x^2 - 1 = (x - 1)(x + 1). A product of two numbers is positive iff the numbers are both negative or both positive, i.e., iff x < 1, x < -1 or x > 1, x > -1. Obviously, x < -1 implies x < 1, and x > 1 implies x > -1, so the product is positive iff x < -1 or x > 1.