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Is |x| + |y| = 0?

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Is |x| + |y| = 0? [#permalink] New post 21 Jun 2012, 18:48
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Is |x| + |y| = 0?

(1) x + 2|y| = 0
(2) y + 2|x| = 0
[Reveal] Spoiler: OA
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Re: Absolute Value [#permalink] New post 21 Jun 2012, 20:54
nades09 wrote:
Is |x| + |y| = 0?

1. x + 2|y| = 0
2. y + 2|x| = 0

Thanks

Hi,

|x|+|y|=0?
or |x|=-|y|?
Both |x| and |y| are positive, so is it possible that |x| is negative of some positive quantity,
thus, only possibility would be |x|=|y|=0? and that's the question.

Using (1),
|y|=-x/2
or x \leq 0, so, it is possible that x = -1, y = 1/2, then |x|+|y| \neq 0
or x =0, y = 0, then |x|+|y| = 0, Insufficient.

Using (2);
|x|=-y/2
or y \leq 0, so, it is possible that y = -1, x = 1/2, then |x|+|y| \neq 0
or y = 0, x = 0, then |x|+|y| = 0, Insufficient.

Combining both,
x \leq 0, then |x|=-y/2
implies, -x=-y/2 or 2x = y.......(a)

Similarly, y \leq 0, then |y|=-x/2
implies, -y=-x/2 or x = 2y........(b)
From (a) & (b)
4y=y, thus y=0 & x=0
Also, |x|+|y| = 0. Sufficient.

Answer is (C).

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Re: Is |x| + |y| = 0? [#permalink] New post 22 Jun 2012, 00:35
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Is |x| + |y| = 0?

Since absolute value is non-negative the from |x| + |y| = 0 we have that the sum of two non-negative values equals to zero, which is only possible if both of them equal to zero. So, the question basically asks whether x=y=0

(1) x + 2|y| = 0. It's certainly possible that x=y=0 but it's also possible that x=-2 and y=1. Not sufficient.

Notice that from this statement |y|=-\frac{x}{2}, so -\frac{x}{2} equals to a non-negative value (|y|), so -\frac{x}{2}\geq{0} --> x\leq{0}.

(2) y + 2|x| = 0. It's certainly possible that x=y=0 but it's also possible that y=-2 and x=1. Not sufficient.

Notice that from this statement |x|=-\frac{y}{2}, so -\frac{y}{2} equals to a non-negative value (|x|), so -\frac{y}{2}\geq{0} --> y\leq{0}.

(1)+(2) We have that x\leq{0} and y\leq{0}, hence equations from the statements transform to: x-2y=0 and y-2x=0. Solving gives x=y=0. Sufficient.

Answer: C.

Hope it's clear.

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Re: Is |x| + |y| = 0? [#permalink] New post 24 Jun 2012, 11:22
Yes, its clear now, Thanks!
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Re: Is |x| + |y| = 0? [#permalink] New post 24 Jun 2012, 12:38
nades09 wrote:
Is |x| + |y| = 0?

(1) x + 2|y| = 0
(2) y + 2|x| = 0


(1) Not sufficient, but implies that x is not positive.
(2) Again, not sufficient, but implies y is not positive.

When considering (1) and (2) together, we can add the two equations side-by-side and obtain x + 2|y| + y + 2|x| = 0,
and (x + |x|) + |x| + (y + |y|) +|y| = 0 + |x| + 0 + |y| = |x| + |y| = 0.
We used the fact that if x is not positive (it is negative or 0), then x + |x| = 0.

Correct answer - C.
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Re: Is |x| + |y| = 0? [#permalink] New post 04 Jul 2013, 00:24
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Re: Is |x| + |y| = 0? [#permalink] New post 04 Jul 2013, 03:58
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nades09 wrote:
Is |x| + |y| = 0?

(1) x + 2|y| = 0
(2) y + 2|x| = 0



Since the Question stem is asking if sum of 2 absolute values (which are positive) equal to 0. We know that sum of 2 positive nos can be zero if both are zero. Hence Question asks if x=y=0

from St1 we have x+2|y|=0
Now if y is less than equal to 0 than we have x-2y=0 or x=2y or x=0 if y is 0
If y>0 then x=-2y

Since using 1 we have more than 1 possible option therefore
A,D ruled out

From st 2 we have y+2|x|=0
If x is less than or equal to zero than y-2x=0 or y=2x or if x=0 then y=0
If x>0 then y=-2x

Again more than 1 solution so Option B ruled out

Combining both statement we get

x=2y ,x=-2y, y=2x and y=-2x and x=0, y=0

Since x=0,y=0 is common from both equation we can say say that |x|+|y|=0
Note that x and y have to be 0 to satisfy all above equations.
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Re: Is |x| + |y| = 0? [#permalink] New post 04 Jul 2013, 12:02
Is |x| + |y| = 0?

