If xy ≠ 0, is x/y greater than y/x ?​

"Is \(\frac{x}{y} > \frac{y}{x}?\)" We don't want to simplify this yet without knowing the signs of x and y. Knowing if they are the same sign or opposite sign is enough to take one step further in simplifying.

Statement 1:

If both positive, the question is if \(x^2 > y^2\) and \(x > y > 0\) guarantees \(x^2 > y^2\) is true.

If x positive and y negative, the question is if \(x^2 < y^2\), which we can say "no" to when x has a higher absolute value than y. Insufficient.

Statement 2:

\(x^2 > y^2\) but we don't know the signs of x and y. Therefore this can give us a yes or no depending on the signs. Insufficient.

Combined:

We have \((x - y)(x + y) > 0\) and \(x - y > 0\), it must follow that \(x + y > 0\). Then add the two shorter inequalities together to get \(2x > 0\) but note that we cannot tell the sign of y. Then we still have the problem from statement 2, insufficient.

Ans: E