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 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 of a system of linear inequalities represent?

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The solution of a system of linear inequalities represents all the points that satisfy all inequalities in the system, often visualized as a shaded region on a 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 (≤ or ≥), while a dashed line indicates that points on the line are not included ( < or >).

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How can 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 point's coordinates into each inequality. If the point satisfies all inequalities, it is a solution.

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What is the standard form of a linear equation?

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The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, and A should be non-negative.

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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 y = mx + b, where m is the slope and b is the y-intercept.

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What does the term 'feasible region' mean in the context of linear inequalities?

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The feasible region is the area on a graph where all the constraints (inequalities) of a linear programming problem are satisfied.

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What is a corner point in the context of linear inequalities?

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A corner point (or vertex) is a point in the feasible region where two or more boundary lines intersect.

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How do you find the corner points of a feasible region?

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To find the corner points, solve the equations of the boundary lines to find their intersection points.