A bounded region is a closed, finite area on a coordinate plane that is enclosed by one or more linear inequalities. It represents the set of all points that satisfy the given system of linear inequalities.
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The bounded region is the area on the coordinate plane where all the linear inequalities in a system intersect, forming a closed, finite shape.
The vertices of the bounded region are the points where the linear inequalities intersect, and these points are the solutions to the system of linear inequalities.
The bounded region can have various shapes, such as a triangle, a rectangle, or a polygon, depending on the number and orientation of the linear inequalities in the system.
The bounded region represents the set of all possible solutions to the system of linear inequalities, and it is used in optimization problems to find the optimal solution.
The size and shape of the bounded region can be affected by the coefficients and constants in the linear inequalities, as well as the number of inequalities in the system.
Review Questions
Explain the relationship between a system of linear inequalities and the bounded region on the coordinate plane.
The bounded region is the area on the coordinate plane where all the linear inequalities in a system intersect, forming a closed, finite shape. The vertices of the bounded region are the points where the linear inequalities intersect, and these points are the solutions to the system of linear inequalities. The size and shape of the bounded region are determined by the coefficients and constants in the linear inequalities, as well as the number of inequalities in the system.
Describe the process of graphing a system of linear inequalities to determine the bounded region.
To graph a system of linear inequalities and determine the bounded region, you first need to graph each individual linear inequality by shading the appropriate half-plane. Then, you need to find the area where all the half-planes overlap, which represents the bounded region. The vertices of the bounded region are the points where the linear inequalities intersect, and these points are the solutions to the system of linear inequalities.
Analyze how the characteristics of the linear inequalities in a system can affect the size, shape, and location of the bounded region.
The size, shape, and location of the bounded region can be affected by the coefficients and constants in the linear inequalities, as well as the number of inequalities in the system. For example, if the coefficients in the linear inequalities are larger, the bounded region may be smaller, and if there are more inequalities in the system, the bounded region may have a more complex shape. Additionally, the orientation and direction of the linear inequalities can also influence the position of the bounded region on the coordinate plane.