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655-705 (Hard)|   Absolute Values|         
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Sanny7
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GMAT Club World Cup 2026 (Day 2): If |x - y| = y - x, x y, and y < 0 07 Jul 2026, 08:39
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Given |x-y|=y-x, to satisfy this while x != y, x should be negative

lets take x = -a for simplicity, given y is also negative so lets take y = -b

term 1 = x/|x| = -a/a = -1
term 2 = |-y|/-y = b/b = 1
term 3 = x-y/|y-x| = (-a+b)/(|-b+a|) = 1
term 4 = |(-a)(-b)|/ab = 1

-1+1+1+1 = 2
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If X-Y>0 then solving the given equation gives X=Y, And it is already given that X is not equal to Y.
It implies X-Y<0 and hence X is negative and more negative than Y.

Base on this there are 4 parts
1st part will be -1
2nd part will be +1
3rd part will be -1
4th Part will be +1

There sum will be ZERO. ANS: B
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Since it is given |x-y| = y-x and an absolute value is always positive, it follows that y-x>0 or y>x
Question also says y<0. So overall x<y<0

Now I personally find assuming integers as a faster way to solving these type of questions. Let x=-10 and y=-4

= x/|x| + |-y|/-y + (x-y)/|y-x| + |xy|/xy
= -10/|-10| + |-(-4)|/-(-4) + (-10-(-4))/|-4-(-10)| + |-10*-4|/(-10*-4)
= -10/10 + 4/4 + -6/6 + 40/40
= -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, 08:42
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|x-y| will be equal to (y-x) when x<y. Since y<0, both x and y are negative.
x/|x| = x/(-x) =-1
|-y|/(-y) = y/(-y)=1
(x-y)/|y-x|=-1
since x and y are negative, xy is positive
|xy|/xy=1
Answer is -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, 08:43
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∣x−y∣=y−x
Only if that expression is less than or equal to zero.

x−y ≤ 0 = x ≤ y

x is not equal to y

This mean x < y

y < 0
Since y is negative and x is less than y, both variables are negative. x<y<0

The sign of each component.

First Part: x/∣x∣ = -1

Second Part: ∣−y∣ / -y = 1
​
Third Part: x-y / ∣y−x∣ = -1

​Since x<y, we know that y−x is a positive number, so ∣y−x∣ = y−x.

Fourth Part: ∣xy∣ / xy
​

Since both x and y are negative, their product xy is positive.

∣xy∣=xy =1

Step 3: Sum the Terms
Substituting our simplified terms back into the original expression:

−1+1+(−1)+1=0

Answer: B. 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, 08:45
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Given:
1. |x-y|=y-x ----> this equation can be true only when y is greater than x.
2. y<0 ----> that means y is a negative number
Hence, it can be conduced from above statements that if y is a negative number x must be more negative than y so that y-x is positive.

A negative number when divided by its absolute value gives -1
Examining the values one by one:
1. x/|x|=-1 (since x is negative)
2. |-y|/-y =+1 ( since y is negative -(-y)= +y )
3. x-y/|y-x|=-1 ( since y>x, x-y must be is negative and as seen above, negative number divided by its absolute value is -1 )
When we add them: -1+1-1+1=0
Therefore the answer is B
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Start with the condition:

|x-y| = y-x

what its really saying:|x-y| = -(x-y), this is only possible if x-y is negative. (the two negatives cancel to give a positive).
This means x-y<0, so x<y, we are told i the question y<0, so x<y<0

Now evaluate each term by sign.

First term is -1, second term in +1, third term is -1 and the last term in +1. Total 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, 08:46
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|x-y| =y-x
This is only true when y-x>=0, so y>x
Also, y<0. So we can conclude that x and y both are negative.
x<y<0

x/|x| = -/+ = -1
|-y|/-y = +1
(x-y)/|y-x| = x-y/y-x = -1
|xy|/xy = +1
After addition of above terms, we get = -1+1-1+1=0

B) 0
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Since |x-y| can be positive, even if there is a negative value, it will become positive. On the other side, y - x cannot be negative as y < 0.
Therefore, x needs to be negative to equalize the equation.
Assume, x = -2, and y = -1
| - 2 + 1| = -1 + 2
=> |-1| = 1
=> 1 = 1
Now, plug the numbers in the equation, we will get (-2)/|-2| + |-(-1)|/-(-1) + [-2-(-1)/|-1-(-2)| + [|(-2)*(-1)|/(-2)*(-1)]
= -1 + 1 -1 + 1
= 0
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|x-y| = y-x and 0> y => 0 > y > x
therefore
x = -|x|
|-y| = -y
0 > x-y = -|y-x|
both x and y < 0 so xy > 0 => |xy| = xy
==> sum of the equation is 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, 08:58
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We know that |x-y| = y-x and y<0 and we need to find a value of a expression where each fraction can only be +1 or -1. so all we need to know is if they are positive or negative and whose mod value is greater.
We already know that y<0, and y-x>0(as it's mod of a number)
therefore x<0 as y is negative as we need to subtract a negative number so as to make the whole expression positive and |x|>|y| as y-x>0
now going through the fractions
1: -1(as x<0)
2: +1(as -y>0)
3: -1(numerator is negative)
4: +1(as denominator is positive -*- = +)
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if |x-y|=y-x, then y-x > 0 as |x-y|>0 always
Thus, y>x, as y<0 so x<0,
Therefore \(\frac{x}{|x|}+\frac{|-y|}{-y}+\frac{x-y}{|y-x|}+\frac{|xy|}{xy}\) = -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, 09:11
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The right answer is B (0)

