Welcome! Check out our free B-School Guides to learn how you compare with other applicants.

## absolute value

tagged by:

This topic has 3 expert replies and 23 member replies
Goto page
apex231 Really wants to Beat The GMAT!
Joined
11 May 2009
Posted:
127 messages
Thanked:
5 times
Sat Sep 17, 2011 10:12 am
I have a question.

Statement 1) x + 2|y| = 0, which can be written as

x + 2y = 0
x - 2y = 0

For both these equations to satisfy, shouldn't x and y be both zero? Doesn't this make stmt 1 sufficient?

Same with Statement 2.
2)y + 2|x| = 0

Need free GMAT or MBA advice from an expert? Register for Beat The GMAT now and post your question in these forums!
Sharma_Gaurav Really wants to Beat The GMAT!
Joined
07 Jul 2009
Posted:
160 messages
Followed by:
1 members
Thanked:
1 times
Mon Jan 09, 2012 4:50 pm

stmt 1 and 2 are not SUFF alone.
Stmt1) variables can have either zero or non zero values,
stmt 2) same as above
but when we combined them , x,y can only have 0 value in order to satisfy the equation

preethikrishna Just gettin' started!
Joined
17 Jan 2012
Posted:
8 messages
Thu Feb 02, 2012 6:50 am
(C)

Joined
18 Jun 2012
Posted:
4 messages
Sun Jul 08, 2012 5:00 am
C
both should be 0

jcnasia Just gettin' started!
Joined
21 Jun 2012
Posted:
19 messages
Thanked:
4 times
Mon Jul 09, 2012 12:44 am
Here's a fun proof to show that if x + 2|y| = 0 and y + 2|x| = 0 then |x| + |y| = 0.

x + 2|y| = 0 and y + 2|x| = 0
=> x + y + 2|x| + 2|y| = 0 //add the two equations to each other
=> -1/2(x + y) = |x| + |y| //rearrange the equation

-1/2(x + y) <= |-1/2(x + y)| //since absolute value of a number is never less than the number
=> -1/2(x + y) <= 1/2(|x + y|) //rearrange inequality

1/2(|x + y|) <= 1/2(|x| + |y|) //since |x + y| is always less than or equal to |x| + |y| (see below for proof of this)
1/2(|x| + |y|) <= |x| + |y| //since 1/2 of a non-negative number is always less than the whole number

By combining all these inequalities, we get...
-1/2(x + y) <= 1/2(|x + y|) <= 1/2(|x| + |y|) <= |x| + |y|

1/2(|x| + |y|) = |x| + |y| //since -1/2(x + y) = |x| + |y| and 1/2(|x| + |y|) is between these two values
=> 0 = 1/2(|x| + |y|) //rearrange the equation
Therefore: 0 = |x| + |y| //rearrange the equation

Well, to me, it's a fun proof, but you should probably use one of the previous solutions on the actual gmat so you don't waste time.

For all x and y, is |x + y| <= |x| + |y|?

There exists x and y such that |x + y| > |x| + |y| //Assume the opposite and look for a contradiction
|x + y|^2 > (|x| + |y|)^2 //square two positive numbers and the inequality will hold
(x + y)^2 > x^2 + 2|x||y| + y^2 //since a square of a number is always positive
x^2 + 2xy + y^2 > x^2 + 2|x||y| + y^2 //algebra
xy > |x||y| //subtract x^2 + y^2 from both sides
xy > |xy|
But this is a contradiction since a number (xy) is never greater than its absolute value (|xy|).
Therefore, our original statement is true: for all x and y, |x + y| <= |x| + |y|

Joined
20 Jul 2012
Posted:
3 messages
Target GMAT Score:
720
Sun Aug 05, 2012 5:54 am
|x| + |y| = 0 translates into x = y = 0 as addition of two positive values can be zero only when each is zero.
statements are not sufficient individually that's easy to see.
however combining the two statement and getting an indicative answer is something that needs a little skill.

I have to practice more.

eagleeye GMAT Destroyer!
Joined
28 Apr 2012
Posted:
521 messages
Followed by:
48 members
Thanked:
329 times
Test Date:
August 18, 2012
GMAT Score:
770
Sun Aug 05, 2012 6:11 am
|x| + |y| = 0 translates into x = y = 0 as addition of two positive values can be zero only when each is zero.
statements are not sufficient individually that's easy to see.
however combining the two statement and getting an indicative answer is something that needs a little skill.