(1) x + 2|y| = 0

There are two ways we can solve. One is to get the positive and negative cases of y. On the other hand, we can isolate |y| as the question is looking for the value of |y|+|x|

x + 2|y| = 0
2|y| = -x
|y| = -x/2
if |y| = -x/2 then -x/2 must be positive which means x is negative. Of course, x could also be zero meaning we don't know if the absolute value of x and y = 0
INSUFFICIENT

(2) y + 2|x| = 0
This is a similar statement to the above one, except we have the absolute value of x instead of y.
y + 2|x| = 0
2|x| = -y
|x| = -y/2
As with the above statement -y/2 = an absolute value so y must be negative. However, it could also be = to zero.
INSUFFICIENT

So....from 1 and 2 we know that x<=0 and y<=0 which means that:
x + 2|y| = 0
x-2y = 0

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

y-2x=x-2y
3y=3x
y=x=0
SUFFICIENT
(C)
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Re: Is |x| + |y| = 0? [#permalink] New post 13 Oct 2013, 02:05
Bunuel wrote:
Is |x| + |y| = 0?

Since absolute value is non-negative the from |x| + |y| = 0 we have that the sum of two non-negative values equals to zero, which is only possible if both of them equal to zero. So, the question basically asks whether x=y=0

(1) x + 2|y| = 0. It's certainly possible that x=y=0 but it's also possible that x=-2 and y=1. Not sufficient.

Notice that from this statement |y|=-\frac{x}{2}, so -\frac{x}{2} equals to a non-negative value (|y|), so -\frac{x}{2}\geq{0} --> x\leq{0}.

(2) y + 2|x| = 0. It's certainly possible that x=y=0 but it's also possible that y=-2 and x=1. Not sufficient.

Notice that from this statement |x|=-\frac{y}{2}, so -\frac{y}{2} equals to a non-negative value (|x|), so -\frac{y}{2}\geq{0} --> y\leq{0}.

(1)+(2) We have that x\leq{0} and y\leq{0}, hence equations from the statements transform to: x-2y=0 and y-2x=0. Solving gives x=y=0. Sufficient.

Answer: C.

Hope it's clear.

P.S. Please read and follow: rules-for-posting-please-read-this-before-posting-133935.html


Hi Bunuel, why do the "equations from the statements transform to: x-2y=0 and y-2x=0"? Shouldn't it be -x + 2y = 0 and -y +2x = 0? Hope you can help clarify. :)
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Re: Is |x| + |y| = 0? [#permalink] New post 13 Oct 2013, 03:11
Expert's post
pauc wrote:
Bunuel wrote:
Is |x| + |y| = 0?

Since absolute value is non-negative the from |x| + |y| = 0 we have that the sum of two non-negative values equals to zero, which is only possible if both of them equal to zero. So, the question basically asks whether x=y=0

(1) x + 2|y| = 0. It's certainly possible that x=y=0 but it's also possible that x=-2 and y=1. Not sufficient.

Notice that from this statement |y|=-\frac{x}{2}, so -\frac{x}{2} equals to a non-negative value (|y|), so -\frac{x}{2}\geq{0} --> x\leq{0}.

(2) y + 2|x| = 0. It's certainly possible that x=y=0 but it's also possible that y=-2 and x=1. Not sufficient.

Notice that from this statement |x|=-\frac{y}{2}, so -\frac{y}{2} equals to a non-negative value (|x|), so -\frac{y}{2}\geq{0} --> y\leq{0}.

(1)+(2) We have that x\leq{0} and y\leq{0}, hence equations from the statements transform to: x-2y=0 and y-2x=0. Solving gives x=y=0. Sufficient.

Answer: C.

Hope it's clear.

P.S. Please read and follow: rules-for-posting-please-read-this-before-posting-133935.html


Hi Bunuel, why do the "equations from the statements transform to: x-2y=0 and y-2x=0"? Shouldn't it be -x + 2y = 0 and -y +2x = 0? Hope you can help clarify. :)


We have that x\leq{0} and y\leq{0}, thus |x|=-x and |y|=-y. Therefore x + 2|y| = 0 becomes x-2y=0 and y + 2|x| = 0 becomes y-2x=0.

Hope it's clear.
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Re: Is |x| + |y| = 0?   [#permalink] 13 Oct 2013, 03:11
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