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# Is |x + y| < |x| + |y|?

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Intern
Joined: 21 Nov 2013
Posts: 39
Is |x + y| < |x| + |y|?  [#permalink]

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23 Jan 2014, 12:56
1
11
00:00

Difficulty:

35% (medium)

Question Stats:

73% (01:51) correct 27% (02:01) wrong based on 353 sessions

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Is |x + y| < |x| + |y|?

(1) | x | ≠ | y |
(2) | x – y | > | x + y |

Appreciate views on how to solve this. Thank you.
I did this by picking numbers choosing various numbers positive, negative, fractions, 0.
Took make almost four minutes. Any other way?
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8678
Location: Pune, India
Re: Is |x + y| < |x| + |y|?  [#permalink]

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23 Jan 2014, 21:02
7
13
BabySmurf wrote:
Is |x + y| < |x| + |y|?

(1) | x | ≠ | y |
(2) | x – y | > | x + y |

Appreciate views on how to solve this. Thank you.
I did this by picking numbers choosing various numbers positive, negative, fractions, 0.
Took make almost four minutes. Any other way?

This question can be done very easily if you are familiar with the properties of absolute value.
For all real x and y, $$|x + y| \leq |x| + |y|$$

$$|x + y| = |x| + |y|$$ when x and y have the same sign (e.g. x = 4, y = 6 OR x = -2, y = -3) or at least one of them is 0.
When x and y have opposite signs, $$|x + y| < |x| + |y|$$

So the question asks us whether x and y have opposite signs or not.

(1) | x | ≠ | y |
Doesn't tell us anything about the signs of x and y. Not sufficient.

(2) | x – y | > | x + y |
When will | x – y | be greater than | x + y |? Only when x and y have opposite signs e.g. | x – y | = |3 - (-2)| = 5 but | x + y | = |3 - 2| = 1
If x and y have the same sign or one of them is 0, | x + y | will be greater or equal to | x – y |.
Hence this statement tells us that x and y have opposite signs. So it is sufficient alone.

_________________

Karishma
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Re: Is |x + y| < |x| + |y|?  [#permalink]

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23 Jan 2014, 13:19
5
BabySmurf wrote:
Is |x + y| < |x| + |y|?

(1) | x | ≠ | y |
(2) | x – y | > | x + y |

Appreciate views on how to solve this. Thank you.
I did this by picking numbers choosing various numbers positive, negative, fractions, 0.
Took make almost four minutes. Any other way?

for |x+y| <|x|+|y|
we have four cases
1.x +ve ,y+ve
in that case
|x+y| =|x|+|y|
2.x-ve ,y -ve
in that case
|x+y| =|x|+|y|
3.x+ve and y -ve
in that case
|x+y| <|x|+|y|
4.x -ve and Y +ve
in that case
|x+y| <|x|+|y|
and if x=0 or y =0 (i.e. any one equal to zero)
|x+y| =|x|+|y|

so if x and y are of opposite signs then only |x + y| < |x| + |y|
now 1 gives
|x|=|y|
with this we cant answer the question as we donot know signs of x and y
2.| x – y | > | x + y |
this implies that x and y are of opposite signs. So with this we can answer the question.
Key point here is you should understand immediately after looking at the inequality that its valid only if x and y are of opposite signs.
Give me kudos if this helps
##### General Discussion
Intern
Joined: 21 Nov 2013
Posts: 39
Re: Is |x + y| < |x| + |y|?  [#permalink]

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23 Jan 2014, 13:42
cherukuri1011 wrote:
BabySmurf wrote:
Is |x + y| < |x| + |y|?

(1) | x | ≠ | y |
(2) | x – y | > | x + y |

Appreciate views on how to solve this. Thank you.
I did this by picking numbers choosing various numbers positive, negative, fractions, 0.
Took make almost four minutes. Any other way?

for |x+y| <|x|+|y|
we have four cases
1.x +ve ,y+ve
in that case
|x+y| =|x|+|y|
2.x-ve ,y -ve
[....]

