Solving Systems of Inequalities and Review

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What is a system of inequalities?

Back

A system of inequalities is a set of two or more 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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How do you graph a system of inequalities?

Back

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 region that satisfies the inequality. The solution set is where the shaded regions overlap.

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

Back

A point is a solution to a system of inequalities if it satisfies all inequalities in the system when substituted into them.

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

Back

To determine if a point is a solution, substitute the x and y coordinates 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?

Back

The boundary line represents the values where the inequality changes from true to false. Points on the line are included if the inequality is ≤ or ≥, and excluded if it is < or >.

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What does the notation y > 2x - 4 represent?

Back

The notation y > 2x - 4 represents all points (x, y) that lie above the line y = 2x - 4.

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What is the solution set of the inequality y < 2x - 4?

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The solution set of the inequality y < 2x - 4 includes all points (x, y) that lie below the line y = 2x - 4.

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How do you identify a point that is NOT part of the solution set?

Back

To identify a point that is NOT part of the solution set, substitute the point's coordinates into the inequalities. If it does not satisfy at least one inequality, it is not part of the solution set.

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

Back

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

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What does it mean if a point (0, 0) is a solution to a system of inequalities?

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If (0, 0) is a solution, it means that when you substitute x = 0 and y = 0 into all inequalities, they all hold true.