Linear Inequalities & Systems of Linear Inequalities

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What is a linear inequality?

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A linear inequality is an inequality that involves a linear function, typically expressed in the form ax + by < c, ax + by ≤ c, ax + by > c, or ax + by ≥ c, where a, b, and c are constants.

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How do you determine if a point is a solution to a linear inequality?

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To determine if a point (x, y) is a solution to a linear inequality, substitute the x and y values into the inequality. If the inequality holds true, then the point is a solution.

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What is the difference between a solid line and a dashed line when graphing linear inequalities?

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A solid line indicates that the points on the line are included in the solution set (used for ≤ or ≥), while a dashed line indicates that the points on the line are not included (used for < or >).

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What does the solution set of a system of linear inequalities represent?

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The solution set of a system of linear inequalities represents all the points that satisfy all inequalities in the system simultaneously.

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How can you graph a linear inequality in two variables?

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To graph a linear inequality, first graph the corresponding linear equation as a line. Then, use a test point to determine which side of the line to shade, indicating the solution set.

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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 coordinate plane into two regions: one that satisfies the inequality and one that does not. The nature of the line (solid or dashed) indicates whether points on the line are included in the solution.

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What does the notation y ≥ -3 represent graphically?

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The notation y ≥ -3 represents all points on or above the horizontal line y = -3, including the line itself.

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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 are considered together, and the solution is the intersection of the solution sets of each inequality.

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How do you identify a point that is not a solution to a system of inequalities?

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To identify a point that is not a solution, substitute the point's coordinates into each inequality in the system. If the point does not satisfy at least one of the inequalities, it is not a solution.

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What is the role of test points in graphing linear inequalities?

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Test points are used to determine which side of the boundary line to shade when graphing a linear inequality. A point that satisfies the inequality indicates that the region containing that point is part of the solution set.