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Is |a| + |b| > |a + b| ?

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Is |a| + |b| > |a + b| ? [#permalink] New post 28 Nov 2010, 06:05
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Is |a| + |b| > |a + b| ?

(1) a^2 > b^2
(2) |a| * b < 0
[Reveal] Spoiler: OA

Last edited by Bunuel on 04 Dec 2012, 00:57, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Absolute value Problems [#permalink] New post 28 Nov 2010, 06:48
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chiragatara wrote:
Hi Bunuel,
I am not very much comfortable with Absolute value Problems, e.g.

Is |a| + |b| > |a + b| ?
(1) a2 > b2
(2) |a| × b < 0

Is there any GYAN Material for improvement? Kindly help.


Hi, Don't worry. You will learn the things and GMAT absolute value tricks sooooon.

The be low is my approcah for any modulus qtn in GMAT.

Remember.
The meaning of |x-y| is "On the number line, the distance between X and +Y"
The meaning of |x+y| is "On the number line, the distance between X and -Y"
The meaning of |x| is "On the number line, the distance between X and 0".

On the # line, Left to 0 are all the -ve #s and right to 0 are all +ve #s

Original Qtn:
Is |a| + |b| > |a + b| ?

menas "is the SUM of the distance between a and 0, and the distance bewtween b and 0 > the distance bewtween a+b and 0"

let us take some cases to simplify the qtn.

case1: if a and b both are RIGHT to 0 on the # line then a+b will also be RIGHT to zero and even more far away from 0 than a or/and b is/are.

Then |a| + |b| > |a + b| is FLASE
and infact LHS is always = RHS
e.g: a = 1 and b = 3 then a+b = 4
a is 1 unit away right to 0
b is 3 units away right to
a+b is 4 units away right to 0
==>
the SUM of the distance between a and 0, and the distance between b and 0 = 1+3 = 4
and the distance b/w a+b and 0 is 4

-------------0----------b-----a---------a+b---

case2: if a and b both are LEFT to 0 on the # line then a+b will also be LEFT to zero and even more far away from 0 than a and/or b is.

Then |a| + |b| > |a + b| is FLASE
and infact LHS is always = RHS
e.g: a = -1 and b = -3 then a+b = -4
a is 1 unit away left to 0
b is 3 units away left to
a+b is 4 units away left to 0
==>
the SUM of the distance between a and 0, and the distance between b and 0 = 1+3 = 4
and the distance b/w a+b and 0 is 4

----a+b------a---------b-------0-------------

HENCE,

for |a| + |b| > |a + b| or
|a| + |b| < |a + b| or
|a| + |b| not= |a + b| to be true,
a and b, on the # line, should NOT be on the same side with respect to 0.

So BASICALLY the qtn is ASKING IF a AND b ARE on DEFFERENT SIDES, with respect to 0, on THE # LINE>

stmnt1: a^2 > b^2
from this we can say that the magnitude of a is > the magnitude of b. magnitude means with out sign.
==> it says that |a| > |b|
FROM THIS,
a and b can be on the same side on the # line with respect to 0

-------0------b----------------a , obviosly a is toofar ways from 0 than b is ==> |a| > |b|

OR,

a can be on a different side than b
-----a-------------0---b------ again, a is toofar ways from 0 than b is ==> |a| > |b|

so we have both the cases...hence stmnt 1 is NOT suff.


stmnt2: |a| × b < 0

|a| is always +ve hence b shud be -ve for the product to be -ve.

so all that stmnt 2 says is b is LEFT to 0 on the # line but it does not say whether a is on LEFT or RIGHT to 0. hence NOT suff.

stmnts 1&2 together
stmnt:1 says that a is too far from zero than b is.
stmnt2 says that b is LEFT to 0 on the # line
hence using both too, we can not say if a and b are on different sides to 0 on the #line or on the on the same sides
hence NOT suff.

ANSWER..."E"

Regards,
Murali.
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Re: Absolute value Problems [#permalink] New post 28 Nov 2010, 07:13
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Expert's post
chiragatara wrote:
Hi Bunuel,
I am not very much comfortable with Absolute value Problems, e.g.

