Half-plane

Definition

A half-plane is a region of the coordinate plane that is divided by a line. It represents all the points on one side of the line, including the line itself. This concept is particularly important in the context of graphing linear inequalities in two variables and systems of linear inequalities.

5 Must Know Facts For Your Next Test

1. The half-plane represents all the points on one side of the boundary line, including the line itself.
2. When graphing a linear inequality, the half-plane is the region of the coordinate plane that satisfies the inequality.
3. The boundary line is the solution to the linear equation within the inequality and separates the two half-planes.
4. The shaded region on the graph represents the half-plane that satisfies the linear inequality.
5. Systems of linear inequalities are graphed by identifying the overlapping region of the individual half-planes, which represents the solution set.

Review Questions

• Explain how the concept of a half-plane is used when graphing a single linear inequality in two variables.
  • When graphing a linear inequality in two variables, the half-plane represents the region of the coordinate plane that satisfies the inequality. The boundary line, which is the solution to the linear equation within the inequality, divides the coordinate plane into two half-planes. The shaded region on the graph corresponds to the half-plane that contains all the points that satisfy the given linear inequality.
• Describe the role of half-planes in the process of graphing a system of linear inequalities.
  • In a system of linear inequalities, each individual inequality is represented by a half-plane. To graph the system, you need to identify the overlapping region of these individual half-planes. The solution set to the system of linear inequalities is the intersection of the half-planes, which represents the area where all the inequalities are satisfied simultaneously.
• Analyze how the concept of half-planes can be used to determine the feasible region for a system of linear inequalities.
  • The feasible region for a system of linear inequalities is the area where all the inequalities are satisfied. This feasible region is determined by the intersection of the individual half-planes represented by each inequality. By identifying the overlapping area of the half-planes, you can determine the feasible region, which contains all the points that satisfy the constraints of the system. The feasible region is an essential concept in optimization problems, where you need to find the optimal solution within the set of feasible solutions.

Related terms

Linear Inequality:

A linear inequality is a mathematical expression that involves a linear equation and an inequality symbol (such as <, >, ≤, or ≥).

Coordinate Plane:

The coordinate plane is a two-dimensional grid used to represent and plot points, lines, and other geometric shapes.

Boundary Line:

The boundary line is the line that separates the two half-planes and is the solution to the linear equation within the inequality.