GMAT prep question

I could not think of any algebraic solution for this one. Plugged in some numbers. Hope I got it correct. If not then please help.

Stmt1: x^2 + y^2 > z^2
Stmt2: x + y > z

Let us look at Stmt2 first.
x = 3, y = 4, z = 6
x + y = 7 which is greater than z = 6 but x^4 + y^4 = 81 + 256 = 337
and z^4 = 1296. So the answer to the question is No.
But we could have x = 3, y = 4 and z = 1. In this case the asnwer would be Yes.
So Stmt 2 is insufficient. Left with A, C and E.

For plugging in the values into stmt1, we could use the sqrt of the values (as x, y, and z need not be integers) used for stmt 2 to arrive at the same conclusion that its insufficient.

Combining the 2 also we cannot conclusively say whether x^4 + y^4 > z^4.

Hence E.

Please comment and correct me if I am wrong.

I think the answer should be E.

question stem: x^4 + y^4 > z^4 ?

Statement I

x^2 + y^2 > z^2

if you use integers x^4 + y^4 > z^4 yes.
if you use fractions x^4 + y^4 > z^4 no.

Rule: when fractions are squared or increased with positive exponents the value of fraction decreases.

Insufficient.

Statement II

x+y>z

if you use integers x^4 + y^4 > z^4 yes.
if you use fractions x^4 + y^4 > z^4 no.

same reasoning as above.

Insufficient.

Combining I & II

if you use integers x^4 + y^4 > z^4 yes.
if you use fractions x^4 + y^4 > z^4 no.

x=0.5, y=0.7, z=0.8

x+y>z
0.5+0.7>0.8
1.2>0.8
x+y>z

x^2 + y^2 > z^2
0.25 + 0.49 > 0.64
0.74>0.64
x^2 + y^2 > z^2

x^4 + y^4 > z^4
0.0625 + 0.2401 < 0.4096
0.3026 < 0.4096

Hence E is the answer.

I don't want to say that the above solutions are not correct, but is there any elegant(pure algebraic) solution to this problem?

Thanks.