Systems of Linear Inequalities

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What is a system of linear inequalities?

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A system of linear inequalities is a set of two or more linear inequalities that involve the same variables. The solution to the system is the set of all points that satisfy all inequalities simultaneously.

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How do you determine if a point is a solution to a system of linear inequalities?

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To determine if a point is a solution, substitute the coordinates of the point into each inequality in the system. If the point satisfies all inequalities, it is a solution.

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What does it mean for a point to be a solution to a system of inequalities?

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A point is a solution to a system of inequalities if it lies in the region defined by the inequalities on a graph, meaning it satisfies all the inequalities.

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What is the graphical representation of a linear inequality?

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The graphical representation of a linear inequality is a half-plane divided by a line. The line represents the boundary, and the half-plane indicates the region where the inequality holds true.

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What is the difference between a solid line and a dashed line in the graph of linear inequalities?

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A solid line indicates that the points on the line are included in the solution (≤ or ≥), while a dashed line indicates that the points on the line are not included ( < or > ).

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How can you graph a system of linear inequalities?

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To graph a system of linear inequalities, first graph each inequality as a line (solid or dashed), then shade the appropriate half-plane for each inequality. The solution region is where the shaded areas overlap.

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What is the significance of the intersection of shaded regions in a graph of linear inequalities?

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The intersection of shaded regions represents the set of all solutions that satisfy all inequalities in the system.

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What is a feasible region in the context of linear inequalities?

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A feasible region is the area on a graph where all constraints (inequalities) are satisfied. It is the intersection of all the shaded regions.

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How do you identify a solution from a graph of linear inequalities?

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To identify a solution from a graph, look for points within the feasible region. Any point in this region is a solution to the system.

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What role do boundary lines play in systems of linear inequalities?

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Boundary lines separate the feasible region from the infeasible region. They help determine which side of the line satisfies the inequality.