Question

When do we have an open circle on the number line for an inequality? When do we have a closed circle on the number line for an inequality?

Answer 1

In the context of inequalities represented on a number line, we use open and closed circles to denote whether a number is included in the solution set or not.

1. **Open Circle**: An open circle is used when the number is **not included** in the solution set. This occurs in cases of strict inequalities:
- For example, in the inequality \( x < 3 \) or \( x > 5 \), we would place an open circle at 3 and 5 on the number line to indicate that those values are not part of the solution.

2. **Closed Circle**: A closed circle is used when the number **is included** in the solution set. This occurs in cases of non-strict inequalities:
- For example, in the inequality \( x \leq 4 \) or \( x \geq 2 \), we would place a closed circle at 4 and 2 on the number line to indicate that those values are included in the solution.

In summary:
- Use an **open circle** for \( < \) or \( > \) (not inclusive).
- Use a **closed circle** for \( \leq \) or \( \geq \) (inclusive).