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My answer is B). 0
Based on |x-y|= y-x; we get
y-x>=0
y>=x. But question tells that x is not equal to y and y is negative.
We can conclude condition: y>x, where both x and y are negative numbers and y is a smaller negative say If x is -2 then y can be -1
Substituting x as -2 and y as -1 in the equation, we get= -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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|x-y| = y-x means the modulus opens with a negative sign, which means x-y < 0, meaning x < y
Now since y < 0, x is also less than 0 since x < y
So both x and y are negative

With this in mind, we resolve all modulus values in the expression
x/|x| -> mod x opens as -x, so -1

|-y|/-y -> |y|/-y, mod y opens as -y, so 1

x-y/|y-x| -> x-y < 0 , so y-x > 0, so mod opens as y-x, so value is -1

|xy|/xy -> since xy both negative, xy will be positive, so mod opens as xy, hence value is 1

Adding all 4, we get 0, hence B)
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|x-y|=y-x
Then, x-y<0. x<y
x<y<0.
Let, x=-2, y=-1

(-2)/2 + 1/1 + (-2+1)/|-1+2| + 2/2
= -1+1-1+1= 0

B
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in order to satisfy the given function condition, let x=-4 and y=-3 since y<0 and x not equal to y. Putting the given values in final equation, the answer is 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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Given,
\(|x-y|=y-x\)
Opening the modulus;

Case 1: Let (x-y)>0
\(|x-y|=y-x\) becomes (x-y) = (y-x)
or, 2x=2y
or, x=y
Not possible (Since given in the question that \(x \ne y\))

Case 2: (x-y) <0
\(|x-y| = y-x \) becomes -(x-y)= (y-x)
or, y-x = y-x
or, 0 = 0
Hence, true for all the values of x & y.
Thus, x-y < 0
or, x < y
Also, since y<0 (this means x is also negative and having more magnitude)

Now, to easily solve the given equation, we can assume any negative values of x & y ( since the case 2 is valid for every x & y)
Let, y= -1 and x =-2
Then the equation becomes:
\( \frac {-2}{|-2|} + \frac {|-(-1)|}{-(-1)} + \frac {-2-(-1)}{|-1-(-2)|} + \frac {|-2 * -1|}{-2 * -1}\)
which simplifies to:
-1 + 1 + (-1) + 1 = 0
Hence the final answer is B. 0
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if |x-y| = y - x that means that x - y < 0.
We can say that x<y. Also given that y<0 we can assume just two numbers here.
say x = -2 and y = -1.
Put this in the expression and we get the expression as 0.
Ans B IMO.
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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In this question if we note that y is negative(GIVEN) and |x-y| = y - x. We could infer that 'x' shall be negative too because when we open the mod |x-y| it equates to y-x and y<0 so it would change the sign to '+' |x+y| and then in the result we get y-x. So x must be negative and smaller than 'y' to prove this below is the explanation too: x-y cannot be zero as x is not equal to y also y is negative. So x-y<0 which gives x<y.
Since y is negative so x<y<0
Not just replace the x with -x and y with -y.
The first term would be -1, second term would be 1, third term would be -1 and the last term would be 1.
When calculated it would equal to zero.

Let me know if there is any confusion
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|x-y|=y-x, then y-x>=0, but as y<0, we can say x<y<0

x/|x| + |-y|/|y| + (x-y)/|y-x| +|xy|/xy
= -1 + 1 + 1 + 1
2
Ans D
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On solving
X/x=1
|-y|/-y= y/-y =-1
X-y/|y-x| = x-y/-(x-y) = -1
|xy|/xy= 1
1-1-1+1= 0
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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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given that \(|x-y| = y-x \) , if this needs to be true then x-y < = 0
so x<=y
also given \(x \ne y\)
so x < y
given y<0
so x<y<0
both are negative,
\(\frac {x} {|x|}\)
x<0 , \(|x|= -x\)
so , \(\frac {x} {-x}\) = -1

