Boundary Line

Definition

A boundary line is a conceptual dividing line that separates two distinct regions or areas. It serves to define the limits or borders of a particular space or domain, establishing the boundaries within which certain conditions or rules apply.

5 Must Know Facts For Your Next Test

1. The boundary line in the graph of a linear inequality represents the set of points that satisfy the inequality with equality, dividing the coordinate plane into two half-planes.
2. In a system of linear inequalities, the boundary lines define the feasible region, which is the set of all points that satisfy all the inequalities simultaneously.
3. The slope of the boundary line is determined by the coefficient of the $x$ variable in the linear inequality, and the $y$-intercept is determined by the constant term.
4. The type of inequality (greater than, less than, greater than or equal to, less than or equal to) determines whether the feasible region is above, below, or on the boundary line.
5. The intersection of two or more boundary lines represents the solution to a system of linear inequalities, where the coordinates of the intersection point satisfy all the inequalities.

Review Questions

• Explain the role of the boundary line in the graph of a linear inequality.
  • The boundary line in the graph of a linear inequality represents the set of points that satisfy the inequality with equality. It divides the coordinate plane into two half-planes, with the feasible region being the area that satisfies the inequality. The type of inequality (greater than, less than, greater than or equal to, less than or equal to) determines whether the feasible region is above, below, or on the boundary line.
• Describe how boundary lines define the feasible region in a system of linear inequalities.
  • In a system of linear inequalities, the boundary lines define the feasible region, which is the set of all points that satisfy all the inequalities simultaneously. The intersection of the boundary lines represents the solution to the system, where the coordinates of the intersection point satisfy all the inequalities. The feasible region is the area enclosed by the boundary lines, and any point within this region satisfies the constraints of the system.
• Analyze the relationship between the boundary line and the solution to a system of linear inequalities.
  • The intersection of the boundary lines in a system of linear inequalities represents the solution to the system, where the coordinates of the intersection point satisfy all the inequalities. The boundary lines define the feasible region, and the solution is the point within this region that optimizes the objective function, such as maximizing or minimizing a linear expression. Understanding the properties of the boundary lines, such as their slopes and $y$-intercepts, is crucial for determining the solution and the feasible region in the context of linear programming problems.

Related terms

Feasible Region: The feasible region is the set of all points that satisfy a given system of linear inequalities, represented by the area enclosed by the boundary lines.

Constraint: A constraint is a restriction or limitation that defines the boundaries of a problem or system, often represented by the boundary lines in the graph of a linear inequality or system of linear inequalities.

Intersection Point: The intersection point is the point where two or more boundary lines meet, representing the solution to a system of linear inequalities.