Solve the inequality LaTeX: \displaystyle \frac{3}{x^{2} - 16}<\frac{3}{x^{2} + 7 x + 12}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{3}{\left(x + 3\right) \left(x + 4\right)} + \frac{3}{\left(x - 4\right) \left(x + 4\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 4\right) \left(x + 3\right) \left(x + 4\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{3 x + 9 - (3 x - 12)}{\left(x - 4\right) \left(x + 3\right) \left(x + 4\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{21}{\left(x - 4\right) \left(x + 3\right) \left(x + 4\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \emptyset , or the zeros of the denominator LaTeX: \displaystyle \left\{-4, -3, 4\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -4\right) \cup \left(-3, 4\right)