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04 Jul 2019, 08:18
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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
04 Jul 2019, 09:08
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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
For the product of two numbers to be positive, they must have the same sign. Since the square of a number is always greater than or equal to 0, then \(ab^2 > 0\) implies that \(a>0\) and \(b \neq 0\).
Since from the above \(a>0\), then \(ac < 0\) implies that \(c < 0\).
So, we have: \(a > 0\), \(c < 0\), and \(b \neq 0\).
Let's check the options:
I. \(ab >0\). Not true if \(b < 0\).
II. \(b^2c < 0\). Always true since \(b^2 > 0\) and \(c < 0\).
III. \(a*c^2 > 0\). Always true since \(a > 0\) and \(c^2 > 0\).
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
For the product of two numbers to be positive, they must have the same sign. Since the square of a number is always greater than or equal to 0, then \(ab^2 > 0\) implies that \(a>0\) and \(b \neq 0\).
Since from the above \(a>0\), then \(ac < 0\) implies that \(c < 0\).
So, we have: \(a > 0\), \(c < 0\), and \(b \neq 0\).
Let's check the options:
I. \(ab >0\). Not true if \(b < 0\).
II. \(b^2c < 0\). Always true since \(b^2 > 0\) and \(c < 0\).
III. \(a*c^2 > 0\). Always true since \(a > 0\) and \(c^2 > 0\).
Answer: D
04 Jul 2019, 09:51
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Bunuel
Given \(ab^2 > 0\) and \(ac < 0\) implies that
a>0, c<0, and b could take any sign.
I. a*b could be negative ; a*b could be negative too. DISCARD
II. \(b^2*c\) is always negative. TRUE
III. \(a*c^2\) is always positive. TRUE
Ans. (D)
