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16 Sep 2014, 01:22
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If \(ab^2 > 0\) and \(ac < 0\), which of the following must be true?
I. \(ab >0\)
II. \(b^2c < 0\)
III. \(a*c^2 > 0\)
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
I. \(ab >0\)
II. \(b^2c < 0\)
III. \(a*c^2 > 0\)
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
16 Sep 2014, 01:22
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Official Solution:
If \(ab^2 > 0\) and \(ac < 0\), which of the following must be true?
I. \(ab >0\)
II. \(b^2c < 0\)
III. \(a*c^2 > 0\)
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
The first condition, \(ab^2 > 0\), to hold, \(a\) and \(b^2\) must have the same sign and since \(b^2\) cannot be negative, then both must be positive. Hence, \(ab^2 > 0\) implies that \(a > 0\) and \(b \neq 0\).
The second condition, \(ac < 0\) to hold, \(a\) and \(c\) must have opposite signs. Since from above we established that \(a > 0\), then \(c < 0\).
Thus, we have established that \(a > 0\), \(b \neq 0\), and \(c < 0\). Let's evaluate each option:
I. \(ab >0\). Since we don't know the sign of \(b\) and only know that \(b \neq 0\), this option may or may not be true.
II. \(b^2c < 0\). Since \(b \neq 0\) and \(c < 0\), this option must be true.
III. \(a*c^2 > 0\). Since \(a > 0\) and \(c \neq 0\), this option must be true.
Answer: D
If \(ab^2 > 0\) and \(ac < 0\), which of the following must be true?
I. \(ab >0\)
II. \(b^2c < 0\)
III. \(a*c^2 > 0\)
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
The first condition, \(ab^2 > 0\), to hold, \(a\) and \(b^2\) must have the same sign and since \(b^2\) cannot be negative, then both must be positive. Hence, \(ab^2 > 0\) implies that \(a > 0\) and \(b \neq 0\).
The second condition, \(ac < 0\) to hold, \(a\) and \(c\) must have opposite signs. Since from above we established that \(a > 0\), then \(c < 0\).
Thus, we have established that \(a > 0\), \(b \neq 0\), and \(c < 0\). Let's evaluate each option:
I. \(ab >0\). Since we don't know the sign of \(b\) and only know that \(b \neq 0\), this option may or may not be true.
II. \(b^2c < 0\). Since \(b \neq 0\) and \(c < 0\), this option must be true.
III. \(a*c^2 > 0\). Since \(a > 0\) and \(c \neq 0\), this option must be true.
Answer: D
06 Sep 2023, 02:15
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
