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?
Answers
GPT-4o mini
On a number line, an open circle and a closed circle are used to represent the solutions to inequalities:
1. **Open Circle**: An open circle is used when the value at that point is **not included** in the solution set. This is typically used for inequalities that are strictly greater than or less than:
- Example: \( x < 3 \) or \( x > 5 \)
- In this case, the circle at 3 and 5 would be open because 3 and 5 themselves are not part of the solution.
2. **Closed Circle**: A closed circle is used when the value at that point is **included** in the solution set. This is typically used for inequalities that are greater than or equal to or less than or equal to:
- Example: \( x \leq 4 \) or \( x \geq 2 \)
- In this case, the circle at 4 and 2 would be closed because 4 and 2 themselves are included in the solution.
In summary:
- Use an **open circle** for inequalities that do not include the endpoint (\(<\) or \(>\)).
- Use a **closed circle** for inequalities that include the endpoint (\(\leq\) or \(\geq\)).
1. **Open Circle**: An open circle is used when the value at that point is **not included** in the solution set. This is typically used for inequalities that are strictly greater than or less than:
- Example: \( x < 3 \) or \( x > 5 \)
- In this case, the circle at 3 and 5 would be open because 3 and 5 themselves are not part of the solution.
2. **Closed Circle**: A closed circle is used when the value at that point is **included** in the solution set. This is typically used for inequalities that are greater than or equal to or less than or equal to:
- Example: \( x \leq 4 \) or \( x \geq 2 \)
- In this case, the circle at 4 and 2 would be closed because 4 and 2 themselves are included in the solution.
In summary:
- Use an **open circle** for inequalities that do not include the endpoint (\(<\) or \(>\)).
- Use a **closed circle** for inequalities that include the endpoint (\(\leq\) or \(\geq\)).