Boundary Line

Definition

A boundary line is a conceptual dividing line that separates regions or areas based on certain criteria. In the context of graphing linear inequalities and systems of linear inequalities, the boundary line represents the line that separates the solutions that satisfy the inequality from those that do not.

5 Must Know Facts For Your Next Test

1. The boundary line of a linear inequality is the line that represents the equality version of the inequality, where the comparison operator is replaced by an equal sign.
2. The boundary line is always a straight line and is included in the solution set if the inequality is non-strict (≤ or ≥), but excluded if the inequality is strict (< or >).
3. When graphing a system of linear inequalities, the boundary lines of the individual inequalities form the boundaries of the feasible region.
4. The feasible region is the area where all the linear inequalities in the system are satisfied simultaneously, and it is typically represented as a polygon or a set of polygons.
5. The boundary lines of the inequalities in a system play a crucial role in determining the shape and size of the feasible region.

Review Questions

• Explain the relationship between the boundary line and the solution set of a linear inequality.
  • The boundary line of a linear inequality represents the set of points where the inequality is satisfied with an equal sign. For a non-strict inequality (≤ or ≥), the boundary line is included in the solution set, meaning the points on the boundary line are considered solutions. For a strict inequality (< or >), the boundary line is excluded from the solution set, and the solutions are the points on one side of the boundary line.
• Describe the role of boundary lines in the graphing of a system of linear inequalities.
  • When graphing a system of linear inequalities, the boundary lines of the individual inequalities form the boundaries of the feasible region. The feasible region is the area where all the linear inequalities in the system are satisfied simultaneously, and it is typically represented as a polygon or a set of polygons. The boundary lines of the inequalities play a crucial role in determining the shape and size of the feasible region, as the intersection of the half-planes defined by the individual inequalities defines the feasible region.
• Analyze how the inclusion or exclusion of the boundary line affects the solution set of a linear inequality or a system of linear inequalities.
  • The inclusion or exclusion of the boundary line can significantly impact the solution set of a linear inequality or a system of linear inequalities. For a non-strict inequality (≤ or ≥), the boundary line is included in the solution set, meaning the points on the boundary line are considered solutions. However, for a strict inequality (< or >), the boundary line is excluded from the solution set, and the solutions are the points on one side of the boundary line. In a system of linear inequalities, the feasible region is determined by the intersection of the half-planes defined by the individual inequalities, and the boundary lines play a crucial role in shaping the feasible region. The inclusion or exclusion of the boundary lines can change the size, shape, and even the existence of the feasible region.

Related terms

Linear Inequality: A linear inequality is a mathematical statement that involves a linear expression and a comparison operator (such as <, >, ≤, or ≥) that divides the coordinate plane into two half-planes.

System of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities that must be satisfied simultaneously, creating a feasible region in the coordinate plane.

Feasible Region: The feasible region is the area in the coordinate plane that satisfies all the linear inequalities in a system, represented by the intersection of the half-planes defined by the individual inequalities.