union of (- infinity, -1] U [-4, infinity)

The union of the two intervals \( (-\infty, -1] \) and \( [-4, \infty) \) represents all the values that are included in either or both of the intervals. Let's analyze it step by step:

1. The interval \( (-\infty, -1] \) includes all values from negative infinity up to and including \(-1\).
2. The interval \( [-4, \infty) \) includes all values from \(-4\) and greater.

When taking the union of these two intervals, you are combining all the numbers they cover. Notice that the second interval already includes all the values from \(-4\) to infinity, which overlaps and subsumes the portion of the first interval from \(-4\) onward. Hence, the union will cover:

\[
(-\infty, \infty)
\]

This represents all real numbers.

Would you like more details or have any questions about this? Here are some related questions:

1. What is the intersection of the intervals \( (-\infty, -1] \) and \( [-4, \infty) \)?
2. How do you represent the complement of a union of intervals?
3. What are the rules for combining intervals in set theory?
4. How does interval notation compare to inequalities?
5. What are other common symbols used in set theory?

**Tip**: When taking the union of intervals, it's helpful to plot them on a number line to visually see the overlap or coverage.

Explore how to solve the union of intervals (-∞, -1] U [-4, ∞) with a step-by-step explanation. This topic covers key concepts in set theory and interval notation. Perfect for high school students in Grades 9-12.

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