Algebra 1
posted by Austin .
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
xy > 2
or a bounded one, as in
x+y > 3
xy > 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, xy > 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?