Are You Trying Too Hard?

By definition of absolute value , $\left | x^4 - x^3\right |$ and $\left | x^2-x \right |$ are both positive. Therefore, it follows that $-\left | x^4 - x^3 \right |$ and $- \left | x^2-x \right |$ are both negative. So, in the inequality we have

$\text{Negative} + \text{Negative} < \text{Positive}$

Obviously this is always true, even for any value of $x$ . So $x$ can be anything in the interval $\boxed{(-\infty , \infty)}$ .