Systems of Inequalities

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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 that share the same variables. The solution to the system 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 the graph of a linear inequality represent?

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The graph of a linear inequality represents all the solutions to the inequality, typically shown as a shaded region on one side of the boundary line.

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What is the difference between a strict inequality and a non-strict inequality?

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A strict inequality (e.g., < or >) does not include the boundary line, while a non-strict inequality (e.g., ≤ or ≥) includes the boundary line in the solution set.

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How can 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. Then, shade the appropriate region for each inequality and identify the overlapping shaded region as the solution set.

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What is the significance of 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 overlap, representing all possible solutions to the system of inequalities.

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How do you write a system of inequalities from a word problem?

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To write a system of inequalities from a word problem, identify the variables, determine the constraints, and express these constraints as inequalities.

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What is the role of the boundary line in graphing inequalities?

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The boundary line separates the graph into two regions. Points on one side satisfy the inequality, while points on the other side do not. The line itself is included if the inequality is non-strict.

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

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A solution is unbounded if the feasible region extends infinitely in one or more directions, meaning there is no maximum or minimum limit to the values of the variables.

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How do you solve a system of inequalities by graphing?

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To solve by graphing, graph each inequality, shade the appropriate regions, and find the intersection of the shaded areas to determine the solution set.