Graph the solution set of each system of inequalities. { -3...

Key Concepts

Boundary Lines

The boundary line of a linear inequality is obtained by replacing the inequality sign with an equality. This line splits the coordinate plane into two half-planes. Whether the line is drawn as solid or dashed depends on whether the inequality is inclusive (? or ?) or strict (< or >), respectively.

System of Linear Inequalities

A system of linear inequalities consists of multiple linear inequalities that are considered simultaneously. The solution to the system is the set of all points that satisfy every inequality in the system, which, in a graphical context, corresponds to the intersection of the regions defined by each individual inequality.

Graphing Linear Inequalities

Graphing linear inequalities involves drawing the boundary lines (which represent the equality part of the inequality) on the coordinate plane, and then shading the region that satisfies the inequality. This process requires understanding which side of the boundary line to shade, often determined by substituting a test point into the inequality.

Feasible Region

The feasible region is the area on the graph where all the individual inequalities of the system overlap. This region represents all the solutions that satisfy every inequality in the system, and it is crucial in fields such as linear programming and optimization.

Test Point Method

The test point method is used to determine which side of the boundary line should be shaded. By choosing a specific point not on the line (commonly the origin if it is not on the boundary) and substituting it into the inequality, one can verify whether the point is part of the solution set, thereby deciding the correct region for shading.

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