Answer
The inequality is valid for values less than -1 and values more than 1 (not including them) i.e. $(-\infty,-1)\cap (1,\infty)$.
Work Step by Step
First, we are going to find the x-intercepts by equating to zero:
$x^4=1$
$x^4-1=0$
$(x^2-1)(x^2+1)=0$
$(x^2-1)=0$
$(x+1)(x-1)=0$
$x_1=-1$
$x_2=1$
These are the critical points. We are going to take three values: one less than -1; one between -1 and 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:
$(-2)^4>1$
$16>1 \rightarrow \text{ TRUE}$
Second test with a value between -1 and 1:
$0^4>1$
$0>1 \rightarrow \text{ FALSE}$
Third test with a value more than 1:
$2^4>1$
$16>1 \rightarrow \text{ TRUE}$
These tests show that the inequality $x^4>1$ is valid for values less than -1 and values more than 1 (not including them) i.e. $(-\infty,-1)\cap (1,\infty)$
