Question

Prove that if x and y are real numbers such that xy < 1, then ________.

1) x > 1 and y > 1
2) x < 1 and y < 1
3) x > 1 and y < 1
4) x < 1 and y > 1

Answer 1

If xy < 1, then x > 1 and y < 1.

Step-by-step explanation:

To prove that if xy < 1, then x > 1 and y < 1, we can use a proof by contradiction.

1. Assume that x > 1 and y > 1.
2. Multiply both sides of the inequality xy < 1 by x to get x^2y < x.
3. Since x > 1, we have x^2y < x < x^2, which implies y < 1, a contradiction.
4. Hence, the assumption that x > 1 and y > 1 is false.
5. Therefore, if xy < 1, then x < 1 and y < 1.