Boundary line

Definition

A boundary line is a line that separates the regions of solutions for a system of linear inequalities in two variables. It is derived from the corresponding equation of the inequality, and it plays a crucial role in determining the areas where the inequalities hold true. The boundary line can either be solid or dashed, indicating whether points on the line are included in the solution set or not.

5 Must Know Facts For Your Next Test

1. The boundary line is found by converting the inequality into an equation and graphing it as if it were a linear equation.
2. A solid boundary line indicates that points on the line are included in the solution set, while a dashed boundary line means those points are excluded.
3. The slope and y-intercept of the boundary line can help determine how to graph it accurately on a coordinate plane.
4. The region above or below the boundary line depends on the direction of the inequality (greater than or less than).
5. When graphing multiple inequalities, the intersection of their respective regions forms the feasible region, with its boundaries defined by these lines.

Review Questions

• How does the type of boundary line (solid vs. dashed) affect the solution set of a linear inequality?
  • The type of boundary line directly impacts whether certain points are part of the solution set for a linear inequality. A solid boundary line indicates that all points on that line are included in the solution set, meaning they satisfy the corresponding linear inequality. Conversely, a dashed boundary line signifies that points on that line are not included in the solution set, meaning they do not satisfy the inequality. Understanding this distinction is essential when graphing inequalities and determining valid solutions.
• What steps would you take to graph a system of linear inequalities and identify the feasible region?
  • To graph a system of linear inequalities and find the feasible region, first convert each inequality into its corresponding boundary line by turning it into an equation. Next, decide whether to use a solid or dashed line based on whether the inequality includes equality. After plotting these lines on a coordinate plane, shade the appropriate regions based on each inequality (above for 'greater than' and below for 'less than'). The intersection of all shaded areas will form the feasible region, which contains all possible solutions to the system.
• Evaluate how changing an inequality from 'less than' to 'greater than' affects both the boundary line and feasible region.
  • Changing an inequality from 'less than' to 'greater than' fundamentally alters both its boundary line and feasible region. The boundary line remains in place as it represents the same linear equation; however, its classification changes from solid to dashed or vice versa if equality is involved. The shading direction also flips—whereas previously shading was below the line for 'less than', it now moves above for 'greater than'. This shift effectively modifies which areas represent valid solutions, demonstrating how sensitive systems of inequalities are to changes in their conditions.

Related terms

linear inequality: A linear inequality is an expression that shows the relationship between two variables, indicating that one variable is less than, greater than, less than or equal to, or greater than or equal to another.

solution set:

The solution set is the collection of all possible points (x, y) that satisfy the given linear inequalities.

feasible region:

The feasible region is the area on a graph where all the constraints of a system of inequalities overlap, representing all possible solutions.