Solve the inequality LaTeX: \displaystyle \frac{6}{x^{2} - 16}<\frac{8}{x^{2} + 11 x + 28}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{8}{\left(x + 4\right) \left(x + 7\right)} + \frac{6}{\left(x - 4\right) \left(x + 4\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 4\right) \left(x + 4\right) \left(x + 7\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{6 x + 42 - (8 x - 32)}{\left(x - 4\right) \left(x + 4\right) \left(x + 7\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{74 - 2 x}{\left(x - 4\right) \left(x + 4\right) \left(x + 7\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{37\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-7, -4, 4\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -7\right) \cup \left(-4, 4\right) \cup \left(37, \infty\right)