Problem 1: Is it the case that for all rational numbers x with x < 1> 1? If so prove it, if not characterize those rational x such that 1 > 1. Yes, it is the case that for all rational numbers x with x < 1> 1. Let x be a rational number such that x < 1 xss=removed> p. Since x < 1> 1. We have: 1/x = 1/(p/q) = q/p. Since p < q> 1, which means that 1/x > 1. Therefore, for all rational numbers x with x < 1> 1. To characterize those rational numbers x such that 1/x > 1, we can say that x must be a rational number such that 0 < x> 1 is the set of rational numbers in the open interval (0, 1)