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I am sorry to say that the following logic is wrong.
[S-R] > [T-U]
opening the inequality gives:
A. S-R > T-U
B. S-R < U-T

for example take s = -16, r = -10, t = -15, u = -10
|s-r| = | -16+10 | = |-6| = 6
|t-u| = | -15+10 | = |-5| = 5
this satisfies the condition |s-r| > |t-u|
but
s-r = -16+10 = -6
t-u = -15+10 = -5
This does not satisfy ( s-r > t-u ) so we cannot use the above logic.

Let us assume that |R-T| > |R|-|T| and not greater than or equal to
then
|R|-|T| > |S|-|U| let us rearrange this equation
-|S|+|R| > |T|-|U|
so |S|-|R| < |T|-|U| this gives the answer to the question
but the problem is
|R-T| can be equal to |R|-|T| and |S-U| can be greater then |S|-|U| then
we cannot say for sure |R|-|T| > |S|-|U|
The same rule applies to condition 2, and for this reason alone we cannot combine the derivatives of conditions 1 and 2.
Thus we can conclude that neither of the conditions can be used to solve the inequality.

I could not agree with you more. I am not sure I would have solved the question in two minutes. We definitely need shortcuts. I just feel that arriving at correct answer does not mean the method of solving the problem is correct. A slight twist in the question can make the solution invalid and can give you wrong answers.
I tried to put the solution because new comers can learn little bit about the absolute inequialities. I have learnt a lot from people like you by going through the post and I would like to contribute wherever I can.

I think this problem should be approached in the following way. I know that the method looks very long but I think if understood once, it should not take long to do similar problems. In a way, it is just an extension of Racers approach.

We know that |x| = x, if x > 0 but |x| = -x if x < 0.

So any inquality |x| > |y| can result in the following possibilities:

x is +ve and y is +ve => x > y
x is -ve and y is -ve => -x > -y
x is +ve and y is -ve => x > -y
x is -ve and y is +ve => -x > y

Applying this logic, what we need to find here is Is |S-R|>|T-U|? In other words,

Is S-R > T-U? or
Is R-S > U-T? OR
Is S-R > U-T? OR
Is R-S > T-U? ========================(Q)

Now applying the logic to statement 1,

|R-T|>|S-U|, is this sufficient to say that ANY ONE of the above 4 possibilities is correct?

Let us see, this statement will give following possibilities

R-T > S-U => R-S > T-U (This is one of the possibilities in Q above. So the answer is YES) T-R > U-S => S-R > U-T (So the answer is YES again) R-T > U-S => R+S > T+U (Can not say ) T-R > S-U => T+U > S+R (Can not say)

Statement 1 NOT SUFF

Similarly statement 2 will give 4 possibilities, 2 of which will answer the question in YES and remaining 2 will not answer. So statement 2 NOT suff.

TOGETHER

Both the statments taken togethet will also result in the 4 possibilities that are mentioned in BOLD above.