key term - Closed Circle

Definition

A closed circle is a graphical representation of the solution set for an inequality, where the endpoints of the solution set are included in the solution. This concept is particularly relevant in the context of solving linear inequalities and rational inequalities.

5 Must Know Facts For Your Next Test

1. A closed circle is used to represent a solution set where the boundary points are included in the solution.
2. The inclusion of the boundary points is indicated by using a closed circle, such as a solid circle or a square, at the endpoints of the solution set.
3. In the context of linear inequalities, a closed circle is used when the inequality symbol includes the equal sign (≤ or ≥).
4. For rational inequalities, a closed circle is used when the boundary point is a value that satisfies the inequality, including the case where the denominator is equal to zero.
5. The graphical representation of a closed circle helps visualize the solution set and understand the behavior of the inequality near the boundary points.

Review Questions

• Explain the significance of a closed circle in the context of solving linear inequalities.
  • In the context of solving linear inequalities, a closed circle is used to represent the solution set when the inequality symbol includes the equal sign (≤ or ≥). This means that the boundary points, where the inequality is satisfied with equality, are part of the solution set. The closed circle at the endpoint(s) indicates that the values at the boundary are included in the solution, which is important for understanding the complete solution set and making decisions based on the inequality.
• Describe how a closed circle is used in the graphical representation of the solution set for a rational inequality.
  • When solving rational inequalities, a closed circle is used to represent the boundary point(s) where the inequality is satisfied, including the case where the denominator is equal to zero. The closed circle indicates that the value at the boundary point is part of the solution set. This is crucial for understanding the behavior of the rational function near the boundary and determining the complete solution set. The graphical representation with closed circles helps visualize the solution set and identify the values that satisfy the rational inequality.
• Analyze the role of closed circles in the context of solving linear and rational inequalities, and explain how they contribute to a comprehensive understanding of the solution set.
  • Closed circles play a vital role in the context of solving both linear and rational inequalities. In linear inequalities, the closed circle at the endpoint(s) signifies that the boundary point(s) are included in the solution set when the inequality symbol includes the equal sign (≤ or ≥). This is important for accurately representing the complete solution set and making decisions based on the inequality. Similarly, in rational inequalities, the closed circle at the boundary point(s) indicates that the value(s) where the denominator is equal to zero are part of the solution set. This graphical representation helps visualize the behavior of the rational function near the boundary and contributes to a comprehensive understanding of the solution set. Overall, the use of closed circles in the graphical representation of inequalities is crucial for accurately depicting the solution set and informing decision-making processes.

Related terms

Solution Set:

The set of all values that satisfy a given inequality or equation.

Inequality Symbol: Symbols used to represent the relationship between two quantities, such as less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥).

Boundary Point:

The value(s) that represent the endpoint(s) of the solution set for an inequality.