Systems and 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 ordered pairs that satisfy all inequalities in the system.

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What does it mean for an ordered pair to be in the solution set of an inequality?

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An ordered pair (x, y) is in the solution set of an inequality if it makes the inequality true when substituted into it.

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Substitute the x-value and y-value of the ordered pair into the inequality. If the resulting statement is true, then the ordered pair is a solution.

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What is the graphical representation of a system of inequalities?

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The graphical representation consists of the shaded regions that satisfy each inequality, with the solution set being the intersection of these regions.

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What is the significance of the boundary line in a linear 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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How do you solve a system of equations graphically?

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To solve graphically, plot each equation on the same coordinate plane and identify the points where the graphs intersect. These points are the solutions to the system.

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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 points, while a non-strict inequality (≤ or ≥) includes the boundary points.

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

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

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What does it mean for a solution set to be empty?

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An empty solution set means that there are no ordered pairs that satisfy all inequalities in the system.

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What is the role of the vertex in a quadratic inequality?

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The vertex of a quadratic function is the highest or lowest point on the graph, which can help determine the direction of the inequality and the region that satisfies it.