If 0 > x > y, then which of the following must be true?

Official Solution:


If \(0 > x > y\), then which of the following must be true?

I. \(x^2 – y^2 < 0\)

II. \(\frac{1}{x^2} < 1\)

III. \(\frac{x + y}{x} > 0\)


A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III


We are given that \(0 > x > y\), so both \(x\) and \(y\) are negative and \(y\) is further from 0 than \(x\).

Check the options:

I. \(x^2 – y^2 < 0\):

• \(x^2 < y^2\)

• \(|x| < |y|\)

• The above must be true because we know that \(y\) is further from 0 than \(x\). Or, since both \(x\) and \(y\) are negative, then \(|x| < |y|\) means \(-x < -y\), which gives \(x > y\) and we know from the stem that this is true.

II. \(\frac{1}{x^2} < 1\)

• The above is not necessarily true. For example, consider \(x=-\frac{1}{2}\)

III. \(\frac{x + y}{x} > 0\)

• \(\frac{x}{x}+\frac{y}{x} > 0\)

• \(\frac{y}{x} > -1\). Since both \(x\) and \(y\) are negative, then \(\frac{y}{x} > 0 >-1\), so this option must be true.


Answer: D