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19 Nov 2021, 03:40
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If \(a^4b^3c^7 > 0\), then which of the following must be true?
I. \(\frac{b}{c} > 0\)
II. \(ab > 0\)
III. \(abc > 0\)
A. I only
B. II only
C. III only
D. I and II only
E. I and III only
I. \(\frac{b}{c} > 0\)
II. \(ab > 0\)
III. \(abc > 0\)
A. I only
B. II only
C. III only
D. I and II only
E. I and III only
19 Nov 2021, 03:40
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Official Solution:
If \(a^4b^3c^7 > 0\), then which of the following must be true?
I. \(\frac{b}{c} > 0\)
II. \(ab > 0\)
III. \(abc > 0\)
A. I only
B. II only
C. III only
D. I and II only
E. I and III only
From \(a^4b^3c^7 > 0\) we can deduce the following:
\(•\) None of the unknowns is 0.
\(•\) \(b\) and \(c\) have the same sign (\(bc > 0\)).
I. \(\frac{b}{c} > 0\). Since \(b\) and \(c\) have the same sign, then this must be true.
II. \(ab > 0\). We don't know anything about relationship between signs of \(a\) and \(b\). So, this option is not necessarily true.
III. \(abc > 0\). We know that \(bc > 0\) but the only thing we know about \(a\) that it's non-zero: it can be positive as well as negative. So, this option is not necessarily true.
Answer: A
If \(a^4b^3c^7 > 0\), then which of the following must be true?
I. \(\frac{b}{c} > 0\)
II. \(ab > 0\)
III. \(abc > 0\)
A. I only
B. II only
C. III only
D. I and II only
E. I and III only
From \(a^4b^3c^7 > 0\) we can deduce the following:
\(•\) None of the unknowns is 0.
\(•\) \(b\) and \(c\) have the same sign (\(bc > 0\)).
I. \(\frac{b}{c} > 0\). Since \(b\) and \(c\) have the same sign, then this must be true.
II. \(ab > 0\). We don't know anything about relationship between signs of \(a\) and \(b\). So, this option is not necessarily true.
III. \(abc > 0\). We know that \(bc > 0\) but the only thing we know about \(a\) that it's non-zero: it can be positive as well as negative. So, this option is not necessarily true.
Answer: A
20 Mar 2025, 03:22
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I like the solution - it’s helpful.
