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If ab^2 > 0 and ac < 0, then which of the following must be true?

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If ab^2 > 0 and ac < 0, then which of the following must be true?  [#permalink]

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New post 23 Jan 2019, 02:52
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E

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Re: If ab^2 > 0 and ac < 0, then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:23
Bunuel wrote:
If \(ab^2 > 0\) and \(ac < 0\), then 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


Official Solution:


If \(ab^2 > 0\) and \(ac < 0\), then 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 and since the square of a number is always more than or equal to 0, then \(ab^2 > 0\) implies that \(a>0\) and \(b \neq 0\)

Since \(a>0\), from above, then \(ac < 0\) implies that \(c < 0\)

So, we have that: \(a > 0\), \(c < 0\) 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
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Re: If ab^2 > 0 and ac < 0, then which of the following must be true?  [#permalink]

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New post 23 Jan 2019, 04:02
Bunuel wrote:
If \(ab^2 > 0\) and \(ac < 0\), then 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


IMO D

If \(ab^2 > 0\) and \(ac < 0\)

From above you can imply the following
a is +ive, b can be +ive or -ive and c is -ive

I. \(ab >0\), True or False , by using the above

II. \(b^2c < 0\), Always true

III. \(a*c^2 > 0\), Always true
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Re: If ab^2 > 0 and ac < 0, then which of the following must be true?  [#permalink]

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New post 23 Jan 2019, 04:27
Bunuel wrote:
If \(ab^2 > 0\) and \(ac < 0\), then 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


Given:
\(ab^2 > 0\) - Since b^2 must be positive only, a would be positive too (to get positive product)

and \(ac < 0\) - Since a is positive (obtained above), c must be negative to get negative product

So we have 'a' positive, 'c' negative and no idea about 'b'.

I. \(ab >0\)
Since we have no idea about b, we don't know whether ab will be positive or not.

II. \(b^2c < 0\)
b^2 will be positive (note that none of these is 0) and c is negative so product will be negative. This is true.

III. \(a*c^2 > 0\)
a is positive and c^2 will be positive too. So product is positive. This is true too.

Answer (D)
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Re: If ab^2 > 0 and ac < 0, then which of the following must be true?  [#permalink]

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New post 23 Jan 2019, 05:41
Bunuel wrote:
If \(ab^2 > 0\) and \(ac < 0\), then 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



Given,

\(ab^2>0.\)

a must be positive. a>0. It is not possible to tell whether b is positive or negative as \(b^2>0\). don't know anything about b.

ac<0. we know that a>0. So, c<0.

I. \(ab >0\)

a>0. don't know anything about b. could be true. MUST not one.

II. \(b^2c < 0\)

b>0 and c<0. Must be true.

III. \(a*c^2 > 0\)

a>0 and \(c^2>0\). Must be true.

D is the correct answer.
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Re: If ab^2 > 0 and ac < 0, then which of the following must be true?  [#permalink]

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New post 23 Jan 2019, 06:40
Bunuel wrote:
If \(ab^2 > 0\) and \(ac < 0\), then 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 expression ab^2>0 & ac<0 to be valid
a +ve , b -ve or +ve and c -ve

in that case only
case
II. \(b^2c < 0\)

III. \(a*c^2 > 0\)
would be valid
IMO D
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Re: If ab^2 > 0 and ac < 0, then which of the following must be true?   [#permalink] 23 Jan 2019, 06:40
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