Answer
The inequality is valid for values more than 1 (not including it) i.e. $ (1,\infty)$.
Work Step by Step
First, we are going to find the x-intercepts by equating to zero:
$x^3=1$
$x^3-1=0$
$(x-1)(x^2+x+1)=0$
The expresion $x^2+x+1$ cannot equal zero, so that leaves us with the only x-intercept:
$(x-1)=0$
$x=1$
This is the only critical point. We are going to take two values: one less than 1 and one more than 1 to test in the original equation and check if the inequality is true or not:
First test with a value less than 1:
$(0)^3>1$
$0>1 \rightarrow \text{ FALSE}$
Second test with a value more than 1:
$2^4>1$
$16>1 \rightarrow \text{ TRUE}$
These tests show that the inequality $x^3>1$ is valid for values more than 1 (not including it) i.e. $ (1,\infty)$