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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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 the inequalities in the system when substituted into them.

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How do you graph a system of inequalities?

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1. Graph each inequality as if it were an equation. 2. Use a dashed line for < or > and a solid line for ≤ or ≥. 3. Shade the appropriate region for each inequality. 4. The solution is where the shaded regions overlap.

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What is the significance of the boundary line in a system of inequalities?

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The boundary line represents the points where the inequality changes from true to false. Points on the line are included if the inequality is ≤ or ≥.

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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 the graph that satisfies all the inequalities in the system. It represents all possible solutions.

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

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

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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 is the role of the coefficients in a linear inequality?

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Coefficients determine the slope and position of the boundary line in the graph of the inequality.

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What does it mean to 'solve' a system of inequalities?

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To solve a system of inequalities means to find all the points (x, y) that satisfy all inequalities in the system.

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How do you write a system of inequalities from a word problem?

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Identify the constraints and relationships described in the problem, then express them as inequalities using variables.