A system of linear inequalities is a set of two or more inequalities that involve the same variables, representing constraints on those variables. These inequalities define a region in the coordinate plane, known as the feasible region, where all solutions to the system can be found. The solutions to a system of linear inequalities can be represented graphically, and they often overlap, creating a multi-dimensional space where various conditions are satisfied simultaneously.
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In a system of linear inequalities, if an inequality includes 'greater than or equal to' or 'less than or equal to', the boundary line is solid; otherwise, it is dashed.
To determine if a point is part of the solution set for a system of linear inequalities, substitute the coordinates of the point into each inequality.
The solution to a system of linear inequalities can consist of infinitely many points located within the feasible region.
When graphing, each inequality divides the plane into two half-planes, and the feasible region is found where these half-planes intersect.
Systems of linear inequalities can have no solution (if the lines are parallel), one solution (if they intersect at one point), or infinitely many solutions (if they overlap).
Review Questions
How can you determine if a point lies within the feasible region defined by a system of linear inequalities?
To check if a point is in the feasible region of a system of linear inequalities, substitute the x and y coordinates of the point into each inequality. If the point satisfies all inequalities, then it lies within the feasible region. If it fails to meet even one inequality, then it is outside that region.
What are the graphical characteristics that distinguish between solid and dashed boundary lines in linear inequalities?
In graphing linear inequalities, a solid boundary line indicates that points on the line are included in the solution set, which occurs with 'greater than or equal to' (\(\geq\)) or 'less than or equal to' (\(\leq\)). In contrast, a dashed boundary line shows that points on the line are not included in the solution set, corresponding to 'greater than' (\(>\,) or 'less than' (\(<\)). This distinction helps visualize which parts of the graph satisfy each inequality.
Evaluate how altering one inequality in a system affects its feasible region and potential solutions.
Changing one inequality in a system can significantly impact its feasible region. If you alter an inequality to be less restrictive or more restrictive, it can expand or shrink the feasible region. This means that new intersection points may form with other lines or eliminate existing ones. Consequently, this may lead to different potential solutions being valid or invalid, directly affecting the overall solution set for that system.
Related terms
Feasible Region: The area in the coordinate plane that represents all possible solutions to a system of linear inequalities.
Boundary Line: The line that represents the equation of a linear inequality when converted to an equality; it separates the coordinate plane into different regions.
Graphing Method: A technique used to solve a system of linear inequalities by graphing each inequality and identifying the overlapping area that satisfies all conditions.