Graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and la

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Graph each system of linear inequalities. Tell whether the...

Key Concepts

System of Linear Inequalities

A system of linear inequalities consists of multiple linear inequalities that are considered simultaneously. Each inequality divides the plane into two regions, and the solution to the system is the intersection of these regions, representing all points that satisfy every inequality in the system.

Graphical Representation of Inequalities

Graphing linear inequalities involves drawing the boundary lines (derived from the corresponding equality) and shading the region of the line that satisfies the inequality. This method provides a visual depiction of the solution set for each inequality on the coordinate plane.

Feasible Region

The feasible region is the area on the graph where all the shaded regions overlap. It represents all the potential solutions that satisfy the entire system of inequalities. This region can take the form of a polygon, or it might be unbounded, extending infinitely in one or more directions.

Bounded vs Unbounded Regions

In the context of linear inequalities, a bounded region is completely enclosed within finite limits on the plane, while an unbounded region extends infinitely in at least one direction. Determining whether a feasible region is bounded or unbounded is essential, particularly in optimization problems.

Corner Points

Corner points, or vertices, are the intersection points of the boundary lines of the inequalities. They are critical in many applications, such as linear programming, because the optimal solution to an objective function (if it exists) is often found at one of these vertices.

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