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 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 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 does the graph of a linear inequality represent?

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The graph of a linear inequality represents all the solutions to the inequality, typically shown as a shaded region on one side of the boundary line.

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What is the difference between a solid line and a dashed line in the graph of 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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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 the shaded regions of the inequalities overlap, representing all possible solutions.

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How can you graph a linear inequality?

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1. Graph the corresponding linear equation as a line. 2. Use a solid line for ≤ or ≥ and a dashed line for < or >. 3. Shade the appropriate side of the line based on the inequality.

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What does it mean for a solution to be 'bounded' in a system of inequalities?

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A solution is bounded if the feasible region is enclosed by the lines of the inequalities, meaning there are limits to the values of the variables.

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What is an unbounded solution in a system of inequalities?

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An unbounded solution occurs when the feasible region extends infinitely in one or more directions, meaning there are no limits on the values of the variables.

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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 their shaded areas overlap. This region represents the solutions to both inequalities.

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What is the significance of the corner points in a feasible region?

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Corner points (or vertices) of the feasible region are significant because they are potential candidates for optimal solutions in linear programming problems.