Is xy + z = z, is |x - y| > 0 ?

First of all: \(xy+z=z\) --> \(xy=0\) --> \(x=0\) or \(y=0\) or both, \(x=y=0\).

Next: absolute value is always non-negative \(|some \ expression|\geq{0}\), so \(|x-y|\geq{0}\). Question asks whether \(|x-y|>0\), so basically question ask whether \(x=y=0\), as only case when \(|x-y|\) is not more than zero is when it equals to zero, and this happens when \(x=y=0\).

(1) \(x\neq{0}\) --> then \(y=0\), so \(x\neq{y}\) --> \(|x-y|>0\). Sufficient.

(2) \(y=0\) --> if \(x=0\) then \(|x-y|=0\) and the answer to the question is NO, but if \(x\neq{0}\) then \(|x-y|>0\) and the answer to the question is YES. Two different answers, hence not sufficient.

Answer: A.
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A...
from eq xy+z=z , we know xy is 0...
for lx-yl>0 x-y ≠ 0...
si) x ≠ 0 so y=0 and x-y will be a -ive or +ive no... suff
sii) y=0... x can be 0 or any other no ...not suff
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Ian / Karishma

I may be loosing it all But this is how I think

If xy = 0. This certainly is no brainer x = 0 or y = 0

But the real problem is going to happen if x and y are both zero. Since x = 0 / y and since y and x are both zero we can infer that x is 0/0 or undefined. However we have assumed x=0. Hence I think we cannot assume that x and y are both zero. Can we ?

Not sure. Thoughts??

gmat1220: As per your request, let me provide the logic here.

Both, gurpreetsingh and your high school, are correct.

Given that x*y = 0.
Think of it in this way: I have two numbers. I don't know their values. But when I multiply them, I get 0. So at least one of the numbers have to be 0. Both can also be 0 since 0*0 = 0. Nothing says that they cannot be equal.

Why were you taught 'Either x or y is 0 in high school?'
Because 'Either or' implies 'At least one'. It is counter intuitive to how we think about 'Either or'. In language, we generally think 'Either or' means either A or B but not both. This is incorrect implication in logic. 'Either or' means at least one of A and B. Both are also possible. (If you do not agree, check out http://en.wikipedia.org/wiki/Logical_disjunction)

Next, how do you explain 0 = 0/0? If you remember, in many DS questions where you have equations with denominators, you are specifically given that the denominator is not 0. e.g.
Given x = y/(x - z), x not equal to z,...
You can only divide something by x if you know that it is not zero.
xy = 0 cannot be re-written as x = 0/y because you do not know whether y is 0 or not. You do not divide an equation by a variable until and unless you know that the variable is not 0.

Hope that clears up the doubts.
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If xy + z = z, is |x - y| > 0? or given xy = 0 (either x or y or both 0), is |x-y|> 0? Squaring, is x^2 + y^2 -2xy > 0 ?

(1) x # 0 (Not equal to): x^2 is +ve. y = 0. So, x^2 + y^2 -2xy > 0
(2) y = 0: if x = 0, NO. If x = 1, x^2 + y^2 -2xy > 0. YES.

A.

If xy + z = z, is |x-y| > 0 ?

xy + z = z;
Cancel z: xy = 0. This implies that either x or y (or both) is 0.

(1) x ≠ 0. Since x is not 0, then y = 0. Thus the question becomes is |x| > 0. The absolute value of a non-zero number (x) is always positive. So, we have a definite YES answer to the question. Sufficient.

(2) y = 0. The question becomes is |x| > 0. If x = 0, then the answer is NO but if x ≠ 0, then the answer is YES. Not sufficient.

Answer: A.

Hope it's clear.
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.

Is xy + z = z, is |x-y| > 0 ?

(1) x ≠ 0
(2) y = 0

in the original condition we have xy+z=z, xy=0 and the question asks for |x-y|>0?
Since we have 2 variables and 1 equation, in order to match the number of variables and equations we need 1 more equation. Since there is 1 each in 1) and 2), D is likely the answer.

In case of 1), y = 0 thus the answer is always yes. Therefore it is sufficient.
In case of 2), x = 2 and y = 0 thus the answer is yes, but if x = 0 and y = 0 the answer is no. Therefore it is not sufficient.
The answer is A
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Hi Bunuel..well this analysis is clear and many thanx for it, but i went through a different route..
modulus of any number is always greater than zero...isn't it? so either satement alone could be the answer.

In a recent exapmple this was the approach adopted

the question was something like this (actual number may diifer)Is -7*mod(4x-y)<14??
It was explained that modulus of 4x-y would always be positive and when multiplied by a negative number it will always be negative...and THERE IS NO NEED TO ACTUALLY SOLVE...the inequality...

Why can't we appply this approach here as well?

Desparately need cl;arification

In case of 1)
, y = 0 thus the answer is always yes.
Therefore it is sufficient.


In case of 2),
x = 2 and y = 0
thus the answer is yes,
but if x = 0 and y = 0
the answer is no.

Therefore it is not sufficient.
The answer is A