There is one final topic that we need to address as far as solution sets go before leaving this section. Consider the following equation and inequality.
x^{2 }+ 1 = 0
x^{2 } = 0
If we limit ourselves to just real solutions (that we won't always do) then there is no solution to the equation. Squaring x makes x greater than equal to zero, after that adding 1 onto i.e that the left side is guaranteed to be at least 1. In other terms, there is no real solution to this equation. For the similar basic reason there is no solution to the inequality. Squaring any real x makes it positive or zero and thus will never be negative.
We required a way to mention the fact that there are no solutions here. In solution set notation we say that the solution set is empty & denote it with the symbol : ∅ . This symbol is frequently called the empty set.
In the discussion of empty sets we supposed that were only looking for real solutions. Whereas i.e. what we will be doing for inequalities, we won't be limiting ourselves to real solutions with equations. Once we get around to solving out quadratic equations (x^{2} + 1 = 0) we will let solutions to be complex numbers & in the case looked at above there are complex solutions to x^{2} + 1 = 0 . If you don't know how to search these at this point i.e. fine we will be covering that material in some sections. At this point simply accept that x^{2} + 1 = 0 does have complex solutions.
Lastly, as noted above we won't be utilizing the solution set notation much in this course. This is a nice notation & does have some use on occasion especially for complicated solutions. Though, for the vast majority of the equations & inequalities which we will be looking at will have simple sufficient solution sets that it's just easier to write the solutions and let it go at that. Thus, that is what we will not be using the notation for our solution sets. Though, you have to be aware of the notation & know what it means.