\(\frac {|-y|} {-y}\)
y<0, -y>0
\(|-y|= -y\)
so, \(\frac {-y} {-y}\) = 1

\(\frac {x-y} {|y-x|}\)
since x<y, y-x is positive
so \(|y-x| = y-x\)
so, \(\frac {x-y} {y-x} = -1\)

\(\frac {|xy|} {xy}\)
since, x<0, y<0
xy>0
so, |xy| = xy
\(\frac {|xy|} {xy} = 1\)

-1+1+(-1) +1 = 0
B
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Given, |x-y| = y-x. This means y>x.
Also given y<0. Meaning x is also less than 0. thus both are negetive.
For the answer sake, let x=-4 and y=-2.
Hence the equation, considering absolute values becomes : -4/4 + 2/-(-2) + (-4-(-2))/|-2-(-4)| + |-2 X -4|/-2 X-4 = -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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|x-y| = y-x, it means y>x and x is not equal to y as given in question statement itself.
Also, y<0 given. It means x is also <0.

Accordingly, x/|x| will be -1, |-y|/-y = 1, x-y/|y-x| = -1 and |xy|/xy = 1
-1+1-1+1 = 0

Ans. 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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|x-y|=y-x
.: x-y<0
.: x<y
we know, y<0
.: x<0
.: |x|/x=-1
|-y|/-y=-(-1)=1
|y-x|=|-1||x-y|=|x-y|
.: (x-y)/ |y-x|= -1
|xy|/xy=1 (Since, both x & y are negative)

.: -1+1-1+1=0
Ans is 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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From the given conditions we can figure out that y is negative as y < 0.
Also, since |x-y|=y-x, from this we can gather that x has to be negative since y-x needs to be a positive number for the equation to satisfy.

Thus:
1. x/|x| = -1
2. |-y|/-y = 1
3. (x-y)/|y-x| = 1
4. |xy|/xy = 1

Therefore, -1+1+1+1 = 2
Answer is (D)
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As |x-y| = y-x, x-y < 0 (Using the property of modulus)
Thus, x < y, and since y < 0, x < 0
For simplification, let's use numbers in place of the variables x and y.
Let's assume y = -1 and x = -2 (Remember x needs to be smaller than y)

Now, let's solve for the individual components of the sum by individually, by using the above information:
x/|x| = -2/2 = -1
|-y|/-y = 1/1 = 1
x-y/|y-x| = -1/1 = -1
|xy|/xy = 2/2 = 1

Adding these all up, -1+1-1+1, we get the answer as 0.

Hence, the correct answer is Option B - 1.

Hope this helps! :)
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Not sure if this is the smartest or the best method but here goes:

We know that |x-y| = y-x ---------(1) (y is negative and x is not equal to y)

Since the question is in terms of x and y, it would help to know the sign of x.

If x is zero OR +ve,
We will get RHS = -ve in (1) which is not possilble (as RHS is an abosolute value)
Therefore, x<0 ---------- (2)

Now that we know the sign, it can help to know a relation between magnitude of x and y

if |x| < |y|
It will again give RHS as -ve
(since we've established x & y both are -ve, the equation will essentially become -|y| + |x|. And if we remove a smaller number from y than itself, RHS will stay -ve)

Therefore, we can deduce that |x| > |y| ---------- (3)

Armed with (2) and (3), we attack the question. Let's look at each term individually
x/|x| = -1 (since x<0, from (2)
|-y|/(-y) = +1 (since y<0)
(x-y)/|y-x| = -1 (since x&y<0; x-y=-(y-x) and we know from (1) that y-x is positive. Therefore, this term = -((y-x)/|y-x|)
|xy|/xy= +1 (since both x & y<0)

This gives us:
-1+1+(-1)+1
=0

Ans: B
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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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y - x > 0

y > x

x/|x| = -1

|-y|/-y = +1

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

|xy| / xy = 1

-1 + 1 - 1 + 1 = 0

Option B
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