Solve the inequality LaTeX: \displaystyle \frac{4}{x^{2} - 25}<\frac{2}{x^{2} + x - 30}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{2}{\left(x - 5\right) \left(x + 6\right)} + \frac{4}{\left(x - 5\right) \left(x + 5\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 5\right) \left(x + 5\right) \left(x + 6\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{4 x + 24 - (2 x + 10)}{\left(x - 5\right) \left(x + 5\right) \left(x + 6\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{2 x + 14}{\left(x - 5\right) \left(x + 5\right) \left(x + 6\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{-7\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-6, -5, 5\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-7, -6\right) \cup \left(-5, 5\right)