Solve the inequality LaTeX: \displaystyle \frac{9}{x^{2} - 16}<\frac{4}{x^{2} - 3 x - 28}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{9}{\left(x - 4\right) \left(x + 4\right)} - \frac{4}{\left(x - 7\right) \left(x + 4\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 7\right) \left(x - 4\right) \left(x + 4\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{9 x - 63 - (4 x - 16)}{\left(x - 7\right) \left(x - 4\right) \left(x + 4\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{5 x - 47}{\left(x - 7\right) \left(x - 4\right) \left(x + 4\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{\frac{47}{5}\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-4, 4, 7\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-4, 4\right) \cup \left(7, \frac{47}{5}\right)