Is |a| + |b| > |a + b| ?
(1) a2 > b2
(2) |a| × b < 0

Is there any GYAN Material for improvement? Kindly help.


You should notice that inequality |a|+|b|>|a+b| holds true if and only a and b have opposite sign, as only in this case absolute value of positive+negative will be less than |positive|+|negative|. For example |-2|+|3|>|-2+3|. In all other cases |a|+|b|=|a+b|.

So, basically the question is whether a and b have opposite sign.

(1) a^2 > b^2 --> can not determine whether a and b have opposite sign. Not sufficient.
(2) |a|*b<0 --> just tells us that b<0, but we don't know the sign of a. Not sufficient.

(1)+(2) b<0 and a^2>b^2 --> still can not get the sign of a. Not sufficient.

Answer: E.

For theory check About Value chapter of Math Book created by Walker: math-absolute-value-modulus-86462.html

Absolute value PS questions: search.php?search_id=tag&tag_id=58
Absolute value DS questions: search.php?search_id=tag&tag_id=37

Hard absolute value questions: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope it helps.
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Re: Absolute value Problems [#permalink] New post 15 Jun 2011, 23:32
clean E here.
b < 0 but a can be < > 0
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Re: Absolute value Problems [#permalink] New post 24 Aug 2011, 19:09
|a| + |b| > |a+b|?

if a and b are of same sign then LHS = RHS. Hence No.
if a and b are of opposite sign then LHS > RHS . Then Yes.

1. Not sufficient

|a|>|b|

we cannot say anything about the sign of a and b.

2. Not sufficient

b<0, but we dont know the sign of a.

together,
|a|>|b| and b<0 => a can be positive or negative
Hence not sufficient.

Answer is E.
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Re: Absolute value Problems [#permalink] New post 30 Aug 2011, 01:08
|a| + |b| > |a + b|

the expression will be true if "a,b have opposite signs e.g. a<0, b>0 etc"

st1) insufficient. a or b could have same or different sign
st2) this proves that b<0. but it is also possible that a<0. so, insufficient.

E is the ans
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Re: Absolute value Problems [#permalink] New post 04 Dec 2012, 00:54
Is |a| + |b| > |a + b| ?
(1) Both positive |5| + |5| > |5 + 5| NO!
(2) Both negative |-5| + |-5| > |-10| NO!
(3) Oppositve signs |5| + |-5| > |5-5| YES!

(1) a2 > b2
|a| > |b|
This doesn't tell us anything about the sign. INSUFFICIENT.

(2) |a| × b < 0
This tells us b is negative. INSUFFICIENT.

Answer: E
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Ds Modules and inequalities [#permalink] New post 06 Jun 2013, 07:03
Is |a| + |b| > |a + b| ?

(1) a^2>b^2

(2)|a| * b<0

got me confusing for a while and choose E.

used the std rule of | A + B | < |A| + | B | when XY > 0

Since it is difficult to find the sign of A in any case selected E.

Please let me know if the approach to this problem is correct ?
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Re: Ds Modules and inequalities [#permalink] New post 06 Jun 2013, 08:43
Normally I solve it by plugging numbers.

1. a^2>b^2
lets take a=-3 and b=-2; so, |a|+|b|=5 & |a+b|=5
Lets take a=-3 and b=2; so |a|+|b| = 5 & |a+b|=1
two different values, hence insufficient.

2. |a|*b<0
This implies b<0.
Lets take a=-3 and b=-2; so |a| + |b| = 5 & |a+b|=5
Lets take a=3 and b=-2; so |a| + |b| = 5 & |a+b|=1
two different values, hence insufficient.

Taking both together:
we can use the same numbers in statement 2. Hence, insufficient.

Answer: E
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Re: Ds Modules and inequalities [#permalink] New post 06 Jun 2013, 09:01
Expert's post
Manhnip wrote:
Is |a| + |b| > |a + b| ?

(1) a^2>b^2

(2)|a| * b<0

got me confusing for a while and choose E.

used the std rule of | A + B | < |A| + | B | when XY > 0

Since it is difficult to find the sign of A in any case selected E.

Please let me know if the approach to this problem is correct ?


