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

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Intern
Joined: 21 Nov 2013
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Kudos [?]: 31 [0], given: 3

Is |x + y| < |x| + |y|? [#permalink]  23 Jan 2014, 12:56
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Difficulty:

35% (medium)

Question Stats:

73% (02:02) correct 27% (01:37) wrong based on 83 sessions
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?
[Reveal] Spoiler: OA
Intern
Joined: 26 May 2013
Posts: 32
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)
Followers: 0

Kudos [?]: 18 [0], given: 10

Re: Is |x + y| < |x| + |y|? [#permalink]  23 Jan 2014, 13:19
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
Intern
Joined: 21 Nov 2013
Posts: 41
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Kudos [?]: 31 [0], given: 3

Re: Is |x + y| < |x| + |y|? [#permalink]  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.
Intern
Joined: 26 May 2013
Posts: 32
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)
Followers: 0

Kudos [?]: 18 [0], given: 10

Re: Is |x + y| < |x| + |y|? [#permalink]  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
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 5870
Location: Pune, India
Followers: 1485

Kudos [?]: 8005 [0], given: 190

Re: Is |x + y| < |x| + |y|? [#permalink]  23 Jan 2014, 21:02
Expert's post
2
This post was
BOOKMARKED
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.

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Math Expert
Joined: 02 Sep 2009
Posts: 29205
Followers: 4751

Kudos [?]: 50291 [1] , given: 7540

Re: Is |x + y| < |x| + |y|? [#permalink]  23 Jan 2014, 23:53
1
KUDOS
Expert's post
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.
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Re: Is |x + y| < |x| + |y|? [#permalink]  27 Jul 2015, 18:34
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Re: Is |x + y| < |x| + |y|?   [#permalink] 27 Jul 2015, 18:34
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