https://file.aiquickdraw.com/mathbot/file/172797505009860sdswju.jpg

For the graphs shown above, find the following:
a) Write a compound inequality to describe the union of the points shown in the graphs.
Answer:

b) Write a compound inequality to describe the intersection of the points shown in the graphs.
Answer:

Let's analyze the number lines provided in the image.

### a) Compound Inequality for the **Union** of the Points

From the graphs:
- The **top line (green)** shows the interval from \(-5\) to \(-1\], meaning all values from \(-5\) to \(-1\), including \(-1\).
- The **bottom line (red)** shows the interval from \(0\) to \(5\), excluding \(0\) but including \(5\).

The **union** of these two graphs includes all the points covered by either interval, which ranges from \(-5\) to \(-1\] and from \((0, 5]\). Thus, the compound inequality for the union is:

\[
-5 \le x \le -1 \quad \text{or} \quad 0 < x \le 5
\]

### b) Compound Inequality for the **Intersection** of the Points

The **intersection** of the two intervals is the set of points that are covered by both graphs simultaneously. 

- The **green line** goes from \(-5\) to \(-1\] and the **red line** goes from \(0\) to \(5\). There is no overlap between these two intervals.

Thus, the intersection is **empty**.

The compound inequality for the intersection can be expressed as:

\[
\text{No solution (empty set)}
\]

### Summary:
- **Union**: \(-5 \le x \le -1\) or \(0 < x \le 5\)
- **Intersection**: No solution (empty set)

Do you want details or have any questions? 

Here are five related questions that may help deepen your understanding:

1. How do you determine the union of two different intervals on a number line?
2. What does an "open" versus a "closed" circle on a number line signify?
3. How do you represent the complement of an interval?
4. Can an intersection ever be represented by a single interval?
5. How does a union differ from an intersection in terms of set theory?

**Tip:** When working with unions and intersections, always pay attention to open and closed endpoints; they determine whether the boundary values are included in the interval.

For the graphs shown above, find the following:
a) Write a compound inequality to describe the union of the points shown in the graphs.
b) Write a compound inequality to describe the intersection of the points shown in the graphs.

Learn how to find the compound inequalities for the union and intersection of two number line intervals. Understand the difference between open and closed endpoints and how to represent empty intersections. Suitable for students in Grades 8-10.

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