How many solution sets do systems of linear inequalities have?

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yamaguchityler's profile pic

Posted on

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.

crmhaske's profile pic

Posted on

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

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