Mod(x-y)= y-x. where y<0 and x not equal to y. This holds only when x <0 and y>x.
y-x is a +ve number. only way for y-x to be +ve when x<0 and y>x. With this understanding, put any two -ve integer in place x and y so that y>x and explore the equation. The answer would always be 0
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|x-y|=y-x, it is given that y<0 and x not equal to y.
absolute value is always greater than 0 so
y-x>0 , since y <0 x must be <0 ( basically |x|>|y|, x is more negative value than y).
It is easier to assume with numbers and solve x=-5 and y=-2
Value of the expression: -5/|-5|+|2|/2+ -5+2/|3|+|10|/10= -1+1-1+1=0
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0>y>x
So -1+1-1+1
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|x-y|= y-x. If x-y>0, then |x-y|=x-y=y-x. This means x=y. But we are given x is not equal to y.
Hence x-y <0. Which means x<y. Also, since y<0, x is also <0. We have both negative numbers.

Hence solving the above, we -1 -1 -1 +1 = -2.

Hence answer is 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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option B is correct
From the given info, it is clear that both x and y are negative, and x is further below zero than y. In simple words, x < y < 0. (How am I saying this?) Because we know that, |A| = -A if and only if A <= 0.
Now here we are given that |x-y| = -(x-y) = y-x, so we can say that x-y < 0 => x < y. Ultimately, x<y<0.
Let's assume the values of x and y randomly; let it be x = -2, y = -1.
x/|x| = -2/2 = -1
|-y|/-y = 1/-(-1) = 1
(x-y)/|y-x| = (-2+1)/|-1+2| = -1
|xy|/xy = |-2*-1|/-2*-1 = 2/2 = 1
adding these terms gives us,
-1+1-1+1 = 0
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The question says lx-yl = y-x, that means x-y<0, because the absolute value will change signs only if it is less than or equal to 0 or negative.
x-y<=0
x<=y
x<y<0.

now, from the question,
x<0, so lxl=-x
x=-ve
-y >0 so l-yl=-y
x-y= -(y-x)
ly-xl= y-x
lxyl= xy

Now, putting these in the question
1. x/lxl = x/-x = -1
2. l-yl/-y = 1
3. (x-y)/ ly-xl =-(y-x)/ y-x =-1
4. lxyl/xy = xy/xy = 1

Adding all the sections:
-1+1+-1+1= 0

Hence B is the correct answer.
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i am going with b. we are given that y<0, |x-y| = |y-x|, which leads us to that x-y<0, x<y, hence both are negative numbers.
taking term by term, x/|x| = x/-x = -1 as x is negative
|-y|/-y = 1
x-y/|y-x| = -1 as we know x-y is negative
|xy|/xy = 1
now summing them up gives us 0.
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first of all idk how to use the fraction method of representing answers in this chat, so please bear with the format of the explanation.
It is given that y< 0 but we do not have any range or sign for x. it can either be that x> 0 or x<0. We need to determine this from the premise.
Also it is given x != y (x not equal to y)

|x-y| = y-x, so wkt y<0, so for |x-y| to be equal to y-x, x-y has to be less than zero, x-y<0 (it can be equal to zero as x!=y)
so |x-y| = -(x-y) = y-x, so x-y<0 and x<y

Now that we have the range values for x and y, we go to the expression

for this question we will use the Plug in method as it is easier. let x= -2 and y = -1

x/|x| = -2/|-2| = -2/2 = -1
|-y|/-y = |-(-1)|/-(-1) = 1/1 = 1
(x-y)/|(y-x) = (-2-(-1))/|(-1-(-2)| = -1/|+1| = -1
|xy|/xy --> since x and y both are -ve and hence their product is +ve, so
this is |(-2)*(-1)|/(-2*-1) = +1

so adding all the four terms together we get --> (-1) + 1 + (-1) + 1 = 0
answer is 0 (option B)

Honestly my brain went fuzzy when I saw so many mods and had to refer to my notes to actually solve this problem. I hope my solution has made your life easier in solving this mod problem.
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