1. The problem asks to identify the set expression $x \leq -4 \cup x \geq 3$ as either an intersection, union, or no intersection of two sets. 2. The symbol $\cup$ denotes the union of two sets, meaning all elements that are in either set or both. 3. The first set is $\{x | x \leq -4\}$, which includes all numbers less than or equal to $-4$. 4. The second set is $\{x | x \geq 3\}$, which includes all numbers greater than or equal to $3$. 5. Since these two sets do not overlap (no number can be both $\leq -4$ and $\geq 3$ simultaneously), their union includes all numbers less than or equal to $-4$ and all numbers greater than or equal to $3$. 6. Therefore, the expression represents the union of two sets. Final answer: Union