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655-705 (Hard)|   Absolute Values|         
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confusedchild
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 13:19
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the question says |x-y| = y-x and x =/ y and y<0

so |x-y| = y-x = -(x-y)
means x-y <0
because of the absolute value rule that:
|A| = A for A>0
|A| = - A for A<0
Example:
|5| = 5 ; but |-5| = - (-5) = 5

basis that i can say that x-y<0
for this to be true, x<y<0
So, both x & y are negative

Now the equation we have to solve:
x/|x| + |-y|/-y + x-y/|y-x| + |xy|/xy

Term 1: -1/1 = -1
Term 2: |- (-1)|/- (-1) = 1/1 = 1
Term 3: -1/1 = -1 ; |y-x| will be positive coz x<y<0, so let's take an example of x=-3 and y=-2, then |y-x| = |-2 - (-3)| = |3-2| = 1, it will always be positive.
Term 4: 1/1 = 1 (-ve x -ve = +ve)

Thus -1 + 1 + -1 + 1 = 0

Answer s
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 14:41
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I found the easiest way to solve this question was to pick numbers that satisfied the conditions. I said y=-1 since it needed to be less than zero. Solving for the equation |X-Y|= y-x, I got an answer of -2.

Plugging these values into the equation gave me -1+1 -1+1 or 0 which is answer B. I'm sure there is a more theoretical approach to the question, but I found picking easy numbers the best way for me to solve.
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 16:01
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|x-y| >=0
==> y-x>=0
==> y>=x, but y<0 hence x<0

x/|x| ==> -1 as x<0
|y|/-y ==> +1 as mod is always +ve and y<0
(x-y)/|y-x| = -1 as x<y ==> x-y<0
xy/|xy| = 1 as both numerator and denominator +ve

==> -1 +1 -1 +1 = 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, 16:58
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mod of mod of y-x is x-y so it cancelled out
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, 17:14
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|x-y|=y-x -> x<y
y<0 -> x<y<0
x and y are negative

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

-1+1-1+1 = 0

IMO 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, 17:29
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Given: |x-y| = y-x and y<0

Therefore x-y < 0 => x<y and x<0
We can assume values and solve. Lets say y = -1 and x = -2

Take each term one by one :-
Term 1: x/|x| = -2/2 = -1
Term 2: |-y|/-y = 1/1 = 1
Term 3: (x-y)/|y-x| = -1/1 = -1
Term 4: |xy|/xy = 2/2 = 1

Adding the 4 terms we get answer = 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, 17:32
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From |x - y| = y - x and y < 0 we can deduce that (x - y) is negative and x < y < 0
x is negative, y is negative

calculating each term:
x / |x| = -1
|-y|/(-y) = 1
(x-y)/|y-x| = -1
|xy|/xy = 1

adding all of them:
- 1 + 1 - 1 + 1 = 0

Answer 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, 18:05
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the ans is B over here as y<0 which means it |x| will be -ve, so it should be -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, 18:56
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To solve this question, we must determine whether x ≥ 0 or x < 0.

From the given equation: |x-y| = y-x, where y < 0 and x≠y.
Try plug in y = -1; |x+1| = -1-x
We can see that there are 2 cases
1. If x ≥ 0, then the RHS would be negative, which is impossible to match the absolute value from the LHS.
2. If x < 0, this is possible because the RHS can be positive.

So, we know x must be negative.

x/|x| = neg/pos = -1
|-y|/-y = pos/pos = 1
(x-y)/|y-x| = neg/pos = -1
|xy|/xy = pos/pos = 1

Then, sum them up = 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, 19:25
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|x-y|= y-x

Only
y-x=-(y-x)
For an absolute value to equal the negative of an expression

|a|=-a, the expression a must be negative or zero, because positive is not equal, as said in the question.

a=x-y
Therefore, x-y is less than or equal to 0
Since x is not equal to y
x<y<0
So both x and y are negative.

