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16 Sep 2014, 00:49
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Is \(x^2 * y^5 * z \gt 0\) ?
(1) \(\frac{xz}{y} \gt 0\)
(2) \(\frac{y}{z} \lt 0\)
(1) \(\frac{xz}{y} \gt 0\)
(2) \(\frac{y}{z} \lt 0\)
16 Sep 2014, 00:49
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Official Solution:
Is \(x^2 * y^5 * z \gt 0\) ?
In order for \(x^2 * y^5 * z \gt 0\) to be true, the following conditions must be satisfied:
1. \(x\) cannot be zero. This is because if \(x = 0\), then \(x^2 * y^5 * z \) will equal 0, not greater than it.
AND
2. \(y\) and \(z\) must either be both positive or both negative. This is because if \(y\) and \(z\) have different signs (or if either of them is 0), then \(yz\) will be negative (or 0), not positive as required.
Thus, only if BOTH conditions are met, we will have \((x^2) * (y^5 * z) = (positive) * (positive) = positive\).
(1) \(\frac{xz}{y} \gt 0\).
The first condition is satisfied: \(x \neq{0}\). However, we don't know about the second one: \(\frac{xz}{y} \gt 0\) implies that either all of them are positive (answer YES) or ANY two are negative and the third one is positive, so it's possible for \(y\) and \(z\) to have opposite signs (answer NO). Not sufficient.
(2) \(\frac{y}{z} \lt 0 \).
From this statement, we can deduce that \(y\) and \(z\) have opposite signs, which violates the second condition. Thus, the answer to the question is NO. Sufficient.
Side note for (2): \(\frac{y}{z} \lt 0\) does not imply that \(x^2 * y^5 * z \lt 0\). Instead, it implies that \(x^2 * y^5 * z \le 0\) because it's possible for \(x\) to be equal to zero, and in this case, \(x^2 * y^5 * z = 0\). However, in any case, \(x^2 * y^5 * z\) is not greater than zero, so we can answer NO to the question.
Answer: B
Is \(x^2 * y^5 * z \gt 0\) ?
In order for \(x^2 * y^5 * z \gt 0\) to be true, the following conditions must be satisfied:
1. \(x\) cannot be zero. This is because if \(x = 0\), then \(x^2 * y^5 * z \) will equal 0, not greater than it.
AND
2. \(y\) and \(z\) must either be both positive or both negative. This is because if \(y\) and \(z\) have different signs (or if either of them is 0), then \(yz\) will be negative (or 0), not positive as required.
Thus, only if BOTH conditions are met, we will have \((x^2) * (y^5 * z) = (positive) * (positive) = positive\).
(1) \(\frac{xz}{y} \gt 0\).
The first condition is satisfied: \(x \neq{0}\). However, we don't know about the second one: \(\frac{xz}{y} \gt 0\) implies that either all of them are positive (answer YES) or ANY two are negative and the third one is positive, so it's possible for \(y\) and \(z\) to have opposite signs (answer NO). Not sufficient.
(2) \(\frac{y}{z} \lt 0 \).
From this statement, we can deduce that \(y\) and \(z\) have opposite signs, which violates the second condition. Thus, the answer to the question is NO. Sufficient.
Side note for (2): \(\frac{y}{z} \lt 0\) does not imply that \(x^2 * y^5 * z \lt 0\). Instead, it implies that \(x^2 * y^5 * z \le 0\) because it's possible for \(x\) to be equal to zero, and in this case, \(x^2 * y^5 * z = 0\). However, in any case, \(x^2 * y^5 * z\) is not greater than zero, so we can answer NO to the question.
Answer: B
General Discussion
15 Jun 2017, 12:42
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What if X = 0? In such a case Statement 2 will not be sufficient. It can be either = 0 or > 0. Bunuel Please can you throw some light on my understanding?
