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16 Sep 2014, 00:45
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16 Sep 2014, 00:45
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Official Solution:
Is \(a^7*b^2*c^3 \gt 0\)?
For the inequality \(a^7*b^2*c^3 > 0\) to hold true, the following conditions must be satisfied:
(i) \(a\) and \(c\) should be either both positive or both negative: in this case \(a^7*c^3\) will be positive;
(ii) \(b\) should not be zero: if \(b = 0\), then the entire expression (\(a^7*b^2*c^3\)) will be 0, not greater than 0, irrespective of the values of \(a\) and \(c\).
(1) \(bc \lt 0\).
From this, we know that \(b \ne 0\). However, we do not have information about \(a\) and \(c\). Therefore, this is not sufficient.
(2) \(ac \gt 0\).
This implies that \(a\) and \(c\) are either both positive or both negative. But, we don't know about \(b\): if \(b=0\), then the expression will be equal to 0. Thus, this is also not sufficient.
(1)+(2) We have both necessary conditions met: \(b \ne 0\) and \(a\) and \(c\) have the same sign. Sufficient.
Answer: C
Is \(a^7*b^2*c^3 \gt 0\)?
For the inequality \(a^7*b^2*c^3 > 0\) to hold true, the following conditions must be satisfied:
(i) \(a\) and \(c\) should be either both positive or both negative: in this case \(a^7*c^3\) will be positive;
(ii) \(b\) should not be zero: if \(b = 0\), then the entire expression (\(a^7*b^2*c^3\)) will be 0, not greater than 0, irrespective of the values of \(a\) and \(c\).
(1) \(bc \lt 0\).
From this, we know that \(b \ne 0\). However, we do not have information about \(a\) and \(c\). Therefore, this is not sufficient.
(2) \(ac \gt 0\).
This implies that \(a\) and \(c\) are either both positive or both negative. But, we don't know about \(b\): if \(b=0\), then the expression will be equal to 0. Thus, this is also not sufficient.
(1)+(2) We have both necessary conditions met: \(b \ne 0\) and \(a\) and \(c\) have the same sign. Sufficient.
Answer: C
General Discussion
20 Aug 2016, 04:48
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I think this is a high-quality question and I agree with explanation.
