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If \(|a| > |b| > |c|\), is \(a*b^3*c^3 > a*b^4*c^2\) ?
(1) \(a > b > c\)
(2) \(a + b > 0\)
(1) \(a > b > c\)
(2) \(a + b > 0\)
19 Aug 2022, 06:19
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Official Solution:
If \(|a| > |b| > |c|\), is \(a*b^3*c^3 > a*b^4*c^2\) ?
First of all notice, that \(|a| > |b| > |c|\) means that \(a\) is further from 0 than \(b\), and \(b\) is further from 0, than \(c\). Also, neither \(a\) nor \(b\) can be 0 (the least value of \(|c|\) is 0 and since both \(|b|\) and \(|a|\) are greater than that, then \(a≠0\) and \(b≠0\)).
Next, work on the question itself:
Is \(a*b^3*c^3 > a*b^4*c^2\) ?
Is \(a*b^3*c^3 - a*b^4*c^2>0\) ?
Is \(a*b^3*c^2(c-b) >0\) ?
(1) \(a > b > c\)
The ordering of \(a\), \(b\), and \(c\), on the number line, is as follows:
\(------c--------b---a---\)
Together with \(|a| > |b| > |c|\) we can have one of the following three\(^*\) cases:
\(------c---0----b---a---\)
\(--0---c--------b---a---\)
\(-----(c=0)------b---a---\)
\(^*\) 0 cannot be to the right of \(b\) because in this case \(b\) won't be further from 0, than \(c\).
In any of the above three cases: \(a*b^3*c^2(c-b) =(positive)(positive)(positive \ or \ 0)(negative)=(0 \ or \ negative)\), thus \(a*b^3*c^2(c-b) \) cannot be positive. We have a definite NO answer to the question. Sufficient.
(2) \(a + b > 0\)
This together with \(|a| > |b|\) means that \(a > 0\).
Consider the following cases:
I. If \(b>0\) too, then we'll get the same three cases we had in (1), which gives a NO answer to the question.
II. If \(b |c|\), we can have one of the following three\(^*\) cases:
\(^*\) so, as we can see \( b < 0\) together with \(|b| > |c|\), mean that \(c > b\).
In any of the above three cases: \(a*b^3*c^2(c-b) =(positive)(negative)(positive \ or \ 0)(positive)=(0 \ or \ negative)\), thus \(a*b^3*c^2(c-b) \) cannot be positive. For all the possible cases (I and II), we have a definite NO answer to the question. Sufficient.
Answer: D
If \(|a| > |b| > |c|\), is \(a*b^3*c^3 > a*b^4*c^2\) ?
First of all notice, that \(|a| > |b| > |c|\) means that \(a\) is further from 0 than \(b\), and \(b\) is further from 0, than \(c\). Also, neither \(a\) nor \(b\) can be 0 (the least value of \(|c|\) is 0 and since both \(|b|\) and \(|a|\) are greater than that, then \(a≠0\) and \(b≠0\)).
Next, work on the question itself:
Is \(a*b^3*c^3 > a*b^4*c^2\) ?
Is \(a*b^3*c^3 - a*b^4*c^2>0\) ?
Is \(a*b^3*c^2(c-b) >0\) ?
(1) \(a > b > c\)
The ordering of \(a\), \(b\), and \(c\), on the number line, is as follows:
\(------c--------b---a---\)
Together with \(|a| > |b| > |c|\) we can have one of the following three\(^*\) cases:
\(------c---0----b---a---\)
\(--0---c--------b---a---\)
\(-----(c=0)------b---a---\)
\(^*\) 0 cannot be to the right of \(b\) because in this case \(b\) won't be further from 0, than \(c\).
In any of the above three cases: \(a*b^3*c^2(c-b) =(positive)(positive)(positive \ or \ 0)(negative)=(0 \ or \ negative)\), thus \(a*b^3*c^2(c-b) \) cannot be positive. We have a definite NO answer to the question. Sufficient.
(2) \(a + b > 0\)
This together with \(|a| > |b|\) means that \(a > 0\).
Consider the following cases:
I. If \(b>0\) too, then we'll get the same three cases we had in (1), which gives a NO answer to the question.
II. If \(b |c|\), we can have one of the following three\(^*\) cases:
\(---b---c---0-------\)
\(---b-------0---c---\)
\(---b------(c=0)---\)
\(---b-------0---c---\)
\(---b------(c=0)---\)
\(^*\) so, as we can see \( b < 0\) together with \(|b| > |c|\), mean that \(c > b\).
In any of the above three cases: \(a*b^3*c^2(c-b) =(positive)(negative)(positive \ or \ 0)(positive)=(0 \ or \ negative)\), thus \(a*b^3*c^2(c-b) \) cannot be positive. For all the possible cases (I and II), we have a definite NO answer to the question. Sufficient.
Answer: D
