What Is A Solution Set?

Quick answer:

A solution set is the set of all numbers that are solutions to a given equation or inequality or to a set of equations or inequalities.

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A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset). (Stover, Christopher and Weisstein, Eric W. "Set." From MathWorld—A Wolfram Web Resource. https://mathworld.wolfram.com/Set.html)

A set is finite if there is a countable number of elements and infinite if there are an uncountable number of elements. Examples of finite sets include S={red, yellow, blue}, S={reflections, rotations, translations}, or S={1,2,3}. Examples of infinite sets include `S=NN={n=1,2,3,...}` , S={n|n is an even number}, and so on.

Order is not important; for example, S={1,2,3} is the same set as S'={2,1,3} as they contain the same elements, and multiplicity is the number of times an element appears in the set.

I. Linear equations, or a system of linear equations, in one variable either have no solutions, exactly one solution, or an infinite number of solutions. For example, 2x+2=2x+3 has no solutions. (The solution set is the null set or the empty set: `S={}` or `S= O/ ` .) 2x+2=8 has the solution x=3 so the solution set is `S={3}` . 2x+2=2x+2 is true for all real numbers so the solution set is all real numbers or `S=RR, S={x|x in RR}, S=(-oo,oo)`

II. Polynomials can have multiple solutions. For instance, `x^2=4` has the two solutions -2 and 2, so the solution set is `S={-2,2}`

III. The solution set for inequalities is usually a range of values. For instance `|x-2|<5` has as solutions all real numbers between -3 and 7 or `S={x|-3<x<7}" or " S=(-3,7)`

Sometimes the solution set is the union of two or more sets. For example, `|x-2|>5` is true for all x<-3 and all x>7 or `S=(-oo,-3)uu(7,oo)`

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