I am unsure with "ve" is... Thanks.
Current Student
Joined: 26 May 2013
Posts: 28
Location: United States
Surendra: Cherukuri
Concentration: Operations, Technology
GMAT 1: 670 Q48 V34
GMAT 2: 670 Q47 V35
GMAT 3: 720 Q50 V36
GPA: 4
WE: Research (Telecommunications)
Re: Is |x + y| < |x| + |y|?  [#permalink]

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23 Jan 2014, 14:00
BabySmurf wrote:
cherukuri1011 wrote:
BabySmurf wrote:
Is |x + y| < |x| + |y|?

(1) | x | ≠ | y |
(2) | x – y | > | x + y |

Appreciate views on how to solve this. Thank you.
I did this by picking numbers choosing various numbers positive, negative, fractions, 0.
Took make almost four minutes. Any other way?

for |x+y| <|x|+|y|
we have four cases
1.x +ve ,y+ve
in that case
|x+y| =|x|+|y|
2.x-ve ,y -ve
[....]

I am unsure with "ve" is... Thanks.

Hi BabySmurf,
I used a shorter expression X +ve means x is positive number and x -ve means if x is a negative number
Math Expert
Joined: 02 Sep 2009
Posts: 51229
Re: Is |x + y| < |x| + |y|?  [#permalink]

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23 Jan 2014, 23:53
1
2
BabySmurf wrote:
Is |x + y| < |x| + |y|?

(1) | x | ≠ | y |
(2) | x – y | > | x + y |

Appreciate views on how to solve this. Thank you.
I did this by picking numbers choosing various numbers positive, negative, fractions, 0.
Took make almost four minutes. Any other way?

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Hope this helps.
_________________
Intern
Joined: 16 Dec 2013
Posts: 37
Location: United States
GPA: 3.7
Re: Is |x + y| < |x| + |y|?  [#permalink]

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14 Apr 2016, 08:07
Bunuel wrote:
BabySmurf wrote:
Is |x + y| < |x| + |y|?

(1) | x | ≠ | y |
(2) | x – y | > | x + y |

Appreciate views on how to solve this. Thank you.
I did this by picking numbers choosing various numbers positive, negative, fractions, 0.
Took make almost four minutes. Any other way?

Similar questions to practice:
is-x-y-x-y-87926.html
is-x-y-x-y-1-x-y-2-x-y-x-101660.html
is-x-y-x-y-123108.html
is-x-y-x-y-160531.html
is-x-y-x-z-1-y-z-2-x-86132.html
is-a-b-a-b-105457.html
is-x-y-x-y-1-x-y-2-x-y-132654.html
is-x-y-146991.html
is-x-y-x-y-137050.html
is-x-z-y-x-z-y-154245.html

Hope this helps.

If I square both sides, I am going to get xy<|x||y|. That would give B as an answer.

I have found squaring methods to help in inequalities (though not in every case)...
Board of Directors
Joined: 17 Jul 2014
Posts: 2618
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30
GPA: 3.92
WE: General Management (Transportation)
Re: Is |x + y| < |x| + |y|?  [#permalink]

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13 Sep 2016, 06:05
BabySmurf wrote:
Is |x + y| < |x| + |y|?

(1) | x | ≠ | y |
(2) | x – y | > | x + y |

i picked B.

1. x has a different absolute value than y.
1st case: x=2, b=3 => |x+y|=|x|+|y| no
2nd case: x=2, b=-3 -> |x+y|<|x|+|y|

2. it must be true that at least one of the numbers is negative, in which case |x + y| < |x| + |y| is always true
Non-Human User
Joined: 09 Sep 2013
Posts: 9184
Re: Is |x + y| < |x| + |y|?  [#permalink]

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13 Oct 2017, 20:51
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: Is |x + y| < |x| + |y|? &nbs [#permalink] 13 Oct 2017, 20:51
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