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 involve 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 if a system of linear inequalities has no solution?

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It means that there is no set of values for the variables that can satisfy all inequalities in the system 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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It means that there are countless combinations of variable values 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 dashed or solid line (dashed for < or >, solid for ≤ or ≥). Then shade 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 (≤ or ≥), while a dashed line indicates that points on the line are not included ( < or >).

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How do you determine if a point is a solution to a system of inequalities?

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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 the solution set of a system of linear inequalities?

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The solution set is represented by the shaded region where all inequalities overlap on the graph.

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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.

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How can you write a system of inequalities from a word problem?

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Identify the constraints in the problem, define variables for the quantities involved, and translate the constraints into inequalities.

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What does the term 'feasible region' refer to in linear programming?

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The feasible region is the area on the graph that satisfies all constraints of the system of inequalities.