Is |x - y| > |x + y|?

I'd start solving by analyzing the stem.

First approach:

Is \(|x-y|>|x+y|\)? In which cases \(|x-y|\) will be more than \(|x+y|\)?

If \(x\) and \(y\) have the same sign (both positive or both negative) then absolute value of their sum (\(|x+y|\)) will be more than absolute value of their difference (\(|x-y|\)), as in this case \(x\) and \(y\) will contribute to each other in \(|x+y|\), for example \(|5+3|\) or \(|-5-3|\) and in \(|x-y|\) they will not.

If \(x\) and \(y\) have opposite signs then absolute value of their difference (\(|x-y|\)) will be more than absolute value of their sum (\(|x+y|\)), as in this case \(x\) and \(y\) will contribute to each other in \(|x-y|\), for example \(|5-(-3)|\) or \(|-5-3|\) and in \(|x+y|\) they will not.

So basically the question is: do \(x\) and \(y\) have opposite signs?

Second approach:

Is \(|x-y|>|x+y|\)? As both sides of inequality are non-negative we can safely square them: is \((x-y)^2>(x+y)^2\)? --> is \(x^2-2xy+y^2>x^2+2xy+y^2\)? --> is \(xy<0\)? Again the question becomes: do \(x\) and \(y\) have opposite signs?

Next:

(1) x^2-y^2 = 9 --> we cannot say from this whether \(x\) and \(y\) have opposite signs. Not sufficient.

(2) x-y=2 --> we cannot say from this whether \(x\) and \(y\) have opposite signs. Not sufficient.

(1)+(2) \(x+y=4.5\) and \(x-y=2\) --> directly tells us that \(|x-y|=2<|x+y|=4.5\). Sufficient. (If you solve system of equations you'll see that \(x\) and \(y\) have the same sign: they are both positive).

Answer: C.

PLUGGING NUMBERS:

On DS questions when plugging numbers, goal is to prove that the statement is not sufficient. So we should try to get a YES answer with one chosen number(s) and a NO with another.

(1) \(x^2-y^2=9\) --> try \(x=5\) and \(y=4\) (\(x^2-y^2=25-16=9\)) --> \(|x-y|=1<|x+y|=9\), so we have answer YES. Now let's try another set of numbers to get NO: \(x=5\) and \(y=-4\) --> \(|x-y|=9>|x+y|=1\), so now we have answer NO. Thus this statement is not sufficient.

We can do the same with statement (2) as well.

For this question I don't recommend to use number plugging for (1)+(2), as it's quite straightforward algebraically that taken together 2 statements are sufficient.

Hope it helps.