If |x|-|y|=13, then which of the following can't be the value of x-y?

Same doubt is in my mind.

x and y can be any real numbers. They can be rational or irrational numbers. Never assume anything in GMAT exam, if it's not explicitly mentioned.
If x=-15.5 and y=2.5
|x|-|y|=13, and x-y=-18



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Here we can see that the options are all negative. Hence x<0
According to given question |x|>|y| and x <= -13, y<=0 to get a value of 13.
Therefore x-y is always less than or equal to -13. Hence x-y cannot be -9.

Hence the answer is A.

|x|-|y|=13
if y<0 |y| = -y
|x| = -13-y

x-y = -13-2y<-13


IMO A

How do I know that |x-y|≥|x|-|y| is true? Is this a general rule or how can I see it with the information given?

|x-y|≥|x|-|y|
It's true for every real number.
1. If x and y have opposite signs, |x-y|>|x|-|y|
2. If x and y have same signs and |x|>|y|, then |x-y|=|x|-|y|
3. If x and y have same signs and |x|<|y|, then |x-y|>|x|-|y|

You can solve this question in seconds by remembering the identities.
|x| - |y| <= |x+y|

Here, |x| - |y| =13

So,
13 <= |x -y|
which of the values mod is not suiting this above equation
A. 13 <= |-9| False
B. 13 <= |-13| True
C. 13 <= |-17| True
D. 13 <= |-18| True
E. 13 <= |-29| True

Answer - A

Since |x - y| ≥ |x| - |y|, it must be true that |x - y| ≥ 13. Thus, x - y cannot equal -9.

Alternatively, we can eliminate every answer choice besides A. For instance, if we choose x = -13 and y = 0, we see that x - y = -13, so we eliminate B. If x = -15 and y = 2, then x - y = -17, which eliminates C. If x = -15.5 and y = 2.5, then x - y = -18. Finally, if x = -21 and y = 8, then x - y = -29. Since we can eliminate every answer choice besides A, the correct answer is A.

Answer: A

Asked: If |x|-|y|=13, then which of the following can't be the value of x-y?

Case 1: x < y < 0
|x| - |y| = -x + y = 13
x- y = -13

Case 2: x< 0 < y
|x| - |y| = -x - y = 13
x + y = -13
x - y = -13 - 2y < -13

x- y <= - 13

A. -9: NOT POSSIBLE
B. -13: POSSIBLE
C. -17: POSSIBLE
D. -18: POSSIBLE
E. -29: POSSIBLE

IMO A

The easiest way would be to use the |x - y| >= |x| - |y| relation. But if it doesn't come to mind, we know that when we have absolute values in multiple terms, we can remove the sign by taking sections on the number line.

Given: |x|-|y|=13

Case 1: x and y both are positive. Then x - y = 13.

Case 2: x and y are both negative. Then -x -(-y) = 13 i.e. x - y = -13. So -13 is possible

Case 3: x is positive and y is negative. This will give x - y as a positive number but we don't have any positive options. So ignore.

Case 4: x is negative and y is positive. Then -x - y = 13 or x + y = -13
Since x is negative and y is positive and the sum is negative, absolute value of x must be greater than y.
x could be -14 and y could be 1. This means x - y = -15
x could be -15 and y could be 2. This means x - y = -17
and so on.. We will get all values greater than -13 in this case.

Hence -9 is not possible. Answer (A)