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# If abc ≠ 0, is abc > 0 ?

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Math Expert
Joined: 02 Sep 2009
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If abc ≠ 0, is abc > 0 ?  [#permalink]

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11 Aug 2017, 01:19
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75% (hard)

Question Stats:

47% (00:56) correct 53% (00:57) wrong based on 67 sessions

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If $$abc ≠ 0$$, is $$abc > 0$$ ?

(1) $$|a –b | = |a| - |b|$$

(2) $$|b + c| = |b| + |c|$$

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Magoosh GMAT Instructor
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Re: If abc ≠ 0, is abc > 0 ?  [#permalink]

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11 Aug 2017, 17:06
1
Bunuel wrote:
If $$abc ≠ 0$$, is $$abc > 0$$ ?

(1) $$|a –b | = |a| - |b|$$

(2) $$|b + c| = |b| + |c|$$

A brilliant question!

Thanks,
Mike
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Mike McGarry
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Re: If abc ≠ 0, is abc > 0 ?  [#permalink]

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11 Aug 2017, 17:55
from 2 you get that b,c should have the same sign therefore the question collapses to a>?0 NS.
from 1 you get that a,b should have same sign (try different signs and get contradiction) the question collapses to c>?0 NS.
Combine together, NS as if all tree positive then its true if all negative you get false.
WDYT?

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Posts: 6278
Re: If abc ≠ 0, is abc > 0 ?  [#permalink]

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11 Aug 2017, 20:16
Bunuel wrote:
If $$abc ≠ 0$$, is $$abc > 0$$ ?

(1) $$|a –b | = |a| - |b|$$

(2) $$|b + c| = |b| + |c|$$

Hi..

Clearly each sentence talks of two variables, hence insufficient individually..

Combined..
(1) $$|a –b | = |a| - |b|$$
I tells us that a>b or a=b and also both a and B are of same sign.
You can square also to find it..
$$a^2+b^2-2ab=a^2+b^2-2|a||b|........ 2|a||b|-2ab=0...$$
So a*b is POSITIVE and both are of same SIGN.

(2) [m]|b + c| = |b| + |c|
Here B and C are of same sign.

So combined all three are of same SIGN..
But if all three are positive, and is YES
If all three are NEGATIVE, and is NO
Insuff

C
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If abc ≠ 0, is abc > 0 ?  [#permalink]

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12 Aug 2017, 00:15
#STATEMENT 1
- Plugin value, we can get a&b both positive or a&b both negative.
- Hence, insufficient.

#STATEMENT 2
- Plugin value, we can get b&c both positive or b&c both negative.
- Hence, insufficient.

Anyway statement #1 and #2 only talk about 2 out of 3 variable - INSUFFICIENT from the begining.

#BOTH STATEMENT TOGETHER
We have two combinations :
1. a,b and c all POSITIVE = YES for the question.
2. a,b and c all NEGATIVE = NO for the question.
INSUFFICIENT

Is this correct?
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Re: If abc ≠ 0, is abc > 0 ?  [#permalink]

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12 Aug 2017, 00:19
chetan2u wrote:
Bunuel wrote:
If $$abc ≠ 0$$, is $$abc > 0$$ ?

(1) $$|a –b | = |a| - |b|$$

(2) $$|b + c| = |b| + |c|$$

Hi..

Clearly each sentence talks of two variables, hence insufficient individually..

Combined..
(1) $$|a –b | = |a| - |b|$$
I tells us that a>b or a=b and also both a and B are of same sign.
You can square also to find it..
$$a^2+b^2-2ab=a^2+b^2-2|a||b|........ 2|a||b|-2ab=0...$$
So a*b is POSITIVE and both are of same SIGN.

(2) [m]|b + c| = |b| + |c|
Here B and C are of same sign.

So combined all three are of same SIGN..
But if all three are positive, and is YES
If all three are NEGATIVE, and is NO
Insuff

C

chetan2u

Why you choose C although both statement insufficient?
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Re: If abc ≠ 0, is abc > 0 ?  [#permalink]

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12 Aug 2017, 11:20
Bunuel wrote:
If $$abc ≠ 0$$, is $$abc > 0$$ ?

(1) $$|a –b | = |a| - |b|$$

(2) $$|b + c| = |b| + |c|$$

Let's take 2 & -2 and plug them across the statements (or choose any 2 values). I choose a number and its apposite to make calculations easy.

(1) $$|a –b | = |a| - |b|$$

(a,b) = (2, 2)
$$|2 –2 | = |2| - |2|$$= 0

(a,b) = (-2, -2)
$$|-2 +2 | = |2| - |2|$$= 0

(a,b) = (-2, 2) and (2,- 2) are invalid as RHS does not equal LHS

Conclusion is a & b must have same sign but c can be any number with any sign

Insufficient

(2) $$|b + c| = |b| + |c|$$

(b,c) = (2, 2)
$$|2 + 2| = |2| + |2|$$= 4

(b,c) = (-2,-2)
$$|-2 -2| = |-2| + |-2|$$= 4

(b,c) = (-2, 2) and (2,- 2) are invalid as RHS does not equal LHS

Conclusion is b & c must have same sign but a can be any number with any sign

Insufficient

Combining 1 & 2

Case I: 2, 2 , 2.................Answer is Yes

Case II: -2, -2,-2..............Answer is NO

Insufficient

Re: If abc ≠ 0, is abc > 0 ? &nbs [#permalink] 12 Aug 2017, 11:20
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