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Is |x| + |y| = 0  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 58% (01:47) correct 42% (01:46) wrong based on 297 sessions

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Is |x| + |y| = 0

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

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do not hesitate me giving kudos if you like my post. Originally posted by Warlock007 on 01 May 2011, 10:37.
Last edited by Bunuel on 29 Jul 2016, 00:56, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: Is |x| + |y| = 0?  [#permalink]

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1
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.

Hope it's clear.

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Re: Is |x| + |y| = 0  [#permalink]

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only possible if x=y=0.
a. x=-6, y= 3 or x=y=0 Not sufficient
b. y=-6,x=3 x=y=0 not sufficient.

a+b sufficient.x =y=0
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Re: Is |x| + |y| = 0  [#permalink]

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(1)
-2 + 2|-1| = 0

-2 + 2|1| = 0

0 + |0| = 0

(2)

-2 + 2|-1| = 0

-2 + 2|1| = 0

0 + |0| = 0

(1) and (2) are insufficient

(1) and (2) together:

|0| + |0| = 0

So the expression can be 0 only when x and y are 0, if both x and y are negative/positive, |x| + |y| > 0. I wonder how the OA is D.
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Re: Is |x| + |y| = 0  [#permalink]

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Warlock007 wrote:
Is |x| + |y| = 0
1)x + 2|y| = 0
2)y + 2|x| = 0

I know its an easy one
even i got answer in seconds but still need more perfection in fundamentals of mod questions
m looking forward to a basic fundamental explanation of the same I think c).
all the above are in absolute value. from question stem we see that for a sum of absolute value to be 0, both the terms should be equal to zero. even if one of the terms is not equal to zero, the sum will nto be equal to zero.

statement 1 says: x + (positive no) = 0 i.e. either both zero or x = -(2y)
statement 2 says: y +(pos no) = 0 again either both zero or y = -(2x)

unless it is mentioed that x and y are positive, id say C. becuase it then becomes clear that x & y have to be zero and so |x| + |y| = 0.

have i missed something?
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Re: Is |x| + |y| = 0  [#permalink]

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Concepts tested is absolute value

It can not be D the OA. It is most probably C

Obviously on 1) and 2) we can plug -2 and 1 and 0 and 0 for x and y and get a Yes and No answer. Insuff

However on 1+2) we get x + 4 /x/ = 0 so x is negative and -4 x = -x so x =0 and therefore y is equal to 0. Suff

Hope this is helpful
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Re: Is |x| + |y| = 0  [#permalink]

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1
1
Warlock007 wrote:
Is |x| + |y| = 0
1)x + 2|y| = 0
2)y + 2|x| = 0

I know its an easy one
even i got answer in seconds but still need more perfection in fundamentals of mod questions
m looking forward to a basic fundamental explanation of the same This is how you can reason it out theoretically:

Question: Is |x| + |y| = 0 ?
A mod is either positive or 0. It can never take a negative value. If sum of two mods is 0, they both individually have to be 0 to give a sum 0. So question comes down to: Is x = y = 0?

1)x + 2|y| = 0
Again, |y| will be either 0 or positive. So x will be 0 in first case (when y = 0) and negative in the second case (when |y| is positive) to give a sum of 0. Hence we cannot say whether x = y = 0. Not sufficient.

2)y + 2|x| = 0
Same is the case here. There is not reason why analysis of this equation should be any different from statement 1 since x and y are just interchanged.

Together, either x = y=0 else x and y both are negative. If x and y both are negative, then x = -2|y| and y = -2|x| i.e. in absolute value terms, x is twice of y and y is twice of x which is not possible. Hence, the only way both statements will hold is if x = y = 0. Hence answer (C).
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GMAT 1: 640 Q50 V26 GMAT 2: 660 Q50 V28 GMAT 3: 730 Q50 V38 Re: Absolute Value  [#permalink]

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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.

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Re: Is |x| + |y| = 0?  [#permalink]

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1
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]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Abolute Values: math-absolute-value-modulus-86462.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

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Re: Is |x| + |y| = 0?  [#permalink]

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1
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]

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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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GMAT 1: 700 Q46 V40 Re: Is |x| + |y| = 0?  [#permalink]

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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.

Hope it's clear.

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. Math Expert V
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Re: Is |x| + |y| = 0?  [#permalink]

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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.

Hope it's clear.

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]

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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.

Hope it's clear.

Hi Bunuel,

I solved this question in the following way - still got the right answer (not sure if I am right or it was just a fluke). Please provide input.

|x| + |y| = 0, means both value of x&y needs to be known (preferably zero)

Each statement talks about two variables at the same time -Both Insufficient- So either C or E

Now, Statement 1 is in form of y=mx+c (m=-0.5/+0.5 - slope)
& Statement 2 is in form of y=mx+c (m=-2/+2 - slope)

Since any of the possibilities do not overlap each other, these lines will intersect each other at one point - and thus a solution is possible with the help of both the statements.
Thus C
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Re: Is |x| + |y| = 0  [#permalink]

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Warlock007 wrote:
Is |x| + |y| = 0
1)x + 2|y| = 0
2)y + 2|x| = 0

I know its an easy one
even i got answer in seconds but still need more perfection in fundamentals of mod questions
m looking forward to a basic fundamental explanation of the same addition of two positive values will only result in zero if both the numbers are 0. niether 1 nor 2 sufficiently states that.
But if we combine both of them, we have x=y=0 , the only case which satisfies the equations.
hence C. The OA is wrong!
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Re: Is |x| + |y| = 0?  [#permalink]

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only when x and y both =0 this will be true

1)may possibilities including one with 0+ 0

2)many possibilities including one with 0+0

1+2,

commn is x=y=0

so C
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Re: Is |x| + |y| = 0?  [#permalink]

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_________________ Re: Is |x| + |y| = 0?   [#permalink] 17 Oct 2019, 01:39
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