Systems of Inequalities

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What is a system of inequalities?

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A system of inequalities is a set of two or more inequalities with 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 graph a system of inequalities?

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To graph a system of inequalities, first graph each inequality as if it were an equation. Then, shade the region that satisfies the inequality. The solution to the system is where the shaded regions overlap.

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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 to a system of inequalities if it satisfies all inequalities in the system when substituted into them.

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What is the significance of the boundary line in an inequality?

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The boundary line represents the points where the inequality changes from true to false. If the inequality is strict (e.g., < or >), the line is dashed; if it is inclusive (e.g., ≤ or ≥), the line is solid.

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What is the difference between a strict and non-strict inequality?

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A strict inequality (< or >) does not include the boundary, while a non-strict inequality (≤ or ≥) includes the boundary.

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How can you determine if a point is a solution to a system of inequalities?

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Substitute the coordinates of the point into each inequality. If the point satisfies all inequalities, it is a solution.

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What are the key components of an inequality?

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The key components of an inequality include the variables, the inequality sign (>, <, ≥, ≤), and the constant terms.

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What does the term 'feasible region' refer to in the context of inequalities?

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The feasible region is the area on a graph where all the solutions to a system of inequalities exist. It is the intersection of the shaded regions.

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How do you write an inequality from a word problem?

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Identify the quantities involved, determine the relationships between them, and express these relationships using inequality symbols.

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What is the role of the x and y variables in a system of inequalities?

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In a system of inequalities, x and y typically represent quantities that are being compared or constrained, such as the number of items produced or sold.