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Apparently the function solve does not take the assumptions into account.
One thing you can do is to include the assumptions yourself.
Either by hand:
sage: solve([(x^3 - 4*x) > 0, x > 0], x)
[[2 < x]]
Or more automatically.
sage: assume(x > 0)
sage: assumptions()
[x > 0]
sage: A = (x^3 - 4*x) > 0
sage: A
x^3 - 4*x > 0
sage: solve(assumptions() + [A], x)
[[2 < x]]
You could also add your inequality to the assumptions.
sage: assume(x > 0)
sage: assume(x^3 - 4*x > 0)
sage: assumptions()
[x > 0, x^3 - 4*x > 0]
sage: solve(assumptions(), x)
[[2 < x]]
