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26 May 2017, 04:41
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Is \(x^{4} -x^{3} + x^{2} - x > 0?\)
(1) \(x>1\)
(2) \(|x^{3}| > |x|\)
(1) \(x>1\)
(2) \(|x^{3}| > |x|\)
26 May 2017, 04:41
Kudos
Bookmarks
Official Solution:
For inequality questions, if the range of the question includes the range of the condition, the condition is sufficient.
You can take the 1st step of the variable approach and modify the original condition and the question, so you get Is \(x^{4} - x^{3} + x^{2} - x>0\)?, Is \(x^{3}(x-1) + x(x-1)>0\)?, Is \((x^{3} + x)(x - 1) > 0\)?, Is \(x(x^{2} + 1)(x - 1) > 0\)?. Since \(x^{2} + 1 > 0\) is always true, "Is \(x(x^{2} + 1)(x - 1) > 0\)?" becomes "Is \(x(x - 1) > 0\)?", and then \(x < 0\) or \(1 < x\)?.
In the original condition, there is 1 variable \((x)\) and in order to match the number of variables to the number of equations, there must be 1 equation. Therefore, D is most likely to be the answer.
In the case of con 1), the range of the question includes the range of the condition, hence it is sufficient.
In the case of con 2) you get \(|x^{3}| > |x|\), or \(|x|^{3} > |x|\), and since the absolute value is always positive even if both sides are divided by \(|x|\), the sign of inequality does not change, so \(|x|^{2} > 1\), then \(|x|^{2} = x^{2}\). If so, \(x^{2} > 1\), \(x^{2} - 1^{2} > 0\), or \((x - 1)(x + 1) > 0\), then \(x < -1\) or \(1 < x\). Since the range of the question includes the range of the condition, the condition is sufficient. The answer is D.
Answer: D
For inequality questions, if the range of the question includes the range of the condition, the condition is sufficient.
You can take the 1st step of the variable approach and modify the original condition and the question, so you get Is \(x^{4} - x^{3} + x^{2} - x>0\)?, Is \(x^{3}(x-1) + x(x-1)>0\)?, Is \((x^{3} + x)(x - 1) > 0\)?, Is \(x(x^{2} + 1)(x - 1) > 0\)?. Since \(x^{2} + 1 > 0\) is always true, "Is \(x(x^{2} + 1)(x - 1) > 0\)?" becomes "Is \(x(x - 1) > 0\)?", and then \(x < 0\) or \(1 < x\)?.
In the original condition, there is 1 variable \((x)\) and in order to match the number of variables to the number of equations, there must be 1 equation. Therefore, D is most likely to be the answer.
In the case of con 1), the range of the question includes the range of the condition, hence it is sufficient.
In the case of con 2) you get \(|x^{3}| > |x|\), or \(|x|^{3} > |x|\), and since the absolute value is always positive even if both sides are divided by \(|x|\), the sign of inequality does not change, so \(|x|^{2} > 1\), then \(|x|^{2} = x^{2}\). If so, \(x^{2} > 1\), \(x^{2} - 1^{2} > 0\), or \((x - 1)(x + 1) > 0\), then \(x < -1\) or \(1 < x\). Since the range of the question includes the range of the condition, the condition is sufficient. The answer is D.
Answer: D
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