System of Inequality Review

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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 set is where the shaded regions overlap.

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What does the symbol '≥' mean in inequalities?

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'≥' means 'greater than or equal to'. It indicates that the value on the left is either greater than or equal to the value on the right.

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What does the symbol '≤' mean in inequalities?

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'≤' means 'less than or equal to'. It indicates that the value on the left is either less than or equal to the value on the right.

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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 is the feasible region in a system of inequalities?

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The feasible region is the area on the graph where all the shaded regions of the inequalities overlap. It represents all possible solutions to the system.

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

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

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What is an example of a real-world application of systems of inequalities?

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An example is budgeting, where you might have constraints on spending and income, represented by inequalities, to determine how much can be spent on different categories.

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How can you solve a system of inequalities algebraically?

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To solve algebraically, isolate one variable in one inequality and substitute it into the other inequalities. Then solve for the remaining variable.