Systems 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 inequalities that involve 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 coordinates of the point into each inequality. If the point satisfies all inequalities, it is a solution.

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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 there are no points that satisfy all inequalities simultaneously, often resulting in parallel lines that never intersect.

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What does it mean if a system of inequalities has infinitely many solutions?

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A system has infinitely many solutions if the inequalities overlap in such a way that there are countless points that satisfy all inequalities, often forming a region.

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

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How do you graph the inequality y < 2x + 3?

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To graph y < 2x + 3, first graph the line y = 2x + 3 as a dashed line. Then shade the area below the line to represent all points where y is less than 2x + 3.

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What is the significance of a solid line in graphing inequalities?

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A solid line indicates that points on the line are included in the solution set, meaning the inequality is non-strict (≥ or ≤).

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What is the significance of a dashed line in graphing inequalities?

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A dashed line indicates that points on the line are not included in the solution set, meaning the inequality is strict (> or <).

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How do you solve a system of linear inequalities by graphing?

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To solve by graphing, graph each inequality on the same coordinate plane, then identify the region where the shaded areas overlap. This overlapping region represents the solution set.

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What is the solution to the system of inequalities: y > x + 1 and y < -x + 4?

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The solution is the region where the area above the line y = x + 1 and below the line y = -x + 4 overlap.