Is \((x + y)^2 > x^2 + y^2\)?


(1) \(|x| \leq 0\)

(2) \(\sqrt{y^2} \leq 0\)


Official Solution is HERE.
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the question stem can be reduced to -
\((x + y)^2 > x^2 + y^2\)-------> \(x^2 + y^2 + 2xy > x^2 + y^2\)
solving it, the question stem becomes Is xy>0

Statement 1: implies that either |x| = 0 or |x| <0. |x| cannot be negative. so we Know that |x| = 0
or xy=0. Hence we get a NO for our question stem . Sufficient

Statement 2: \(\sqrt{y^2} \leq 0\) implies that either \(\sqrt{y^2} = 0\) or \(\sqrt{y^2} < 0\)
Square root is always positive in GMAT. Hence y=0.
So xy=0. Hence we get a NO for our question stem . Sufficient

Hence Option D

Target question: Is (x + y)² > x² + y²?

This is a good candidate for rephrasing the target question.

Take: (x + y)² > x² + y²
Expand and simplify: x² + 2xy + y² > x² + y²
Subtract x² and y² from both sides to get: 2xy > 0
Divide both sides by 2 to get: xy > 0

REPHRASED target question: Is xy > 0?

Statement 1: |x| ≤ 0
This inequality should seem odd to us, since the absolute value of a number is always GREATER THAN OR EQUAL to zero.
This inequality is saying the absolute value of some number is LESS THAN OR EQUAL to zero.
Since the absolute value of a number can never be negative, it must us the case that |x| = 0, which means x = 0.
If x = 0, then xy = (0)y = 0, which means we can answer the REPHRASED target question with certainty.
It is NOT the case that xy > 0
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: √(y²) ≤ 0
This inequality should also seem odd to us, because the square root notation instructs us to find the POSITIVE square root of a number, which means √(some number) is always GREATER THAN OR EQUAL to zero.
The given inequality is saying √(some number) is LESS THAN OR EQUAL to zero.
Since √(some number) can never be negative, it must us the case that √(y²) = 0, which means y = 0.
If y = 0, then xy = (x)(0) = 0, which means we can answer the REPHRASED target question with certainty.
It is NOT the case that xy > 0
Since we can answer the REPHRASED target question with certainty, statement 2 is SUFFICIENT

Answer:

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Official Solution:


Is \((x + y)^2 > x^2 + y^2\)?

First let's rephrase the question: is \(x^2 +2xy+ y^2 > x^2 + y^2\)?

First let's rephrase the question: is \(xy>0\)?

(1) \(|x| \leq 0\). The absolute value of a number cannot be negative, it can be 0 or positive. Thus, from this statement we can deduce that \(x=0\). Therefore, \(xy=0\). We have a NO answer to the question. Sufficient.

(2) \(\sqrt{y^2} \leq 0\). This is the same as \(|y| \leq 0\). So, we have the same as above: \(y=0\). Therefore, \(xy=0\). We have a NO answer to the question. Sufficient.


Answer: D.
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MUST KNOW: \(\sqrt{x^2}=|x|\):

The point here is that since square root function cannot give negative result then \(\sqrt{some \ expression}\geq{0}\).

So \(\sqrt{x^2}\geq{0}\). But what does \(\sqrt{x^2}\) equal to?

Let's consider following examples:
If \(x=5\) --> \(\sqrt{x^2}=\sqrt{25}=5=x=positive\);
If \(x=-5\) --> \(\sqrt{x^2}=\sqrt{25}=5=-x=positive\).

So we got that:
\(\sqrt{x^2}=x\), if \(x\geq{0}\);
\(\sqrt{x^2}=-x\), if \(x<0\).

What function does exactly the same thing? The absolute value function: \(|x|=x\), if \(x\geq{0}\) and \(|x|=-x\), if \(x<0\). That is why \(\sqrt{x^2}=|x|\).

Hope it's clear.
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