Doubt in countability of a subset of real number

Let $S=\{x \in \mathbb{R} \mid x>1, \frac{1-x^4}{1-x^3}>22\}$ then $S $ is

1. Empty
2. Countably finite
3. Countably infinite
4. Uncountable.

My approach is:

$\frac{1-x^4}{1-x^3}>22, x>1$$\implies 1+x+x^2+x^3>22+22x+22x^2, x>1 \implies x^3-21x^2-21x-21>0,x>1 $

using the wavy curve method, $S$ is uncountable.

Option 4) is correct. Is my approach is correct?