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What are the solutions to system of inequalities?
Systems of inequalities in n variables with create an n-dimensional shape in n-dimensional space which is called the feasible region. Any point inside this region will be a solution to the system of inequalities; any point outside it will not. If all the inequalities are linear then the shape will be a convex polyhedron in n-space. If any are non-linear inequalities then the solution-space will be a complicated shape. As with a system of equations, with continuous variables, there need not be any solution but there can be one or infinitely many.
How many solution sets do systems of linear inequalities have Must solutions to systems of linear inequalities satisfy both inequalities In what case might they not?
A solution to a linear inequality in two variables is an ordered pair (x, y) that makes the inequality a true statement. The solution set is the set of all solutions to the inequality. The solution set to an inequality in two variables is typically a region in the xy-plane, which means that there are infinitely many solutions. Sometimes a solution set must satisfy two inequalities in a system of linear inequalities in two variables. If it does not satisfy both inequalities then it is not a solution.
How many solution sets do systems of linear inequalities have. Must solutions to systems of linear inequalities satisfy both inequalities. In what case might they not?
There is only one solution set. Depending on the inequalities, the set can be empty, have a finite number of solutions, or have an infinite number of solutions. In all cases, there is only one solution set.
In a system of nonlinear inequalities the solution set is the region where the shaded regions overlap.?
The answer depends on which area is shaded for each inequality. I always teach pupils to shade the unwanted or non-feasible region. That way the solution is in the unshaded area. This is much easier to identify than do distinguish between a region which is shaded three times and another which is shaded four times.
Does checking just one inequality guarantee that a solution is correct?
Check all of the inequalities.
