Solving Systems of Inequalities

Front

What is a system of inequalities?

Back

A system of inequalities is a set of two or more inequalities with the same variables. The solution is the set of all points that satisfy all inequalities in the system.

Front

How do you graph a linear inequality?

Back

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

Front

What does the solution set of a system of inequalities represent?

Back

The solution set represents all the points that satisfy all inequalities in the system, often visualized as a shaded region on a graph.

Front

What is the significance of the boundary line in a linear inequality?

Back

The boundary line divides the coordinate plane into two regions: one that satisfies the inequality and one that does not.

Front

How do you determine if a point is a solution to a system of inequalities?

Back

To determine if a point is a solution, substitute the point's coordinates into each inequality. If the point satisfies all inequalities, it is part of the solution set.

Front

What is the meaning of 'x + y ≤ 300' in a real-world context?

Back

This inequality represents a constraint where the total number of items (x and y) cannot exceed 300, such as the number of seats in a theater.

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What does '7.5x + 5.5y ≥ 2000' represent in a ticket sales scenario?

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This inequality indicates that the total revenue from ticket sales (7.5 dollars for adults and 5.5 dollars for children) must be at least 2000 dollars.

Front

What is the difference between a solid line and a dashed line when graphing inequalities?

Back

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

Front

How can you find the intersection of two inequalities?

Back

To find the intersection, graph both inequalities and identify the region where the shaded areas overlap. This overlapping region represents the solution to the system.

Front

What is a feasible region in the context of linear programming?

Back

The feasible region is the area on a graph that satisfies all constraints of a system of inequalities, representing all possible solutions.