Linear Inequalities and Systems of Linear Inequalities

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What is a linear inequality?

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A linear inequality is a mathematical statement that relates a linear expression to a value using inequality symbols (>, <, ≥, ≤) instead of an equal sign.

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What does the solution set of a linear inequality represent?

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The solution set of a linear inequality represents all the points (x, y) that satisfy the inequality, often visualized as a shaded region on a graph.

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How do you graph a linear inequality?

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To graph a linear inequality, first graph the corresponding linear equation as a solid line (for ≥ or ≤) or a dashed line (for > or <). Then shade the region that satisfies the inequality.

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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 set (≥ or ≤), while a dashed line indicates that points on the line are not included (> or <).

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What does the symbol '≥' mean in a linear inequality?

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The symbol '≥' means 'greater than or equal to', indicating that the values on the left side of the inequality can be greater than or equal to the value on the right.

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What does the symbol '≤' mean in a linear inequality?

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The symbol '≤' means 'less than or equal to', indicating that the values on the left side of the inequality can be less than or equal to the value on the right.

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

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To determine if a point is a solution, substitute the x and y values of the point into the inequality. If the inequality holds true, the point is a solution.

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What is a system of linear inequalities?

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A system of linear inequalities consists of two or more linear inequalities that are considered together, and the solution is the region where all inequalities overlap.

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How do you find the solution set of a system of linear inequalities?

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To find the solution set, graph each inequality on the same coordinate plane and identify the region where all shaded areas overlap.

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What is the significance of the intersection of two lines in a system of inequalities?

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The intersection of two lines in a system of inequalities represents a boundary point that may be included in the solution set, depending on the type of inequalities.