How many solution sets do systems of linear inequalities have?
Do solutions to systems of linear inequalities need to satisfy both inequalities?
In what case might they not?
how many solution sets? from zero to the degree of the equation.
do they have to satisfy both? Yes.
In what case might they not? They have to satisfy both, or it is not a solution.
by definition, a solution set must satisfy all of the inequalities. Otherwise, it's not a solution.
The number of solution sets depends on the particular problem. There may be none at all, as in
x+y > 6
x+y < 0
Or there may be an unbounded one, as in
x+y > 3
x-y > 2
or a bounded one, as in
x+y > 3
x-y > 2
3x+y < 10
a visit to wolframalpha.com will let you play around with some. For the last example, just type
solve x+y > 3, x-y > 2, 3x+y < 10
and it will show the solution sets and graph the region.
Do the equations x = 4y + 1 and x = 4y – 1 have the same solution?
Systems of linear inequalities can have infinitely many solution sets.
To determine the number of solution sets, you need to graph the system of inequalities on a coordinate plane. The points that satisfy all the inequalities represent the solution set.
The solutions to systems of linear inequalities must satisfy all the inequalities simultaneously. This means that any point within the shaded region on the graph represents a solution.
However, there can be cases in which there is no solution or the solution is an empty set. This occurs when the shaded regions representing the individual inequalities do not overlap, resulting in no common solution. In other words, the system of inequalities is inconsistent and has no solution.