System 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 are considered simultaneously. The solution to the system 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. Use a dashed line for '<' or '>' and a solid line for '≤' or '≥'. Then, 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. This means that when the coordinates of the point are substituted into each inequality, the resulting statements are true.

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What is the difference between '>' and '≥' in inequalities?

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'>' means that the value is strictly greater than, while '≥' means that the value is greater than or equal to. This affects whether the boundary line is dashed or solid when graphing.

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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 x and y coordinates of the point into each inequality. If the point satisfies all inequalities, it is a solution.

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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. It divides the graph into two regions: one that satisfies the inequality and one that does not.

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

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A system of inequalities is consistent if there is at least one solution that satisfies all inequalities in the system.

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What is the graphical representation of the solution set for a system of linear inequalities?

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The solution set is represented by the shaded region(s) on the graph where all inequalities overlap. This region contains all points that satisfy the system.

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How do you find the intersection of two linear 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 to the system.

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What is the role of test points in solving systems of inequalities?

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Test points are used to determine which side of the boundary line satisfies the inequality. By selecting a point not on the line, you can check if it makes the inequality true or false.