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# Is |a - b| < |a| + |b| ? (1) a/b < 0 (2) a^2*b < 0

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Director
Joined: 02 Oct 2017
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Is |a - b| < |a| + |b| ? (1) a/b < 0 (2) a^2*b < 0  [#permalink]

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27 Jul 2018, 05:59
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85% (hard)

Question Stats:

52% (02:01) correct 48% (02:05) wrong based on 156 sessions

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Is |a - b| < |a| + |b| ?

(1) a/b < 0

(2) a^2*b < 0

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Math Expert
Joined: 02 Aug 2009
Posts: 7334
Is |a - b| < |a| + |b| ? (1) a/b < 0 (2) a^2*b < 0  [#permalink]

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27 Jul 2018, 07:09
Is |a-b|<|a|+|b| ?

When will be |a-b|=|a|+|b|
This will happen when both X and y are of different signs...
In |a-b|, both a and B will get added if they are different signs and it's Modulus will be equal to |a|+|b|

1)a/b <0
This means both a and B are of opposite sign..
Sufficient

2)a^2*b<0
b<0, but we cannot say if a is negative or positive as a^2 will always be positive
Insufficient

A
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

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Re: Is |a - b| < |a| + |b| ? (1) a/b < 0 (2) a^2*b < 0  [#permalink]

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27 Jul 2018, 09:50
We need to check if a and b have same signs, only then the given statement holds true.

(1) a/b = -ve, therefore a and b have opposite signs. Sufficient.

(2) a^2*b<0, therefore b is negative. But no information about a as square of any number is always positive. Insufficient.

Hence, A.
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Joined: 26 Mar 2013
Posts: 2060
Re: Is |a - b| < |a| + |b| ? (1) a/b < 0 (2) a^2*b < 0  [#permalink]

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18 Aug 2018, 02:57
chetan2u wrote:
]Is |a-b|<|a|+|b|

This will happen when both X and y are of different signs...
In |a-b|, both a and B will get added if they are different signs and it's Modulus will be equal to |a|+|b|

Hi chetan2u

As per you statement above, it holds true if they are different signs. This is not the case.

x=3 & y=1.......|2|<|3|+|1|.......|2|<|4|.............question holds true when x & y same sign

x=-3 & y=-1.......|-2|<|3|+|1|.......|2|<|4|.........question holds true when x & y same sign

x=-3 & y=1.......|-2|<|3|+|1|.......|-4|<|4|.........question does NOT hold true when x & y different sign

x=3 & y=-1.......|2|<|3|+|1|.......|4|<|4|...........question does NOT hold true when x & y different sign
Math Expert
Joined: 02 Aug 2009
Posts: 7334
Is |a - b| < |a| + |b| ? (1) a/b < 0 (2) a^2*b < 0  [#permalink]

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18 Aug 2018, 03:10
Mo2men wrote:
chetan2u wrote:
]Is |a-b|<|a|+|b|

This will happen when both X and y are of different signs...

In |a-b|, both a and B will get added if they are different signs and it's Modulus will be equal to |a|+|b|

Hi chetan2u

As per you statement above, it holds true if they are different signs. This is not the case.

x=3 & y=1.......|2|<|3|+|1|.......|2|<|4|.............question holds true when x & y same sign

x=-3 & y=-1.......|-2|<|3|+|1|.......|2|<|4|.........question holds true when x & y same sign

x=-3 & y=1.......|-2|<|3|+|1|.......|-4|<|4|.........question does NOT hold true when x & y different sign

x=3 & y=-1.......|2|<|3|+|1|.......|4|<|4|...........question does NOT hold true when x & y different sign

Hi...

That statement is for |a-b|=|a|+|B| . I have also explained there. See the coloured portion
Here a and B of different sign will add up and give same result on two sides.
However in other cases, LHS<RHS.
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

GMAT Expert

SVP
Joined: 26 Mar 2013
Posts: 2060
Re: Is |a - b| < |a| + |b| ? (1) a/b < 0 (2) a^2*b < 0  [#permalink]

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18 Aug 2018, 11:45
push12345 wrote:
Is |a - b| < |a| + |b| ?

