Systems of Inequalities (SOI)

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What is a system of inequalities?

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A system of inequalities is a set of two or more inequalities that share the same variables. The solution is the set of all points that satisfy all inequalities in the system.

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How do you graph a linear inequality?

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To graph a linear inequality, first graph the corresponding linear equation as a boundary line. Then, use a dashed line for < or > and a solid line for ≤ or ≥. Finally, shade the region that satisfies the inequality.

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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 satisfies all inequalities in the system, meaning it lies in the shaded region of the graph.

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What is the significance of the boundary line in a linear inequality?

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The boundary line represents the points where the inequality changes from true to false. Points on the line are included in the solution set if the inequality is ≤ or ≥.

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

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

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What is the difference between a solid line and a dashed line in graphing inequalities?

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

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What does the solution set of a system of inequalities represent graphically?

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The solution set represents the region of the graph where all inequalities overlap, indicating all possible solutions that satisfy the system.

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How do you find the intersection of two inequalities?

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To find the intersection, graph both inequalities and identify the region where the shaded areas overlap. This overlapping region represents the solution set.

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

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A feasible region is the set of all possible points that satisfy a system of inequalities, representing all potential solutions to the problem.

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Can a system of inequalities have no solution?

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Yes, a system of inequalities can have no solution if the shaded regions do not overlap, meaning there are no points that satisfy all inequalities.