Feasible Region

Definition

The feasible region is the set of all possible solutions that satisfy a system of linear inequalities in two variables. It represents the area on a coordinate plane where all the constraints or inequalities are met simultaneously.

5 Must Know Facts For Your Next Test

1. The feasible region is the intersection of all the half-planes defined by the linear inequalities in a system.
2. The feasible region is always a convex polygon, which means that any two points within the region can be connected by a line segment that also lies entirely within the region.
3. The vertices of the feasible region are the points where the boundaries of the inequalities intersect.
4. The feasible region represents the set of all possible solutions that satisfy the given constraints or linear inequalities.
5. Identifying the feasible region is crucial in optimization problems, where we need to find the optimal solution within the set of feasible solutions.

Review Questions

• Explain how the feasible region is determined when graphing a system of linear inequalities in two variables.
  • When graphing a system of linear inequalities in two variables, the feasible region is the area on the coordinate plane where all the inequalities are satisfied simultaneously. This is found by graphing each inequality and identifying the common region where all the half-planes intersect. The feasible region is the convex polygon formed by the intersection of these half-planes.
• Describe the relationship between the feasible region and the constraints or linear inequalities in the system.
  • The feasible region is directly determined by the constraints or linear inequalities in the system. Each inequality defines a half-plane, and the feasible region is the intersection of all these half-planes. The vertices of the feasible region correspond to the points where the boundaries of the inequalities intersect, and the edges of the feasible region are the line segments that satisfy all the inequalities. Therefore, the feasible region represents the set of all possible solutions that meet the given constraints.
• Analyze the importance of the feasible region in the context of optimization problems involving systems of linear inequalities.
  • The feasible region is crucial in optimization problems because it defines the set of all possible solutions that satisfy the given constraints. Once the feasible region is identified, the goal is to find the optimal solution within this region, such as the point that maximizes or minimizes a particular objective function. The vertices of the feasible region are particularly important, as they represent the extreme points that must be evaluated to determine the optimal solution. Understanding the properties and characteristics of the feasible region is essential for solving optimization problems effectively.

Related terms

Linear Inequality: A linear inequality is an inequality that can be written in the form $ax + by \geq c$, where $a$, $b$, and $c$ are real numbers.

System of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities that must be satisfied simultaneously.

Constraint: A constraint is a restriction or limitation that must be met in order for a solution to be valid.