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 that involve 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 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 for a point to be a solution to a system of inequalities?

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A point is a solution if it lies in the region where all inequalities overlap when graphed.

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What is the graphical representation of a linear inequality?

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The graphical representation of a linear inequality is a half-plane divided by a line. The line can be solid (inclusive) or dashed (exclusive) depending on whether the inequality includes equality.

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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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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.

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How can you graph a system of linear inequalities?

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1. Graph each inequality as if it were an equation. 2. Use a solid or dashed line as appropriate. 3. Shade the appropriate half-plane for each inequality. 4. Identify the overlapping shaded region.

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What is an example of a system of linear inequalities?

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Example: y < 2x + 3 and y ≥ -x - 1.

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What does it mean for a point to NOT be a solution to a system of inequalities?

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A point is not a solution if it does not satisfy at least one of the inequalities in the system.

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How do you find a point that is a solution to a system of inequalities?

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Choose a point within the overlapping shaded region of the graph and check if it satisfies all inequalities.