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 that share the same variables. The solution is the set of all ordered pairs that satisfy all inequalities in the system.

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How do you determine if an ordered pair is a solution to a system of inequalities?

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To determine if an ordered pair is a solution, substitute the values of the ordered pair into each inequality. If the pair satisfies all inequalities, it is a solution.

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

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A system has an infinite number of solutions if the inequalities overlap in such a way that there are countless points that satisfy all inequalities, often forming a region on a graph.

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What is the graphical representation of a linear inequality?

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The graphical representation of a linear inequality is a half-plane divided by a boundary line. The line is dashed if the inequality is strict (>, <) and solid if it includes equality (≥, ≤).

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How do you graph the inequality y < 1/2x - 2?

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To graph y < 1/2x - 2, first graph the line y = 1/2x - 2 as a dashed line. Then shade the area below the line to represent all points where y is less than the expression.

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

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The boundary line separates the solution set from the non-solution set. Points on the line may or may not be included in the solution, depending on whether the inequality is strict or inclusive.

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How can you solve a system of inequalities by graphing?

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To solve by graphing, graph each inequality on the same coordinate plane, identify the overlapping shaded region, which represents the solution set.

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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 in the solution set.

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How do you rewrite the inequality 3x - 6y > 12 in slope-intercept form?

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To rewrite 3x - 6y > 12 in slope-intercept form (y = mx + b), isolate y: y < 1/2x - 2.

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What is a feasible region in the context of systems of inequalities?

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A feasible region is the area on a graph where all constraints (inequalities) are satisfied. It represents all possible solutions to the system.