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The compound inequality presented in the question is:

\[
x \leq -1 \quad \text{or} \quad x \geq 2
\]

This means the solution includes all values of \(x\) that are either less than or equal to \(-1\) or greater than or equal to \(2\). To solve and represent this on a number line, let's break it down:

1. \(x \leq -1\): This includes all values less than or equal to \(-1\). On a number line, this is represented as a ray extending leftwards from \(-1\), with a closed circle at \(-1\) (indicating that \(-1\) is included).

2. \(x \geq 2\): This includes all values greater than or equal to \(2\). On a number line, this is represented as a ray extending rightwards from \(2\), with a closed circle at \(2\) (indicating that \(2\) is included).

### Analyzing the given options:

- **Option A**: This shows the correct solution: the ray extends leftwards from \(-1\) (closed circle at \(-1\)) and rightwards from \(2\) (closed circle at \(2\)).
- **Option B**: Shows no solution for values less than \(-1\), so this is incorrect.
- **Option C**: The ray extends from \(-2\) instead of \(-1\), so this is incorrect.
- **Option D**: The solution between \(-1\) and \(2\) is omitted, but these values should not be included, so this is incorrect.
- **Option E**: This says the solution set is empty, which is incorrect.
- **Option F**: This is missing the portion \(x \geq 2\), so it's incorrect.

### Correct answer: **Option A**

Would you like more details on graphing inequalities, or do you have any questions?

Here are five related questions you might explore:
1. How do you graph a compound inequality with "and" instead of "or"?
2. What happens if the inequality symbols are strict, like \(x < -1\) or \(x > 2\)?
3. How do compound inequalities work with intervals, and how do you represent them in interval notation?
4. Can compound inequalities result in an empty solution set? If so, how does that look graphically?
5. What are the differences between open and closed circles when graphing inequalities?

**Tip**: Always remember that a closed circle on the number line means the point is included (≤ or ≥), while an open circle means the point is excluded (< or >).

Solve the compound inequality: x ≤ -1 or x ≥ 2. Graph the solution set on a number line.

Learn how to solve and graph the compound inequality x ≤ -1 or x ≥ 2 on a number line. This detailed explanation covers key algebraic concepts, including compound inequalities and graphing with closed intervals. Suitable for students in Grades 8-10.

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