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 system of inequalities?

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To graph a system of inequalities, first graph each inequality as if it were an equation. Use a dashed line for < or > and a solid line for ≤ or ≥. Then, shade the appropriate region for each inequality, and the solution 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, meaning it lies in the overlapping shaded region of the graph.

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What is the first step in solving a word problem involving a system of inequalities?

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The first step is to define the variables that represent the quantities in the problem.

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Write a system of inequalities for the following scenario: A store sells apples for $2 each and oranges for $3 each. If a customer buys no more than 10 fruits and spends no more than $20, what are the inequalities?

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Let x = number of apples and y = number of oranges. The system is: x + y ≤ 10 (total fruits) and 2x + 3y ≤ 20 (total cost).

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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. Points on the line are included if the inequality is ≤ or ≥, and excluded if < or >.

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

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To determine if a point is a solution, 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 graphical representation of the solution to a system of inequalities?

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The graphical representation is the region where the shaded areas of all inequalities overlap, indicating all possible solutions.

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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 do not overlap at all, meaning there are no points that satisfy all inequalities.

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How do you find the feasible region in a system of inequalities?

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The feasible region is found by graphing all inequalities and identifying the area where all shaded regions intersect.