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 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. Use a dashed line for < or > (not included) and a solid line for ≤ or ≥ (included). Then shade the region that satisfies the inequality.

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

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The solution set represents all the points (x, y) that satisfy all inequalities in the system. It is the overlapping shaded region of the graphs of the inequalities.

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What is the difference between '≤' and '<' in linear inequalities?

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'≤' means that the boundary line is included in the solution set (solid line), while '<' means the boundary line is not included (dashed line).

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How do 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 does the term 'feasible region' mean in the context of linear inequalities?

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The feasible region is the area on the graph that represents all possible solutions to the system of inequalities. It is where the shaded regions of all inequalities overlap.

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

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The boundary line separates the graph into two regions: one that satisfies the inequality and one that does not. It is crucial for determining the solution set.

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How can you express a real-world scenario as a system of linear inequalities?

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Identify the constraints and objectives in the scenario, define variables for the quantities involved, and write inequalities based on the relationships and limits described.

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What is the role of the slope in a linear inequality?

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The slope indicates the steepness of the line and the direction of the inequality. A positive slope means the line rises, while a negative slope means it falls.

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What does it mean if a system of linear inequalities has no solution?

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If a system has no solution, it means that the inequalities do not overlap at any point, indicating that there are no values that satisfy all inequalities simultaneously.