A system of linear inequalities can have no solutions, a finite number of solutions (often just a few specific points if further constraints are added), or an infinite number of solutions.
The number of solutions depends on the inequalities themselves and how they interact. Here's a breakdown:
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No Solutions: This occurs when the inequalities contradict each other, meaning there are no values that satisfy all of them simultaneously. For example:
x > 3andx < 1
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Finite Number of Solutions: This is less common with just linear inequalities alone. However, if you add constraints that limit the region defined by the inequalities, or restrict solutions to integers, you might end up with a finite number of solutions. Example:
x + y < 5x > 0y > 0x, y are integers. In this case, the solutions will be a finite set of integer pairs.
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Infinite Number of Solutions: This is the most typical scenario. When you graph a linear inequality, the solution is a shaded region of the coordinate plane. When you have a system of inequalities, the solution is the intersection of those shaded regions. This intersection is almost always an infinite set of points. For example:
x + y < 5x > 0y > 0This represents all the points within a triangle in the first quadrant below the line x+y=5.
In summary, while it's possible to have no solutions or a finite set of solutions under special constraints, the most common outcome for a system of linear inequalities is an infinite number of solutions, represented by a region in the coordinate plane.
