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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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 part of the solution set?

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A point is part of the solution set if it satisfies all the inequalities in the system, meaning it lies in the region defined by the inequalities.

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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 represents the boundary, and the shaded area indicates the solutions.

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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 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. The solution set is where the shaded regions overlap.

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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, representing all possible solutions.

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What does it mean for a point to NOT be part of the solution set?

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A point is not part of the solution set if it does not satisfy at least one of the inequalities in the system, meaning it lies outside the shaded region.

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How can you check if a point is a solution to a specific inequality?

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Substitute the x and y values of the point into the inequality. If the resulting statement is true, the point is a solution.

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What is an example of a linear inequality?

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An example of a linear inequality is y < 2x + 3.