System 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 for a point to be a solution to a system of inequalities?

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A point is a solution to a system of 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 do you graph a linear inequality?

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To graph a linear inequality, first graph the corresponding linear equation as a boundary line. Then, shade the region that satisfies the inequality (above the line for 'greater than' and below for 'less than').

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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 'greater than or equal to' or 'less than or equal to'). A dashed line indicates that points on the line are not included (for 'greater than' or 'less than').

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

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A system of 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 inequalities has infinitely many solutions?

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

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

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You can determine the number of solutions by graphing the inequalities and observing the overlap of the shaded regions. Count the distinct regions where the inequalities intersect.

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What is the significance of the vertices in the solution region of a system of inequalities?

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The vertices of the solution region are significant because they represent potential optimal solutions in linear programming problems and can be used to evaluate the objective function.

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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 where all the constraints (inequalities) of a linear programming problem are satisfied. It is the intersection of all the shaded regions.

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How do you solve a system of inequalities by graphing?

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To solve a system of inequalities by graphing, graph each inequality on the same coordinate plane, shade the appropriate regions, and identify the overlapping area as the solution.