Solve the inequality LaTeX: \displaystyle \frac{7}{x^{2} - 9}<\frac{7}{x^{2} - 10 x + 21}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{7}{\left(x - 3\right) \left(x + 3\right)} - \frac{7}{\left(x - 7\right) \left(x - 3\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 7\right) \left(x - 3\right) \left(x + 3\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{7 x - 49 - (7 x + 21)}{\left(x - 7\right) \left(x - 3\right) \left(x + 3\right)} < 0 . Simplifying gives LaTeX: \displaystyle - \frac{70}{\left(x - 7\right) \left(x - 3\right) \left(x + 3\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\{-3, 3, 7\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-3, 3\right) \cup \left(7, \infty\right)