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# If 0 < ab < ac, is a negative?

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If 0 < ab < ac, is a negative? [#permalink]

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15 Nov 2012, 11:08
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If 0 < ab < ac, is a negative?

(1) c < 0
(2) b > c

[Reveal] Spoiler:
I know that ab and ac > 0, ab < ac

b could be bigger than c and still ab < ac when b is negative but then the expression will not be bigger than 0. If "a" alone is negative also the whole expression will not be bigger than 0.

If a is negative and b is negative then ab will not be < ac.

I don't quite get it. Pls help
[Reveal] Spoiler: OA

Last edited by Bunuel on 16 Nov 2012, 04:02, edited 2 times in total.
Renamed the topic and edited the question.

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15 Nov 2012, 20:19
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KarolPL wrote:
Hello Everyone,

I don't quite get a problem. I've got an inequality:

0 < ab < ac

(1) -> c < 0 -> this one I understand well
(2) -> b > c

Is "a" negative?

I know that ab and ac > 0, ab < ac

b could be bigger than c and still ab < ac when b is negative but then the expression will not be bigger than 0. If "a" alone is negative also the whole expression will not be bigger than 0.

If a is negative and b is negative then ab will not be < ac.

I don't quite get it. Pls help

We are given 0<ab<ac
=> ab and ac both are positive.

Statement 1: c <0
=> for ac to be positive , a must be negative. Sufficient

Statement 2: b >c
Now lets take a look at what is given in question
0<ab<ac
=> ab <ac
=> ab-ac <0
=> a(b-c) <0

But from statement 2 we know b>c thus b-c must be positive.
Therefore for a(b-c) <0 to be true, a must be negative. Sufficient.

Since each statement is sufficient, Ans D it is.

Hope it helps.
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Re: If 0 < ab < ac, is a negative? [#permalink]

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05 Jul 2014, 17:55
Am I on right path here?
Statement 1 is fine but for statement 2 we have ac > ab > 0 and b > c
I want to plug in values and solve this one.

let b = 5 and c = 3 , so b > c is satisfied . this will mean 3a > 5 a > 0 , so a needs to be positive for this to be greater than zero but 3a can never be greater than 5a , if a is positive so we cannot consider this.

b = -3 , c = -5 ac >ab>0 means -5a > -3a > 0 which will hold when a < 0 .
So statement 2 is sufficient as well.

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Re: If 0 < ab < ac, is a negative? [#permalink]

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11 May 2017, 22:56
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If 0 < ab < ac, is a negative? [#permalink]

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13 May 2017, 23:33
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KarolPL wrote:
If $$0 < ab < ac$$, is a negative?

(1) $$c < 0$$
(2) $$b > c$$

(1) $$c<0$$, $$0<ac$$, $$0<(-ve)(-ve).$$ Sufficient

(2) $$b>c$$, $$b-c>0$$, $$ab<ac$$, $$a(b-c)<0$$, $$(-ve)(+ve)<0$$. Sufficient.

OFFICIAL SOLUTION

(D): By the transitive property of inequalities, if 0 < ab < ac, then 0 < ac. Therefore, a and c must have the same sign.

(1) SUFFICIENT: Statement (1) tells you that c is negative. Therefore, a is negative.

(2) SUFFICIENT: Statement (2) is trickier. The statement indicates that b > c, but the question stem also told you that ab < ac. When you multiply both sides of b > c by a, the sign gets flipped. For inequalities, what circumstance needs to be true in order to flip the sign when you multiply by something? You multiply by a negative. Therefore, a must be negative, because multiplying the two sides of the equation by a results in a flipped inequality sign.

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Re: If 0 < ab < ac, is a negative? [#permalink]

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13 May 2017, 23:37
1. If c is neg then a must be negative for ac>0,hence sufficient
2. If b>c, then for ab<ac only if a is negative hence sufficient.
Ans d

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Re: If 0 < ab < ac, is a negative?   [#permalink] 13 May 2017, 23:37
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