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

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What does it mean to graph a linear inequality?

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Graphing a linear inequality involves drawing the boundary line (solid for ≤ or ≥, dashed for < or >) and shading the region that satisfies the inequality.

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What is the difference between a solid line and a dashed line in graphing inequalities?

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

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

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The solution set is the region on the graph where the shaded areas of all inequalities overlap, representing all possible solutions.

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How do you write a linear inequality from a word problem?

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Identify the variables, determine the relationship described, and express it using inequality symbols based on the context (e.g., greater than, less than).

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What is the significance of the intersection of two inequalities?

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The intersection of two inequalities represents the set of points that satisfy both inequalities simultaneously, which is crucial for finding solutions in a system.

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How can you find the number of solutions in a system of linear inequalities?

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The number of solutions can be determined by analyzing the graph: if the shaded regions overlap, there are infinitely many solutions; if they do not overlap, there are no solutions.

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What is a feasible region in the context of linear inequalities?

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A feasible region is the area on a graph that satisfies all constraints of a system of linear inequalities, representing all possible solutions.

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What does it mean for two linear inequalities to be equivalent?

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Two linear inequalities are equivalent if they have the same solution set, meaning they represent the same region on a graph.