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Re: Absolute Value [#permalink]
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.

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

2

This post received KUDOS

Expert's post

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.

Re: Is |x| + |y| = 0? [#permalink]
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. _________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

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. _________________

“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”

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

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

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.

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

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 \(y-2x=0\).

Re: Is |x| + |y| = 0? [#permalink]
28 Oct 2015, 14:04

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