Question

What is the solution of 2/x-1 > 7/x+1

Answer 1

For x > 1 and x < -1, we can multiply by the product of the denominators to get
  2(x+1) > 7(x-1)
  9 > 5x
  x < 9/5

For -1 < x < 1, we can do the same, but we must reverse the sense of the inequality.
  2(x+1) < 7(x-1)
  9 < 5x
  x > 1.8

A graph confirms the solution is
  -∞ < x < -1 ∪ 1 < x < 1.8

Answer 2

$\dfrac{2}{x - 1} > \dfrac{7}{x + 1} \\ \\ \dfrac{2}{x - 1} - \dfrac{7}{x + 1} > 0 \\ \\ \frac{2(x + 1) - 7(x - 1)}{ {x}^{2} - 1} > 0 \\ \\ 2(x + 1) - 7(x - 1) > 0 \\ \\ 2x + 2 - 7x + 7 > 0 \\ \\ 5x < 9 \: \: \\ \\ || \: \: \: x < 1.8 \: \: || \: \: \: \: \: \: \: Ans.$