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

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

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

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What is the graphical representation of a system of linear inequalities?

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The graphical representation consists of shaded regions on a coordinate plane, where each region corresponds to the solutions of the inequalities. The overlapping region represents the solutions to the entire system.

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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. If the inequality is strict (e.g., < or >), the line is dashed; if it is inclusive (e.g., ≤ or ≥), the line is solid.

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What does it mean if a point lies above a boundary line in a linear inequality?

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If a point lies above the boundary line of a linear inequality, it means that the point satisfies the inequality, indicating that it is part of the solution set.

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What does it mean if a point lies below a boundary line in a linear inequality?

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If a point lies below the boundary line of a linear inequality, it means that the point does not satisfy the inequality, indicating that it is not part of the solution set.

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What is the difference between a strict inequality and a non-strict inequality?

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A strict inequality (e.g., < or >) does not include the boundary line, while a non-strict inequality (e.g., ≤ or ≥) includes the boundary line in the solution set.

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How do 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, then shade the appropriate region for each inequality. The solution set is where the shaded regions overlap.

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What is an example of a system of linear inequalities?

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An example is: y < 2x + 3 and y ≥ -x - 1. The solution set includes all points that satisfy both inequalities.