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 inequalities that involve 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 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 point's coordinates into each inequality. If the point satisfies all inequalities, it is a solution.

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What does the graph of a linear inequality represent?

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The graph of a linear inequality represents all the solutions to the inequality, typically shown as a shaded region on one side of the boundary line.

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

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What does the inequality y > 2x - 4 represent graphically?

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It represents the region above the line y = 2x - 4, where all points (x, y) satisfy the inequality.

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How can you graph a system of linear inequalities?

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1. Graph each inequality as if it were an equation. 2. Use solid or dashed lines as appropriate. 3. Shade the region that satisfies each inequality. 4. The solution is where the shaded regions overlap.

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What is the solution set of a system of linear inequalities?

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The solution set is the set of all points that satisfy all inequalities in the system, represented graphically as the overlapping shaded region.

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What does it mean if a point lies on the boundary line of a linear inequality?

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If a point lies on the boundary line, it may or may not be a solution, depending on whether the inequality is strict (>) or inclusive (≥).

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How do you identify the correct graph for a given system of inequalities?

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Analyze the inequalities to determine the boundary lines and the direction of shading. Compare with the provided graphs to find the one that matches.

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What is the significance of the intersection of two linear inequalities?

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The intersection represents the set of points that satisfy both inequalities, which is the solution to the system.