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?
A solution to an inequality is defined as the value, or values, that make the inequality a true statement. Even if there is a solution set for one of the inequalities, in order for there to be a solution to the system of linear inequalities (i.e, 2 or more inequality statements), the solutions need to satisfy both inequalities, otherwise the value is not considered as solution to the system.
To find the number of solution sets, the student graphs the inequality, and shades in the values which satisfy each separate inequality. By visually representing the potential values of each one, the student will quickly notice if there is an overlap. Wherever the shading overlaps is said to be the solution set for the system. If they do not overlap, there is no solution to the system. For example, consider two parallel lines. If the solution to one are the values above the line, and the solution to the other one are the values below the other line, there is no intersection and therefore there is also no solution to the system.