# How many solution sets do systems of linear inequalities have?

yamaguchityler | Certified Educator

Linear inequalities can either have no solution, one specific solution, or an infinite amount of solutions. Thus, the total possible would equal three.

For instance, say we have a variable x. Although we do not know what x is, we can determine it's value depending on what inequalities it has been placed next to.

For instance, say we have X>10. This would indicate that X is obviously a number greater than 10. It also cannot equal 10 because it is not a "greater than or equal to" notation. This simply says that X is larger than 10. Similarly, if we also have X> 15, we now also know that X is larger than 15 as well. However, if this is all we're given, then we will have no idea what X is. There is an infinite number of solutions because X could be any number larger than 15.

When there is no answer, say we have Y>9. We know that Y is not definitely above 9. However, say there is also Y<7. This is impossible and has no answer because Y cannot be both greater than 9, and above 7. Thus, Y is undefined and unknown.

Finally, say we have this 5<X<7. Assuming we are not taking decimals and fractions into place, this would leave us with an answer that X=6. Obviously this changes when taking into account decimals and proportions, however this is for example only. In this case, there is only one answer and X has to equal 6.

A system of linear inequalities can have none, one, or an infinite number of solutions; therefore, there are three.

In order for a number to be a solution to a linear inequality it must satisfy all linear inequalities.  For example, in a system with two linear inequalities:

This has no solutions: x>7 and x<4 because a number cannot be greater than 7 AND less than 4

This has one solution: x>=7 and x<=7 because the only value that satisfies both inequalities is 7

This has an infinite number of solutions: x>1 and x>2 because all numbers greater than 2 satisfy both inequalities