Solve the inequality LaTeX: \displaystyle \frac{7}{x^{2} - 16}<\frac{6}{x^{2} + 2 x - 8}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{6}{\left(x - 2\right) \left(x + 4\right)} + \frac{7}{\left(x - 4\right) \left(x + 4\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 4\right) \left(x - 2\right) \left(x + 4\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{7 x - 14 - (6 x - 24)}{\left(x - 4\right) \left(x - 2\right) \left(x + 4\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{x + 10}{\left(x - 4\right) \left(x - 2\right) \left(x + 4\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{-10\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-4, 2, 4\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-10, -4\right) \cup \left(2, 4\right)