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(1) \(|x| \leq 0\) x can be zero or take positive or negative values. If x = -1, the expression is false. If x = 1, the expression is false. The equation is true only when x=0. At x=0, 2xy is never greater than 0. Sufficient
(2) \(\sqrt{y^2} \leq 0\)[/quote] This is the same as saying \(|y| \leq 0\) y can be zero or take positive or negative values. Again going by the options we went for statement 1, we get y=0 as the only solution. At y=0, the expression 2xy is never greater than 0. Sufficient (Option D) _________________
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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
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
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.
Hi, Why sqrt(y2)≤0 is the same as saying |y|≤0 I know tht sqrt(9) is +/-3 but sqrt(3)^2 is always positive right?
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 \(\sqrt{x^2}=|x|\).
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
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