System of Linear Inequalities

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What is a system of linear inequalities?

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A system of linear inequalities is a set of two or more linear inequalities that involve 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 the solution to a system of linear inequalities represent?

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The solution represents all the points (x, y) that satisfy all inequalities in the system, often visualized as a shaded region on a graph.

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What is the difference between a solid line and a dashed line in graphing inequalities?

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A solid line indicates that points on the line are included in the solution (≤ or ≥), while a dashed line indicates that points on the line are not included ( < 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 point's coordinates into each inequality. If the point satisfies all inequalities, it is a solution.

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What is the standard form of a linear inequality?

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The standard form of a linear inequality is Ax + By < C, Ax + By ≤ C, Ax + By > C, or Ax + By ≥ C, where A, B, and C are constants.

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What does the inequality 7x + 13y ≤ 81 represent in a real-world context?

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It represents a budget constraint where x is the number of breadsticks and y is the number of pizzas, and the total cost cannot exceed $81.

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

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The boundary line separates the solution space from the non-solution space, indicating where the inequalities change from true to false.

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How do you find the intersection of two linear inequalities?

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To find the intersection, graph both inequalities and identify the region where the shaded areas overlap; this region represents the solution to the system.

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

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A feasible region is the area on a graph that satisfies all constraints of a system of linear inequalities, representing all possible solutions.