(1) a/b < 0

(2) a^2*b < 0

We can interpret the question stem as follows:

Is 'some positive number<some positive number'?

So based on the above, we can square the both sides safely without flipping the sign

Is (|a - b|)^2 < (|a| + |b|)^2 ?

a^2 -2ab +b^2 < a^2 +2|a||b|+b^2.......cancel out a^2 & b^2 from both sides

-2ab < 2|a||b|........divide by 2

-ab < |a||b|........This will happen if a & b have same sign only. So we can rephrase question:

Is ab > 0?

(1) a/b < 0

It means that a & b have different sign.........Answer is always NO.

Sufficient

(2) a^2*b < 0

b is negative sign but we can't determine the sign of a because a^2 disguise the sign of a. it might be negative or positive.

Insufficient

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Re: Is |a - b| < |a| + |b| ? (1) a/b < 0 (2) a^2*b < 0  [#permalink]

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18 Aug 2018, 22:54
Hi,

Question: Is I a-b| < |a| + |b| ?

The above modulus expression will be true (answer “Yes” to the question) only if “a” and “b” have same signs.

Say for an example, if a =2 and b= 3,

1 < 5

Say if both are negative, a = -2 and b = -3

1 < 5

For “a” and “b” different signs then the above modulus expression won’t hold true (answer “NO” to the question).

Say for an example, if a = -2 and b = 3

5 = 5

Statement I is sufficient:

a/b < 0

So, a and b should have alternate signs.

So the answer to the question would be “NO”. A definite NO, so sufficient. Because it always be equal if they have different signs.

So sufficient.

Statement II is insufficient:

a^2 * b < 0

Since a^2 is positive number, b has to be negative.

But only thing here, a^2 is positive we cant say whether “a” is positive or negative.

If “a” is positive and since b is negative, they have alternate signs, so the answer to the question would be “NO”.

But if “a” is negative and since b is negative, they have same signs, so the answer to the question would be “YES”.

So, Statement II is insufficient.

So, the answer is A(I alone sufficient)
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Joined: 21 May 2018
Posts: 1
Re: Is |a - b| < |a| + |b| ? (1) a/b < 0 (2) a^2*b < 0  [#permalink]

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25 Aug 2018, 11:05
chetan2u

Can you provide a numeric example for why B is wrong please? I have failed to find any that fit.
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Joined: 26 Mar 2013
Posts: 2060
Re: Is |a - b| < |a| + |b| ? (1) a/b < 0 (2) a^2*b < 0  [#permalink]

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25 Aug 2018, 17:23
Cindydu wrote:
chetan2u

Can you provide a numeric example for why B is wrong please? I have failed to find any that fit.

Hi,

I'm happy to help. Please look below

x=-3 & y=-1.......(-3)^2 *(-1) <0........Substitute in the question: |-2|<|3|+|1|.......|2|<|4|.........Answer is Yes

x=3 & y=-1.........(3)^2 *(-1) <0.........Substitute in the question: |2|<|3|+|1|.......|4|<|4|...........Answer is No

Tow different answer........So B is insufficient

I hope it helps
Math Expert
Joined: 02 Aug 2009
Posts: 7334
Re: Is |a - b| < |a| + |b| ? (1) a/b < 0 (2) a^2*b < 0  [#permalink]

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25 Aug 2018, 17:54
1
Cindydu wrote:
chetan2u

Can you provide a numeric example for why B is wrong please? I have failed to find any that fit.

Hi,

When will be |a-b|=|a|+|b|
This will happen when both X and y are of different signs...
In |a-b|, both a and B will get added if they are different signs and it's Modulus will be equal to |a|+|b|

2)a^2*b<0
b<0, but we cannot say if a is negative or positive as a^2 will always be positive
Insufficient

So take a as 5 and b as -2
|5-(-2)|=|5|+|-2|........|5+2|=5+2......yes
Now take a as -5 and b as -2
|-5-(-2)|=|-5|+|-2|.........|-5+2|=5+2.......|-3|=7.......no
So insufficient
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

GMAT Expert

Re: Is |a - b| < |a| + |b| ? (1) a/b < 0 (2) a^2*b < 0   [#permalink] 25 Aug 2018, 17:54
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