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

Answer is 0
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Soln:
Attachments

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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 19:42
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If |x - y| = y - x and y < 0 -> x - y < 0 -> x < y < 0
both x and y are negative

Terms:
x/|x| = -1
|-y|/(-y) = 1
(x-y)/|y-x| = -1
|xy|/xy = 1

-1 + 1 - 1 + 1 = 0

The answer is 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, 20:01
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It is given that mod(x-y)= y-x . Therefore, solving this will give only one case that is y>x and it is given that y<0. Hence, solving the given equation will give ( -1+1-1+1, i.e equal to 0). Hence, B is the 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, 20:03
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|x-y| = y-x and x not equal to y
x-y < 0
x < y

y < 0
x < y < 0

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

Adding: -1+1-1+1 = 0

The correct answer is 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, 20:23
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since |x-y| = y-x ; it means that the number inside the absolute value is negative and due to the absolute value it becomes a non negative number which is basically y-x
to prove this take the example, let (we are already told y is less than zero) -
x = -3 and y = -2
|-3-(-2)| = |-1| = 1 (which is the same as y-x)

continuing with this example, after opening the absolute value in the given equation, it becomes,
-3 / |-3| + |-2| / -(-2) + (-3+2) / |-2+3| + |-3*-2|/-3*-2
= -3/3 + 2/2 + -1/1 + 6/6
= 0

Answer is 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, 21:04
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As |x-y|=y-x, then x-y<0.
Therefore, x<y.

We already know y<0, thus x<y<0 (i.e., both x and y are negative)
Therefore, the expression's final value becomes:
-1 (as x<0) +1 (both numerator and denominator are positive) -1 (as x-y<0) +1 (as xy>0)
=-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, 21:11
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|x-y| = y - x tells us that y-x >= to 0 based on the definition of absolute value. this also means y>=x and since we know y does not equal x, y>x.

We know y<0, so x<y<0 and both x and y are negative.

Taking the problem term by term x/|x| = -1 since x is negative.

|-y|/(-y) = 1. The absolute value of |-y| is positive and -y is also positive because y is negative.

(x-y) / |y-x|. we now x - y is negative since y > x. y-x is positive as we stated earlier so the answer is -1.

|xy|/xy equals one. The absolute value of xy is positive and so is the product of x and y because both are negative then the numerator and denominator cancel out to give 1

Now we just add all the terms together
-1 + 1 -1 +1 = 0. So the answer is 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, 21:47
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Since | x - y | = y - x, and y < 0, x < y < 0. This follows from the fact that | X | = -X if X < 0 and thus if | x - y | = - (x - y), x - y < 0 or x < y.
Thus, x & y are negative with y > x.

For quick solve, assuming x = -3, y = -1
Thus the expression will be -3/3 + 1/1 + -2/2 + 3/3 = 0

This can also be solved just by looking at the sign of each term since denominator & numerator are same all the places and will be equal to 1 => Negative + Positive - Negative + Positive = 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, 22:12
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First, let's decode the clues about x and y:

The clue |x - y| = y - x only happens if x is smaller than y.

The clue y < 0 means y is negative.

Put them together: both x and y are negative, and x is "more negative" (just imagine x is -5 and y is -2).

Now, let's look at the four fractions knowing that both numbers are negative:

x / |x| : A negative number divided by its positive version is always -1.

|-y| / -y : Since y is negative, -y is actually positive. Any positive number divided by itself is 1.

x-y / |y-x| : Since x is smaller than y, x-y is negative. A negative divided by its positive absolute value is always -1.

|xy| / xy : A negative times a negative is positive. So the top and bottom are the exact same positive number, which equals 1.

Finally, just add those four results together:
-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, 22:45
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I believe the answer to be 0

I can be completely wrong as |x-y|= y-x

can either be

x-y = x-y - 1
x-y= y-x - 2

where 1 does not seem to fit right

and 2 is where y-x>0
and y>x
while its given that y<0

so x<y<0

now i will assume x=-2 and y =-1

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

so

-1 + 1 + -1 + 1
=0

I hope this fits, and fingers crossed i answered right
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