Solve the inequality LaTeX: \displaystyle \frac{2}{x^{2} - 25}<\frac{2}{x^{2} + 4 x - 5}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{2}{\left(x - 1\right) \left(x + 5\right)} + \frac{2}{\left(x - 5\right) \left(x + 5\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 5\right) \left(x - 1\right) \left(x + 5\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{2 x - 2 - (2 x - 10)}{\left(x - 5\right) \left(x - 1\right) \left(x + 5\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{8}{\left(x - 5\right) \left(x - 1\right) \left(x + 5\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\{-5, 1, 5\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -5\right) \cup \left(1, 5\right)