Solutions to Systems of Inequalities- Graphs! 12/3

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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 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 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 does it mean for a point to be a solution to an inequality?

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A point is a solution to an inequality if it makes the inequality true when the coordinates of the point are substituted into the inequality.

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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 solid if the inequality includes equality (≥ or ≤) and dashed if it does not (> or <).

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

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The boundary line represents the points where the inequality changes from true to false. It helps to define the region of solutions.

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How do you graph a system of inequalities?

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1. Graph each inequality as a line. 2. Use a solid line for ≥ or ≤ and a dashed line for > or <. 3. Shade the appropriate half-plane for each inequality. 4. The solution region is where the shaded areas overlap.

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What does it mean for a point to be 'sometimes' a solution?

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A point is 'sometimes' a solution if it satisfies some but not all inequalities in the system.

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

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Substitute the coordinates of the point into each inequality. If it satisfies all inequalities, it is a solution; if it satisfies none, it is not; if it satisfies some, it is 'sometimes' a solution.

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What is the difference between a solution to an equation and a solution to an inequality?

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A solution to an equation is a specific point that makes the equation true, while a solution to an inequality is a range of values that satisfy the inequality.

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What does the term 'feasible region' refer to in the context of systems of inequalities?

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The feasible region is the area on the graph where all the inequalities overlap, representing all possible solutions to the system.