A system of linear inequalities is a set of two or more linear inequalities that must be satisfied simultaneously. It represents a region in the coordinate plane where all the inequalities are true at the same time, known as the feasible region.
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The solution to a system of linear inequalities is the set of all points that satisfy all the inequalities in the system.
The feasible region of a system of linear inequalities is the area in the coordinate plane where all the inequalities are true simultaneously.
The boundaries of the feasible region are determined by the lines representing the individual inequalities.
The feasible region is always a convex polygon, which means it has no holes or indentations.
The vertices of the feasible region are the points where the boundaries of the inequalities intersect.
Review Questions
Explain the process of graphing a system of linear inequalities.
To graph a system of linear inequalities, you first need to graph each individual inequality by plotting its boundary line and shading the appropriate half-plane. The feasible region is the area where all the inequalities are satisfied, which is the intersection of the individual solution sets. The vertices of the feasible region are the points where the boundary lines intersect, and the feasible region is always a convex polygon.
Describe the relationship between the feasible region and the solution set of a system of linear inequalities.
The feasible region of a system of linear inequalities is the set of all points that satisfy all the inequalities in the system. It represents the solution set, which is the collection of all possible values for the variables that make the system of inequalities true. The feasible region is the visual representation of the solution set, and any point within the feasible region is a solution to the system of linear inequalities.
Analyze how the number of inequalities in a system affects the shape and size of the feasible region.
The number of inequalities in a system of linear inequalities directly impacts the shape and size of the feasible region. As more inequalities are added, the feasible region becomes smaller because it must satisfy all the constraints simultaneously. The feasible region is the intersection of the individual half-planes defined by each inequality, and with more inequalities, the area of overlap becomes more restricted. The feasible region is always a convex polygon, but its specific shape and size depend on the number and orientation of the inequalities in the system.
A linear inequality is an inequality that can be written in the form $ax + by \geq c$ or $ax + by > c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables.
The feasible region is the set of all points in the coordinate plane that satisfy all the inequalities in a system of linear inequalities. It is the intersection of the solution sets of the individual inequalities.
Graphing is the process of visualizing the solution set of a system of linear inequalities by plotting the boundaries of each inequality and identifying the feasible region.