34}{\left(x - 2\right) \left(x + 2\right) \left(x + 6\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{- \frac{34}{3}\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-6, -2, 2\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(- \frac{34}{3}, -6\right) \cup \left(-2, 2\right)