Solving 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 linear inequality?

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

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What does it mean if a point is 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, meaning it lies in the shaded region of the graph.

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What is the difference between '≤' and '<'?

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'≤' means 'less than or equal to', indicating that the boundary line is included in the solution. '<' means 'less than', indicating that the boundary line is not included.

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

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The boundary line represents the points where the inequality changes from true to false. It divides the coordinate plane into regions that satisfy or do not satisfy the inequality.

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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 does it mean if a system of inequalities has no solution?

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A system of inequalities has no solution if the shaded regions of the inequalities do not overlap, meaning there are no points that satisfy all inequalities.

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What does it mean if a system of inequalities has infinite solutions?

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A system of inequalities has infinite solutions if the shaded regions overlap completely or if one inequality is contained within another, meaning there are countless points that satisfy all inequalities.

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

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Identify the constraints from the problem, define variables for the quantities involved, and write inequalities that represent those constraints.

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What is the graphical representation of the inequality y > 2x + 3?

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The graph of y > 2x + 3 is a dashed line representing the equation y = 2x + 3, with the region above the line shaded.