Is |x - y| = ||x| - |y|| ? (1) xy > 0 (2) x < y < 0

\(||x| - |y||\) is the distance between the absolute values of x and y. That is, it is the absolute difference between two positive quantities.

Even \( |x - y|\) is a positive value but whether it is the distance between two positive values or not would depend on the signs of x and y.
If both x and y are of same sign, then it will be equal to \(||x|-|y||\). For example, 2 and 5 will give |2-5| or 3 AND -2 and -5 will give |-2-(-5)| or |-2+5| or 3.

So, we are looking for whether xy<0.

Statement I gives the answer directly.
Statement II gives both x and y as negative , so xy<0.

D
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There's a really simple framework for this. There are really five possibilities:
- where x is positive and y is positive ==> statement is true; you can imagine away the absolute values in this case
- where x is positive and y is negative ==> statement is not true (try +1 and -1)
- where x is negative and y is positive ==> statement is not true (try -1 and +1)
- where x is negative and y is negative ==> statement is true*
- one of x, y is zero ==> statement is obviously true

Statement (1) tells you that x and y share the same sign, i.e. (+,+) or (-,-). So it's sufficient.
Statement (2) tells you that x and y are negative, i.e. (-,-). So it's sufficient.

Both are individually sufficient.


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*
Suppose a and b are positive integers; then (-a) and (-b) are negative integers

Let's start with an algebraically true statement: (-a) - (-b) = -a + b = b - a

Let's pose another true statement: |b - a| = |a - b|
And another true statement: |-a| = |a|; |-b| = |b|

So we put these together and say |(-a) - (-b)| = |b - a| = |a-b| = ||a| - |b|| = ||(-a)| - |(-b)||