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 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 that inequality, typically shaded 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 linear 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 region for each inequality. 4. Identify the overlapping shaded region.

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What does it mean for a point to be a solution to a system of inequalities?

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It means that the point satisfies all inequalities in the system, making it a valid solution.

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What is the significance of the boundary line in a linear inequality?

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The boundary line separates the graph into two regions: one that contains solutions and one that does not.

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How do you write a linear inequality from a word problem?

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Identify the variables, determine the relationship (greater than, less than, etc.), and express the conditions mathematically.

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What is an example of a system of linear inequalities?

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y < 2x + 3 and y ≥ -x - 1.