Merging similar topics.

Look at the original post here: is-a-b-a-b-105457.html#p824216 The name of the topics MUST be "Is |a| + |b| > |a + b| ?"
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

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Re: Is |a| + |b| > |a + b| ? [#permalink] New post 30 Jun 2013, 11:46
Is |a| + |b| > |a + b| ?

Is a^2 + b^2 > (a+b)^2?
is a^2 + b^2 > a^2 + 2ab + b^2?
Is 0 > 2ab?
Is ab negative?

(1) a^2 > b^2

This tells us that a^2 is greater than b^2 but it tells us nothing about the signs of a and b. For example, (-3)^2 > (2)^2 but -3 is less than 2. In this case, ab would be negative. On the other hand, (3)^2 > (2)^2 in which case 3 > 2 and ab would be positive.
INSUFFICIENT

(2) |a| * b < 0

This tells us that b must be negative as |a| will always be positive. however, a could be negative in which case ab would be positive or a could be negative in which case ab would be negative.
INSUFFICIENT

1+2) a^2 > b^2 and b is negative: that means the absolute value of |a| > |b|.

|-4| > |-1| ===> 4>1 Valid ===> ab = (-4)*(-1) = 4 (positive)
|4| > |-1| ===> 4>1 Valid ===> ab = (4)*(-1) = -4 (negative)

In other words, ab could be either positive or negative.
INSUFFICIENT

(E)
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Re: Is |a| + |b| > |a + b| ? [#permalink] New post 22 Jul 2013, 12:41
Is |a| + |b| > |a + b| ?

|4| + |2| > |4 + 2| 6>6 Invalid

|-4| + |-2| > |-4 + -2| 6>6 Invalid

|-4| + |2| > |-4 + 2| 6>2 Valid

|4| + |-2| > |4 + -2| 6> 2 Valid

(Take note that the only two valid cases are when a and b have opposite signs)

The question then becomes, do a and b have opposite signs?

(1) a^2 > b^2
|a| > |b| but we don't know the signs of either.
INSUFFICIENT

(2) |a| * b < 0
For this to hold true, regardless of what a is (any number but zero) b must be negative. Like #1, this tells us nothing about the sign of a.
INSUFFICIENT


1+2) a^2 > b^2, |a| * b < 0
we know that b is negative but a could be positive or negative and a^2 > b^2 could still hold true. For example:

-6^2 > -5^2
36>25
-a, -b

6^2 > -5^2
36 > 25
a, -b

INSUFFICIENT

(E)




There is a similar problem in which you plug in numbers to solve (is-xy-0-1-x-3-y-5-x-y-2-0-2-x-y-x-y-86780.html) but when I try to solve it, it becomes a confusing mess. Could someone explain to me where I went wrong?

So, valid when:
|-a| > |b|
|a| > |-b|

(1) a^2 > b^2
The absolute value of a must be greater than the absolute value of b. For #1:
|a| > |b|
|-a| > |b|
|-a| > |-b|

Therefore, according to #1, the values of a and b could make the inequality |4| + |2| > |4 + 2| invalid (for example, if |a|>|b| or valid if |-a| > |b|)
INSUFFICIENT

(2) |a| * b < 0
|a|*b < 0
This is saying [positive or zero] * [b] < 0. If a is positive, b is negative. a and b cannot = 0. b is always negative regardless of what a is as |a| will always be positive or zero.

|-a| * (-b) < 0
|a| * (-b) < 0

The problem is valid when |-a| > |-b|. We don't know if |-a| > |b| as is stated in the stem. For example, |-1| * (-4) < 0 but |-1| is not greater than |-4|
INSUFFICIENT
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Re: Is |a| + |b| > |a + b| ? [#permalink] New post 07 Apr 2014, 04:55
From question we need only to know if ab<0.

Statement 1 tells us that abs A > abs B, which is clearly insufficient.

Statement 2 tells us that b<0.

From both statements we can't tell whether a is negative.

Therefore E is the correct answer choice

Hope this helps
Cheers
J :)
Re: Is |a| + |b| > |a + b| ?   [#permalink] 07 Apr 2014, 04:55
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