Systems of Linear Inequalities Practice

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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 graph a system of linear inequalities?

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To graph a system of linear inequalities, first graph each inequality as if it were an equation. Use a dashed line for inequalities that do not include equality (e.g., < or >) and a solid line for those that do (e.g., ≤ or ≥). Then, shade the appropriate region that satisfies the inequality.

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

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The solution to a system of linear inequalities represents all the points in the coordinate plane that satisfy all the inequalities in the system.

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

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The boundary line in a linear inequality separates the coordinate plane into two regions: one that satisfies the inequality and one that does not. The line itself may or may not be included in the solution set, depending on whether the inequality is strict or inclusive.

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

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A system of linear inequalities has no solution if the shaded regions of the inequalities do not overlap, meaning there are no points that satisfy all inequalities simultaneously.

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What does it mean if a system of linear inequalities has an infinite number of solutions?

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A system of linear inequalities has an infinite number of solutions if the shaded regions overlap in such a way that there are infinitely many points that satisfy all inequalities.

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How can 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 to a system of linear inequalities, 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 role of the objective function in application problems involving linear inequalities?

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The objective function in application problems involving linear inequalities represents the quantity to be maximized or minimized, such as profit, cost, or time, subject to the constraints defined by the inequalities.

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How do you interpret the constraints in a word problem involving linear inequalities?

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Constraints in a word problem involving linear inequalities represent the limitations or restrictions on the variables, such as time, resources, or budget, that must be considered when finding a solution.

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What is the graphical representation of the solution set for the inequalities y ≤ x + 2 and y ≤ -x + 4?

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The graphical representation of the solution set for the inequalities y ≤ x + 2 and y ≤ -x + 4 is the region where the shaded areas of both inequalities overlap, which is typically a polygonal area on the graph.