key term - Closed Circle

Definition

A closed circle is a mathematical concept that refers to a set of elements or points that form a continuous, unbroken loop. This term is particularly relevant in the context of solving linear inequalities and applications involving linear inequalities.

5 Must Know Facts For Your Next Test

1. A closed circle is a feasible region that is bounded by a single linear inequality, resulting in a circular shape.
2. The boundary of a closed circle is the circle itself, which is the set of points that satisfy the equality form of the linear inequality.
3. The interior of a closed circle is the set of points that satisfy the strict inequality, while the boundary is the set of points that satisfy the non-strict inequality.
4. When solving applications with linear inequalities, a closed circle can represent the set of all possible solutions that satisfy the given constraints.
5. The center and radius of the closed circle are determined by the coefficients and constants in the linear inequality.

Review Questions

• Explain how a closed circle is represented in the context of solving linear inequalities.
  • A closed circle is a feasible region that is bounded by a single linear inequality, resulting in a circular shape. The boundary of the closed circle is the circle itself, which represents the set of points that satisfy the equality form of the linear inequality. The interior of the closed circle is the set of points that satisfy the strict inequality, while the boundary is the set of points that satisfy the non-strict inequality. The center and radius of the closed circle are determined by the coefficients and constants in the linear inequality.
• Describe the relationship between a closed circle and the feasible region when solving applications with linear inequalities.
  • In the context of solving applications with linear inequalities, a closed circle can represent the set of all possible solutions that satisfy the given constraints. The feasible region, which is the set of all points that satisfy a system of linear inequalities, can be a closed circle if the problem involves a single linear inequality. The boundary of the closed circle is the set of points that satisfy the equality form of the linear inequality, while the interior of the circle represents the set of points that satisfy the strict inequality.
• Analyze how the properties of a closed circle, such as the center and radius, are determined by the coefficients and constants in the linear inequality.
  • The center and radius of a closed circle are directly determined by the coefficients and constants in the linear inequality that defines the circle. The center of the circle is determined by the values of the coefficients, while the radius is determined by the constant term in the inequality. Specifically, the center of the circle is located at the point $(-b/a, c/a)$, where $a$ and $b$ are the coefficients of the $x$ and $y$ terms, respectively, and $c$ is the constant term. The radius of the circle is given by $r = |c/a|$. Understanding these relationships between the circle's properties and the coefficients in the linear inequality is crucial for solving problems involving closed circles.

Related terms

Linear Inequality:

A linear inequality is an inequality that involves a linear expression, such as $ax + b \geq c$, where $a$, $b$, and $c$ are constants.

Feasible Region:

The feasible region is the set of all points that satisfy a system of linear inequalities. It is often represented as a closed polygon or a closed circle.

Boundary Line:

The boundary line is the line that separates the feasible region from the infeasible region. It is the line that satisfies the equality form of the linear inequality.