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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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. Use a dashed line for < or > (not included) and a solid line for ≤ or ≥ (included). Then shade the region that satisfies the inequality.

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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 inequalities in the system, meaning it lies in the shaded region of the graph.

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

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

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The feasible region is the area on the graph where all shaded regions overlap, representing all possible solutions to the system of inequalities.

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What does it mean for a point to be outside the feasible region?

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A point outside the feasible region does not satisfy at least one of the inequalities in the system, meaning it is not a solution.

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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. This system includes two inequalities that can be graphed to find the feasible region.

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How do you find the intersection of two linear inequalities?

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To find the intersection, graph both inequalities and identify the region where they overlap. This overlapping area represents the solutions to both inequalities.

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What is a boundary line in the context of linear inequalities?

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A boundary line is the line that represents the equation of the inequality. It separates the graph into two regions: one that satisfies the inequality and one that does not.