key term - Open Circle

Definition

An open circle is a graphical representation used in the context of linear inequalities, where the circle on the number line or coordinate plane is not filled in, indicating that the value represented by the circle is not included in the solution set.

5 Must Know Facts For Your Next Test

1. An open circle is used to represent a strict inequality, such as $x < 5$ or $y > 2$, where the value at the point of the open circle is not included in the solution set.
2. In the context of solving linear inequalities, an open circle is used to graph the boundary of the solution set, indicating that the value at that point is not part of the solution.
3. When solving applications with linear inequalities, open circles are used to represent strict constraints, such as the maximum or minimum value that a variable can take.
4. The use of open circles helps distinguish strict inequalities from non-strict inequalities, which are represented by closed circles or solid points on the number line or coordinate plane.
5. Understanding the meaning and usage of open circles is crucial in correctly graphing and interpreting the solution sets of linear inequalities and their applications.

Review Questions

• Explain the purpose of using an open circle when solving linear inequalities.
  • When solving linear inequalities, an open circle is used to represent a strict inequality, where the value at the point of the open circle is not included in the solution set. This helps distinguish strict inequalities, such as $x < 5$ or $y > 2$, from non-strict inequalities, which are represented by closed circles or solid points. The use of open circles is crucial in correctly graphing and interpreting the solution sets of linear inequalities.
• Describe how the use of open circles differs between solving linear inequalities and solving applications with linear inequalities.
  • In the context of solving linear inequalities, open circles are used to graph the boundary of the solution set, indicating that the value at that point is not part of the solution. However, when solving applications with linear inequalities, open circles are used to represent strict constraints, such as the maximum or minimum value that a variable can take. For example, in an application problem, an open circle might be used to represent a strict upper or lower bound on a quantity, where the value at the point of the open circle is not a valid solution.
• Analyze the relationship between open circles, strict inequalities, and the solution set of a linear inequality or application.
  • The use of open circles is directly linked to the concept of strict inequalities, where the value at the point of the open circle is not included in the solution set. This relationship is crucial in both solving linear inequalities and solving applications with linear inequalities. By understanding the meaning of open circles and their connection to strict inequalities, students can correctly graph the solution sets, interpret the constraints, and solve problems involving linear inequalities and their real-world applications.

Related terms

Inequality: A mathematical statement that compares two expressions using an inequality symbol, such as less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥).

Solution Set:

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

Number Line:

A visual representation of the real number system, where numbers are arranged in a linear fashion.