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655-705 (Hard)|   Absolute Values|         
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2448
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 10:24
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y<0, so if y-x>0, then x should also be negative. so, x<0. Also, as y-x = -|y| + |x| >0 = |x|>|y|.
which translates the equation to: -1+1-1+1 = 0.
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 10:46
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Studying the Q stem I can see that x<y<0
So x/|x| = -1
|-y|/(-y) = 1
(x-y)/|y-x| = -1
|xy|/xy = 1
hence Sum = 0 Ans
Bunuel
If \(|x-y|=y-x\), \(x \ne y\), and \(y<0\), what is the value of \(\frac{x}{|x|}+\frac{|-y|}{-y}+\frac{x-y}{|y-x|}+\frac{|xy|}{xy}\)?

A. -2
B. 0
C. 1
D. 2
E. 4


 


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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 10:46
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I think the answer is B. 0

As y <0, and y>x, we can deduce that x<0 (both x and y are negative) and |x-y| = y-x.

Breaking the question into parts below:

As x is negative --->> x/ |x| = x/(-x) = -1
As y is negative ---> |-y|/(-y)= (-y)/(-y) = 1
As y-x>0 ---> (x-y)/|y-x| = (x-y)/(y-x) = -1
As x and y are both negative ---> |xy|/(xy) = 1

Sum: -1+1 -1+1=0 (Answer: B)
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LibrarianTurtle
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 10:51
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1. Y is negative (-) since Y<0
2. Absolute value of X-Y cannot equal negative therefore Y-X must be positive.
3. For Y-X to be positive X has to be negative since Y is negative and X is smaller than Y.
Now we can solve easily.
First Fraction: positive X / Negative X = (-1)
Second Fraction: Absolute value of negative Y (Positive) / negative Y (Positive) = 1
Third Fraction: X-Y is negative since X is smaller than Y and both are negative. Therefore (-)/(+)= -1 since X-Y/Y-X = -1
Last Fraction: (-)x(-) = (+) so denominator is also positive, therefore (+)/(+) = 1
When we sum all the numbers it equals = 0
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 10:58
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Given, |x-y| = y-x, which means x-y<0, x<y. Given y<0, therefore, x<0. Both x and y are negative nos.

|x|/x = -1
|-y|/-y = +1
(x-y)/|y-x| = -1
|xy|/xy=+1

Therefore, answer is zero.
Bunuel
If \(|x-y|=y-x\), \(x \ne y\), and \(y<0\), what is the value of \(\frac{x}{|x|}+\frac{|-y|}{-y}+\frac{x-y}{|y-x|}+\frac{|xy|}{xy}\)?

A. -2
B. 0
C. 1
D. 2
E. 4


 


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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 11:08
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I believe the answer is 0. My thought process for arriving at the answer is mentioned below:
It is given that -
y <0, x and y are not the same, | x-y| = y-x
Since |x-y| = y-x this indicates that the value inside the mod is negative value and y is less than 0 but in mod it is given that y is negative - therefore x also needs to be less than 0 for the equation x-y to be less than 0. Accordingly, x <0
Now the entire equation has 4 parts:
Part 1 = x/|x| = -x/x = -1
Part 2 = |-y|/-y = y/y = 1
Part 3 = x-y/|y-x| = y-x/-(y-x) = -1
Part 4 = |xy|/xy = xy/xy = 1

Adding all the parts, we get an answer of 0
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 11:08
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x-y<0 hence x is smaller than y and y is smaller than 0. Evaluate each term here, x/parentheses x=-1, next term would be 1, then -1, then 1, if you add them addition, you will get -1+1-1+1 which is 0 which is our answer. Hence option B is the correct answer.
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 11:13
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|x-y| = y-x, so y-x > 0 but cannot be zero since x not equal to y, and y < 0

y-x > 0, so y > x. However, y is negative, so x must be negative and smaller than y

Taking x = (-2), and y = (-1)

-2/2 + 1/-(-1) + (-2+1)/|-1+2| + 2/2

-1 + 1 -1 + 1 = 0
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|x-y| = (y-x) = -(x-y) ==> This means (x-y)<=0 bcoz |a| = -a when a<=0
=> x-y<=0 => x<=y => x<y bcoz x not equal to y.

hence x<y<0

i. x/|x| = x/(-x) = -1 [|x| = -x as x<0]
ii. |-y|/(-y) = -y/-y = 1 [|-y| = -y as y<0, ex: |-(-3)| = -(-3) = 3]
iii. (x-y)/|y-x| = (x-y)/(y-x) = -1 [ y-x is positive hence |y-x| = y-x]
iv. |xy|/xy = xy/xy = 1 ( xy = positive as x.y = neg . neg = pos, |xy| = xy ]

i+ii+iii+iv = -1+1-1+1 = 0
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Vivek1707
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 11:18
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Some concepts
|x| = -x when x < 0

Say if you have x = -3 then |x| will be 3

So since our x term is -3, |x| will be = -x and not x

Here |x-y| = y - x this implies that x - y < 0. Additionally we are told that y < 0 then and x < y it means that x < 0 and y < 0

So we have x - y < 0 , x < 0 and y < 0

Coming to the question

|x| = -x since x < 0 Hence x /|x| = x/-x i.e -1

Next, |-y| = -y ---> Since y is already negetive, adding the minus sign to it makes it positive and when the value inside mod is positive the mod will open as it is.
So |-y|/y = -y/-y = 1

Next we have (x - y)/ |y-x| ---> Here we can write (y - x) as -(x-y) again since x-y is negetive, adding a minus sign makes it positive and hence the mod will open as it is
So this will become (x - y) / -(x- y) i.e -1

xy is anyway positive so |xy| / xy = 1

So -1 + 1 -1 + 1 = 0 , Answer B






Bunuel
If \(|x-y|=y-x\), \(x \ne y\), and \(y<0\), what is the value of \(\frac{x}{|x|}+\frac{|-y|}{-y}+\frac{x-y}{|y-x|}+\frac{|xy|}{xy}\)?