I have to practice more.
Since most people have figured out that required condition is |x|=|y|=0, but struggle with combining the statements, here's how to go about it:

1)x + 2|y| = 0
=> x= -2|y|
=> |x| = |-2|y|| (take absolute value on both sides)
=> |x| = 2|y|

2)y + 2|x| = 0
In the same way as above,
y = -2|x|
=> |y| = 2|x|
=> |y| = 2 (2|y|) = 4|y| (substituting the equation above)
=> |y| (1-4)=0
=> |y| = 0.
Hence |x| = 2|y| = 0 also.
So we get |x|=|y|=0. Sufficient. And C is correct
Cheers!

mparakala Rising GMAT Star
Joined
07 May 2012
Posted:
87 messages
Thanked:
1 times
Wed Nov 07, 2012 10:00 am
the only way a sufficent ans can be obtained is by using both (1) and (2)
so if x=0 and y = 0 that satisfies both 1 and 2, then the question can be answered as a "yes"

C

rajeshsinghgmat Really wants to Beat The GMAT!
Joined
08 Jan 2013
Posted:
162 messages
Thanked:
1 times
Fri Jan 25, 2013 1:03 am

kinji@BTG Just gettin' started!
Joined
23 Apr 2011
Posted:
4 messages
Mon Feb 25, 2013 4:14 am
prashant misra wrote:
i am still not able to understand why the answer is C.how have been the two options combined.can anyone explain
|x| + |y| = 0 means that |x| and |y| should individually = 0

Statement 1:
x + 2|y| = 0
x = -2|y|

Putting in the main equation, 2|y| + |y| = 3|y| may be = 0 or may not be =0, Not sufficient.

Similarly for statement 2, Not sufficient.

If we combine Statement 1 and Statement 2:
From Statement 1:
x = -2|y|
From Statement 2:
y = -2|x|

Hence y = -4|y|,
if y <0
y = -4(-y) = 4y => 3y = 0, hence x and y individually = 0

if y > 0
y = -4y => 5y = 0 again x and y individually = 0

Hence in either case, it is sufficient.
Hence both is required and hence C

### GMAT/MBA Expert

GMATGuruNY GMAT Instructor
Joined
25 May 2010
Posted:
6114 messages
Followed by:
1043 members
Thanked:
5294 times
GMAT Score:
790
Mon Feb 25, 2013 9:50 am
colakumarfanta wrote:
Is |x| + |y| = 0?
1)x + 2|y| = 0
2)y + 2|x| = 0
|x| and |y| must be NONNEGATIVE.
Thus, for the sum of |x| and |y| to be 0, |x| and |y| themselves must each be equal to 0.

Question rephrased: Does x=y=0?

Statement 1: x = -2|y|.
It's possible that x=y=0.
It's possible that y=1 and x=-2.
INSUFFICIENT.

Statement 2: y = -2|x|.
It's possible that x=y=0.
It's possible that x=1 and y=-2.
INSUFFICIENT.

Statements combined:
Substituting y=-2|x| into x = -2|y|, we get:
x = -2| -2|x| |
x = -2 |2x|
x = -4|x|.

Since |x| must be NONNEGATIVE, |x|≥0.
If |x| > 0, then both sides of the equation above can safely be divided by |x|:
x/|x| = -4.
Not possible:
If x>0, then x/|x| = 1.
If x<0, then x/|x| = -1.

Since it is not possible that |x|>0, |x| = 0.
Since y = -2|x|, y = -2*0 = 0.
Thus, x=y=0.
SUFFICIENT.

_________________
Mitch Hunt
GMAT Private Tutor and Instructor
GMATGuruNY@gmail.com
If you find one of my posts helpful, please take a moment to click on the "Thank" icon.
Contact me about long distance tutoring!

Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now.
rakhibubbly86 Just gettin' started!
Joined
17 Apr 2013
Posted:
1 messages
Thu Apr 18, 2013 4:46 am
The answer is C because when you combine A and B the example values should satisfy both A and B and 0,0 is the only set which satisfy both A and B.

### Best Conversation Starters

1 varun289 31 topics
2 sana.noor 23 topics
3 killerdrummer 21 topics
4 Rudy414 19 topics
5 sanaa.rizwan 14 topics
See More Top Beat The GMAT Members...

### Most Active Experts

1 Brent@GMATPrepNow

GMAT Prep Now Teacher

199 posts
2 GMATGuruNY

The Princeton Review Teacher

134 posts
3 Jim@StratusPrep

Stratus Prep

106 posts
4 Anju@Gurome

Gurome

47 posts