I used the Pythagorean theorem and a characteristic of triangles to prove that (1) is not suff. and (2) also.

Can anybody guess how?

I used the Pythagorean theorem and a characteristic of triangles to prove that (1) is not suff. and (2) also.

Can anybody guess how?

in my learned opinion, this is one of the most difficult inequality problems that gmatprep has put out there in some time, so you definitely shouldn't feel bad about tanking it.

along the way, we're going to learn 2 VERY important takeaways about data sufficiency number plugging. in fact, the first takeaway is so important that i'll state it 3 times.
here it is for the first time:
takeaway #1: when you plug numbers on a DS problem, YOUR GOAL IS TO PROVE THAT THE STATEMENT IS INSUFFICIENT.
therefore, as soon as you get a 'yes' answer, you should be TRYING to get a 'no' answer to go along with it; and, as soon as you get a 'no' answer, you should be TRYING to get a 'yes' answer to go along with it.

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statement (2)
you need to pick numbers such that x + y > z, per this statement.
first, pick a completely random set of numbers that does this: how about x = 1, y = 1, z = 0.
these numbers give a YES answer to the prompt question, since 1^4 + 1^4 is indeed greater than 0^4.
now remember: your goal is to prove that the statement is INSUFFICIENT.
this means that we have to try for a 'no' answer.
this means that we have to make z^4 as big as possible, while still obeying the criterion x + y > z.
fortunately, this is somewhat simple to do: just make z a big negative number.
try x = 1, y = 1, z = -100
in this case, x + y > z (satisfying statement two), but x^4 + y^4 is clearly less than z^4, so, NO to the prompt question.
insufficient.

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statement (1)
you need to pick numbers such that x^2 + y^2 > z^2, per this statement.
first, pick a completely random set of numbers that does this: how about x = 1, y = 1, z = 0 (the same set of numbers we picked last time).
these numbers give a YES answer to the prompt question, since 1^4 + 1^4 is indeed greater than 0^4.
now remember: your goal is to prove that the statement is INSUFFICIENT.
this means that we have to try for a 'no' answer.
this means that we have to make z^4 as big as possible, while still obeying the criterion x^2 + y^2 > z^2.
unfortunately, this isn't as easy to do as it was last time; we can't just make z a huge negative number, because z^2 would then still be a giant positive number (thwarting our efforts at obeying the criterion).
so, we have to finesse this one a bit, but the deal is still to make z as big as possible while still obeying the criterion.
let's let x and y randomly be 3 and 3.
then x^2 + y^2 = 18. we need z^2 to be less than this, but still as big as possible. so let's let z = 4 (so that z^2 = 16, which is pretty close).**
with these numbers, x^4 + y^4 = 162, which is much less than z^4 = 256. therefore, NO to the prompt question, so, insufficient.

answer = e.

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by the way, you may have noticed that divya didn't get the algebra to work, so she just tossed her electronic hands in the air and said 'i give up'.
now, clearly, NOT FIGURING OUT the algebra doesn't PROVE that a statement is insufficient, but, whether intentionally or not, divya is onto something here. specifically:
takeaway #2: if a statement is sufficient, then you WILL be able to PROVE that it is, algebraically or with some other form of theory. in other words, you'll never get a statement that's sufficient, but for which you can only figure that out by number plugging.

since the algebra just doesn't work out - especially for a student as strong as divya (she has posted some pretty amazing stuff on other threads) - you should have a strong inclination to think that the statements are insufficient.
and you'd be right.

now, clearly, NOT FIGURING OUT the algebra doesn't PROVE that a statement is insufficient, but, whether intentionally or not, divya is onto something here. specifically:
takeaway #2: if a statement is sufficient, then you WILL be able to PROVE that it is, algebraically or with some other form of theory. in other words, you'll never get a statement that's sufficient, but for which you can only figure that out by number plugging.

since the algebra just doesn't work out - especially for a student as strong as divya (she has posted some pretty amazing stuff on other threads) - you should have a strong inclination to think that the statements are insufficient.
and you'd be right.

Is there any good way to decide whether employing algebra OR "plugging in" would be a better way to approach a specific problem?

Trying out algebra on this kind of a tough problem wastes at least a minute or so of valuable time, until we are left confused with the algebra not working out.