Solid Line

Definition

A solid line is a continuous, unbroken line used in graphical representations to depict the boundary or solution set of a linear inequality or system of linear inequalities. It is a fundamental visual element in the contexts of graphing linear inequalities and systems of linear inequalities.

5 Must Know Facts For Your Next Test

1. A solid line is used to represent the boundary or solution set of a linear inequality or system of linear inequalities that is included in the final solution.
2. The region of the coordinate plane that satisfies a linear inequality with a solid line boundary is called a half-plane, and this half-plane is considered part of the solution set.
3. When graphing a system of linear inequalities, the solution set is the intersection of the individual half-planes, and the boundaries of these half-planes are represented using solid lines.
4. The solid line used to represent the boundary of a linear inequality or system of linear inequalities is a continuous, unbroken line that extends infinitely in both directions.
5. The solid line is an essential visual cue in understanding the solution set of a linear inequality or system of linear inequalities, as it clearly delineates the region of the coordinate plane that satisfies the given constraints.

Review Questions

• Explain the role of a solid line in the graph of a linear inequality.
  • In the graph of a linear inequality, a solid line is used to represent the boundary of the half-plane that is included in the solution set. The solid line depicts the set of points that satisfy the inequality, and the region on the correct side of the line is considered part of the final solution. The solid line extends infinitely in both directions, indicating that the inequality holds true for all points on that side of the boundary.
• Describe how solid lines are used in the graphing of a system of linear inequalities.
  • When graphing a system of linear inequalities, solid lines are used to represent the boundaries of the individual half-planes that make up the solution set. The intersection of these half-planes, which are bounded by solid lines, defines the final solution set. The solid lines clearly delineate the regions of the coordinate plane that satisfy all the constraints of the system, allowing for a visual understanding of the overall solution.
• Analyze the relationship between solid lines and the solution set of a linear inequality or system of linear inequalities.
  • The solid line used in the graph of a linear inequality or system of linear inequalities is directly tied to the solution set. The solid line represents the boundary of the half-plane that is included in the final solution, indicating that all points on the correct side of the line satisfy the inequality or system of inequalities. The continuous, unbroken nature of the solid line emphasizes that the solution set is a connected region of the coordinate plane, rather than a set of isolated points. Understanding the significance of the solid line is crucial in interpreting the solution set and making inferences about the constraints imposed by the linear inequality or system.

Related terms

Dashed Line: A dashed line is a discontinuous line used to represent the boundary or solution set of a linear inequality or system of linear inequalities that is not included in the final solution.

Half-Plane: A half-plane is the region of the coordinate plane that is defined by a linear inequality and is bounded by a line, which can be represented by a solid or dashed line.

Solution Set: The solution set of a linear inequality or system of linear inequalities is the set of all points in the coordinate plane that satisfy the given inequality or system of inequalities, and is typically represented using solid and dashed lines.