We are given two sets:

• $E = \{ x \mid x < -7 \}$, which represents all real numbers less than $-7$.
• $F = \{ x \mid x \geq -1 \}$, which represents all real numbers greater than or equal to $-1$.

We are tasked with finding the union of these two sets, $E \cup F$.

Understanding the Union

The union of two sets, $E \cup F$, consists of all elements that are in either $E$, $F$, or in both.

• $E = \{ x \mid x < -7 \}$ includes all real numbers less than $-7$.
• $F = \{ x \mid x \geq -1 \}$ includes all real numbers greater than or equal to $-1$.

Combining the Two Sets

To form the union $E \cup F$, we combine all elements of both sets:

• For $E$, we have all numbers less than $-7$.
• For $F$, we have all numbers greater than or equal to $-1$.

Notice that there is a gap between $-7$ and $-1$, where no elements are included in either set. Therefore, the union of these sets excludes the interval $-7 \leq x < -1$.

Thus, the union $E \cup F$ can be written as: $E \cup F = \{ x \mid x < -7 \} \cup \{ x \mid x \geq -1 \} = (-\infty, -7) \cup [-1, \infty)$

Final Answer:

$E \cup F = (-\infty, -7) \cup [-1, \infty)$

Would you like further clarification on any step or additional examples?

Here are some related questions:

1. What is the intersection $E \cap F$ of these two sets?
2. How would you find the union of two sets if the intervals overlapped?
3. What is the difference between union and intersection in set theory?
4. Can the union of sets include the same number multiple times?
5. What does it mean for a number to belong to a set with a strict inequality (e.g., $x < -7$)?

Tip: When working with unions of sets, always check if the sets overlap or have gaps between them to visualize the combined range.