$(1, \infty)$ Refer to the image below for the graph.
Work Step by Step
$x \ge -4$ means all the real numbers greater than or equal to $-4$. $x\gt 1$ means all the real numbers greater than (to the right of) 1. (refer to the attached image below for the graph) The conjunction used is AND, which means that both conditions/set must be satisfied. This means the solution set will contain the numbers that are common to both sets/intervals. Note that the numbers common to both intervals are the ones to the right of $1$ (but not including 1 itself). Therefore, the solution to the compound inequality is $(1, \infty)$.