Data Sufficiency

I think you've gotten a bit mixed up with your target question here. If we want to simplify this algebraically, we could say:

(x^4)(x^1) > x^4 ?

It's tempting to divide both sides by x^4 here... but what if x = 0 ? We're not allowed to divide by 0. For non-zero values, we can divide by x^4 (which will always be positive, so we don't have to worry about flipping the sign). Then the question becomes:
x^1 > 1? In other words, is x > 1 or is x = 0?

Often on inequalities DS questions, the easiest way to rephrase the question might be to ask: what kinds of numbers are larger when taken to the 5th than the 4th? What kinds of numbers aren't? We can use this number line to test cases:



We can clearly see that the only time we get a "yes" answer is when x > 1, so our target question is: is x > 1 ?

(I couldn't initially tell if those were dashes or negative signs in your notation, so just be careful with that).

1) x^3 > -x
We can't divide here because these are odd exponents. Use the number line to test:



As we can see, this will be true wherever x > 0, so the statement is really saying: x > 0. This doesn't help us to answer the question "is x > 1?" INSUFFICIENT

2) 1/x < x
Again, we can't divide. Let's test cases:



This will be true when x is a negative fraction, or when x > 1. INSUFFICIENT

1 & 2) If we combine the statements, then x has to be positive, so x cannot be a negative fraction. It must be greater than 1. SUFFICIENT.