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 linear 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 linear 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 line is solid in a graph of an inequality?

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A solid line indicates that the points on the line are included in the solution set, meaning the inequality is 'greater than or equal to' (≥) or 'less than or equal to' (≤).

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What does it mean if a line is dashed in a graph of an inequality?

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

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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 shaded region on one side of the line, indicating all the solutions that satisfy the inequality.

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How can you identify a solution from a set of points?

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To identify a solution, substitute each point into the inequalities. The point that satisfies all inequalities is the solution.

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

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The boundary line separates the solution set from the non-solution set. Points on one side of the line satisfy the inequality, while points on the other side do not.

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What is the difference between 'solution' and 'not a solution' in the context of inequalities?

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A 'solution' satisfies all inequalities in the system, while 'not a solution' does not satisfy at least one inequality.

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How do you graph a system of linear inequalities?

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To graph a system, first graph each inequality as a line (solid or dashed), then shade the appropriate region for each inequality. The overlapping shaded area represents the solution set.

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What is an example of a system of linear inequalities?

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An example is: y > 2x + 1 and y ≤ -x + 4.