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 ordered pairs that satisfy all inequalities in the system.

Front

How do you graph a system of inequalities?

Back

To graph a system of inequalities, first graph each inequality on the same coordinate plane. Use a solid line for '≤' or '≥' and a dashed line for '<' or '>'. The solution set is the region where the shaded areas overlap.

Front

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

Back

A point is a solution to a system of inequalities if it satisfies all inequalities in the system, meaning it lies in the overlapping shaded region of the graph.

Front

What is the difference between a solid line and a dashed line in 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 determine if a point is part of the solution set for a system of inequalities?

Back

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

Front

What is the solution set of the inequalities y < 2x + 3 and y ≥ -x - 1?

Back

The solution set is the region where the area below the line y = 2x + 3 and above the line y = -x - 1 overlap.

Front

What does the term 'feasible region' refer to in the context of systems of inequalities?

Back

The feasible region is the area on the graph that represents all possible solutions to the system of inequalities.

Front

How do you find the vertices of the feasible region?

Back

To find the vertices of the feasible region, graph the inequalities, identify the intersection points of the boundary lines, and determine the coordinates of these points.

Front

What is the significance of the vertices in a system of inequalities?

Back

The vertices of the feasible region are significant because they represent potential solutions, and in optimization problems, the maximum or minimum values often occur at these points.

Front

What is an ordered pair in the context of inequalities?

Back

An ordered pair is a pair of numbers (x, y) that represent a point on the coordinate plane, where 'x' is the horizontal coordinate and 'y' is the vertical coordinate.