If xy ≠0, is x^3 + y^3 > 0 ?
(1) x + y > 0
(2) xy > 0
GMAT Official Guide 2019 If xy ≠0, is
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Given: xy ≠0 => None of x and y is 0.BTGmoderatorDC wrote:If xy ≠0, is x^3 + y^3 > 0
(1) x + y > 0
(2) xy > 0
We have to determine whether x^3 + y^3 > 0.
Let's take each statement one by one.
(1) x + y > 0
Case 1: If both x and y are positive, then x^3 + y^3 > 0. The answer is Yes.
Case 2: Say one of x and y is positive and the other is negative. Since x + y > 0, the absolute value of the positive number must be greater than the absolute value of the negative number. Say x is positive and y is negative, thus, |x| > |y|.
=> |x|^3 > |y|^3
=> x^3 > y^3
=> x^3 + y^3 > 0. The answer is Yes. Sufficient.
(2) xy > 0
Case 1: If both x and y are positive, then x^3 + y^3 > 0. The answer is Yes.
Case 2: If both x and y are negative, then either x^3 > y^3 or x^3 < y^3; thus, x^3 + y^3 is not necessarily greater than 0. Insufficient.
The correct answer: A
Hope this helps!
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In the printed OG2019 book, there are some typos in the solution. The last sentence for statement (1) should be as follows:
$$Therefore,\ x^3>\left(-y\right)^3,\ or\ x^3>-y^3,\ or\ x^3+y^3>0.$$
$$Therefore,\ x^3>\left(-y\right)^3,\ or\ x^3>-y^3,\ or\ x^3+y^3>0.$$
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Stuart is a Harvard grad GMAT expert who scored 760 the first time he took the exam, with 99th percentile quant and verbal scores. He has extensive experience teaching for one of the "elite" GMAT prep companies. Through https://www.simplybrilliantprep.com he offers online classes, private tutoring and MBA application consulting for clients worldwide.
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Solution:BTGmoderatorDC wrote: ↑Sun Jul 01, 2018 12:24 amIf xy ≠0, is x^3 + y^3 > 0 ?
(1) x + y > 0
(2) xy > 0
Question Stem Analysis:
We need to determine whether x^3 + y^3 > 0, given that neither x nor y is 0.
Statement One Only:
Since x + y > 0, either both x and y are positive or one of them is positive and the other is negative. If both x and y are positive, then obviously x^3 + y^3 > 0. If one of them is positive and the other is negative, without loss of generality, we can let x be positive and y be negative. In order for x + y > 0, x must be “more” positive than y is negative (in other words, |x| > |y|) . In that case, x^3 will also be “more” positive than y^3 (in other words, |x^3| > |y^3|). Therefore, x^3 + y^3 > 0. Statement one alone is sufficient.
(Alternatively, recall that x^3 + y^3 = (x + y)(x^2 - xy + y^2). Again, if both x and y are positive, then obviously x^3 + y^3 > 0. Now, if one of the values is positive and the other is negative, then xy is negative and hence -xy is positive. Since x^2 + y^2 is positive, x^2 - xy + y^2 will be positive. Therefore, given the fact that x + y is positive, we see that (x + y)(x^2 - xy + y^2) will be positive, i.e., x^3 + y^3 is positive.)
Statement Two Only:
Since xy > 0, either both x and y are positive or both are negative. If both x and y are positive, then obviously x^3 + y^3 > 0. However, if both are negative, then x^3 + y^3 < 0. Statement two is not sufficient.
Answer: A
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