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

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

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It means that there is no set of values for the variables that can satisfy all inequalities simultaneously, often represented by parallel lines that never intersect.

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

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It means that there are countless sets of values for the variables that satisfy all inequalities, often represented by overlapping lines or inequalities.

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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 dashed or solid line, then shade the region that satisfies the inequality (above or below the line).

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

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A dashed line indicates that points on the line are not included in the solution (for < or >), while a solid line indicates that points on the line are included (for ≤ or ≥).

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What is the significance of the solution set in a system of linear inequalities?

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The solution set represents all possible combinations of variable values that satisfy all inequalities in the system, often visualized as a shaded region on a graph.

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How can you determine if a point is a solution to a system of linear 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 graphical representation of the inequality y ≤ 2x - 1?

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The graph includes a solid line for y = 2x - 1, with the region below the line shaded, indicating all points where y is less than or equal to 2x - 1.

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What does the inequality 3b + 6t > 66 represent in a real-world context?

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It represents a situation where the total earnings from babysitting (b) and tutoring (t) must exceed $66.

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What does the inequality b + t < 15 signify in a work scenario?

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It signifies that the total hours worked in babysitting (b) and tutoring (t) must be less than 15 hours per week.