Linear Inequalities and Systems of Inequalities

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What is a linear inequality?

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A linear inequality is a mathematical statement that relates a linear expression to a value using inequality symbols (such as <, >, ≤, or ≥) instead of an equal sign.

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How do you graph a linear inequality?

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To graph a linear inequality, first graph the corresponding linear equation as a boundary line. Then, use a dashed line for < or > and a solid line for ≤ or ≥. Finally, shade the region that satisfies the inequality.

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What does the solution set of a linear inequality represent?

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The solution set of a linear inequality represents all the points (x, y) that satisfy the inequality, forming a region on the graph.

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

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How can you determine if a point is a solution to a linear inequality?

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To determine if a point is a solution, substitute the x and y values of the point into the inequality. If the inequality holds true, then the point is a solution.

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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 region where the shaded areas of all inequalities overlap.

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How do you find the feasible region in a system of inequalities?

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The feasible region is found by graphing all inequalities in the system and identifying the area where all shaded regions overlap.

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What is the significance of the feasible region in application problems?

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The feasible region represents all possible solutions that satisfy the constraints of the problem, allowing for optimal solutions to be identified.

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What is the slope-intercept form of a linear equation?

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The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope and b is the y-intercept.

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How do you convert a linear inequality to slope-intercept form?

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To convert a linear inequality to slope-intercept form, isolate y on one side of the inequality, similar to how you would for an equation.