A. -2
B. 0
C. 1
D. 2
E. 4


 


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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 11:45
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Suppose x-y>=0, then x>=y.
Now, solving, |x-y| = y-x, considering, x >=y. This will result in x = y.
Now, for the case x-y<0, the given equation |x-y| = y-x, will result in 0.

Now x = y and we know y <0, hence, x <0 and |x| > 0 and |y| >0.

Solving, the given equation = x/|x| + |-y|/-y + x-y/ |y-x| + |xy|/xy = -1 + 1 + 0 + 1 = 1 (Ans C)
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 12:00
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Option B - 0 is correct

as |x-y| = y-x and x\ne y , y<0

so taking the absolute value for |x-y| = y-x
-(x-y) = y-x

so now , x-y <0 ( as it is negative)
x< y
Now , x<y<0

Putting the values in equation
x/|x| = 1/-1 =1
|-y|/-y = -1
x-y/|y-x|= -1
|xy|/xy = 1

1+ (-1) + (-1) + 1 = 0





Bunuel
If \(|x-y|=y-x\), \(x \ne y\), and \(y<0\), what is the value of \(\frac{x}{|x|}+\frac{|-y|}{-y}+\frac{x-y}{|y-x|}+\frac{|xy|}{xy}\)?

A. -2
B. 0
C. 1
D. 2
E. 4


 


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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 12:08
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As | x -y | = y - x, we can say that x-y is negative, hence x-y < 0 , hence x < y
Also it is given y<0 hence both x and y are negative terms
considering 2 values for x and y, x = - 5 and y = -3
we solve the equation and get the following
-5/5 + 3/3 + (-2)/2 + 15/15
= -1 + 1 -1 +1
= 0
(B)
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I used smart numbers to solve this question. Both x and y must be negative numbers, and the absolute value of x is larger. (x is negative because y-x must yield a positive number because it's equal to an absolute value, and the absolute value of x is larger than y because on the second part of the equation it turns into a positive value after the minus sign, and therefore must outweigh the negative y.)

For example, I chose numbers -5 for x and -4 for y.

Solving the equation, we have: 1 + (-1) + (-1) + (-1) = -2

So, in my opinion the answer would be A
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Bunuel
If \(|x-y|=y-x\), \(x \ne y\), and \(y<0\), what is the value of \(\frac{x}{|x|}+\frac{|-y|}{-y}+\frac{x-y}{|y-x|}+\frac{|xy|}{xy}\)?

A. -2
B. 0
C. 1
D. 2
E. 4


 


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Since ∣x−y∣=y−x, x must be less than y. Let x=−5.

Now substitute:

x is negative, so ∣x∣ x =−1
−y is positive, so −y ∣−y∣ =1
x - y is negative, so (x-y)/|y-x| = -1
xy is positive, so |xy|/xy = 1

-1 + 1 -1 + 1 = 0

Option B
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RahulSax
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 12:37
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As I solved the Question.....
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Arya25
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 12:38
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for |x-y| = y-x to be true, x<y.
hence we now know x<y<0
substitue values like x=-4, y=-2 then solving this equation, gives ans as 0.
Hence option B
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rs827
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 12:40
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|x-y| = y-x indicates that y>=x;
y!=x ---> y>x;
y<0 ---> x<0 & x<y<0;

Now the question becomes simplified as x/|x| ---> x/-x; as x<0
x/|x| = -1

Similarly, |-y|/-y ---> -y/-y; as -y>0
|-y|/-y = 1

x-y/|y-x| ---> x-y/y-x; as y>x
x-y/|y-x| = -1

|xy|/xy ---> xy/xy as xy is positive
|xy|/xy = 1

Finally, if you add them all up you get = -1+1-1+1 = 0

Option B.
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 12:48
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x-y is negative
If y is negativ, x has to be negativ such that |x| > |y|

-x/x = -1
|y|/-y =1
x-y/|y-x|=-1
|xy|/xy = 1

-1+1-1+1 = 0

Ans B
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 13:14
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If |x-y| = y - x then 2 cases arise.
1. x - y = y - x
2. x - y = - (y - x)

Solving the 1st case gives:
2x = 2y, hence x = y
But this contradicts the given constraint x is not equal to y. Hence the first case goes away.

Using the second case we can conclude the x - y is negative, i.e. x - y < 0. And since y<0 hence x is also negative.

x < 0, y < 0
Coming to the equation
x/|x| = -1
|-y|/-y = 1
x-y/|y-x| = -1
|xy|/xy = 1

Adding all of these we get 0.
Hence 0 is the answer.
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