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 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 inequalities that do not include equality (e.g., < or >) and a solid line for those that do (e.g., ≤ or ≥). Then, shade the appropriate region for each inequality, and the solution set is where the shaded regions overlap.

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What does the solution to a system of inequalities represent?

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

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What is the difference between a linear equation and a linear inequality?

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A linear equation represents a straight line and has an equality sign (e.g., y = mx + b), while a linear inequality represents a region of the graph and includes inequality signs (e.g., y < mx + b or y ≥ mx + b).

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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 to a system of inequalities, substitute the x and y values of the point into each inequality. If the point satisfies all inequalities, it is a solution.

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What is the significance of the boundary line in a graph of an inequality?

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The boundary line in a graph of an inequality indicates the limit of the solution set. If the inequality is strict (e.g., < or >), the line is dashed, meaning points on the line are not included in the solution. If the inequality is inclusive (e.g., ≤ or ≥), the line is solid, meaning points on the line are included.

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How do you rewrite an inequality in 'y=' form?

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To rewrite an inequality in 'y=' form, isolate y on one side of the inequality. This often involves rearranging the terms and simplifying the expression.

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What is the graphical representation of the inequality y ≤ 2x - 1?

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The graphical representation of the inequality y ≤ 2x - 1 is a solid line with a slope of 2 and a y-intercept of -1, with the region below the line shaded.

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What does it mean for a point to be a solution of the system of inequalities?

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A point is a solution of the system of inequalities if it lies within the shaded region of the graph that represents the solution set for all inequalities in the system.

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

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The feasible region in a system of inequalities is found by graphing all inequalities and identifying the area where the shaded regions overlap. This region contains all possible solutions.