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Re: If (x + y)^2 < x^2, which of the following must be true? [#permalink]
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Official Solution:


If \((x + y)^2 < x^2\), which of the following must be true?

I. \(y(y + 2x) < 0\)

II. \(y < x\)

III. \(xy < 0\)


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


First, let's simplify the given inequality:

\(x^2 + 2xy + y^2 < x^2\)

\(2xy + y^2 < 0\)

\(y(2x + y) < 0\)

Now, let's evaluate each option:

I. \(y(y + 2x) < 0\). This option is directly true since it matches our simplified inequality.

II. \(y < x\). From \(y(2x + y) < 0\), we can infer that \(y\) and \(2x + y\) must have different signs. Let's try finding a counterexample: if \(y\) is positive, say 1, then \(2x + y\) must be negative, which can be obtained if \(x\) is, for example, -10, making \(y > x\). Hence, this option is not necessarily true.

III. \(xy < 0\). We derived that \(2xy + y^2 < 0\) earlier. Since \(y^2\) is always non-negative, \(2xy\) must be negative for the inequality to hold. This implies that \(xy\) is negative. Thus, this option is always true.

Therefore, only options I and III must be true.


Answer: E
GMAT Club Bot
Re: If (x + y)^2 < x^2, which of the following must be true? [#permalink]
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