Answer
$ (1,\infty)$.
Work Step by Step
Step 1. For $x^3-1\gt0 \Longrightarrow (x^2+x+1)(x-1)\gt0$, identify boundary points as $x=1$.
Step 2. Form intervals $(-\infty,1),(1,\infty)$.
Step 3. Choose test values for each interval $x=0,2$.
Step 4. Test the inequality to get results: $ False,\ True$
Step 5. We have the solution $ (1,\infty)$.
