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# Constraining solutions of systems of inequalities

## Video transcript

which X values make the ordered pair X comma negative to a solution of the system of inequalities represented by the graph below so let's look at this so we're constraining ourselves to all of the points of the form X comma negative 2 which is another way of saying we're going to constrain ourselves to Y equaling negative 2 if we constrain ourselves to Y equaling negative 2 what do what has to be true of X in order for this point to be a solution to this system of inequalities and so I encourage you to pause the video look at this look at this graph here and then pick one of the choices all right now let's work through it together so let's just be very clear what's going on here let me pick some points so this point right over here this is this is a solution to neither of the inequalities in our system you can think of this is the green system and this is the blue system in order to be a solution set you have to be in the shaded area for that system so this this point right over here it's in the solution set for neither of the inequalities in the system this point right over here it still would be it would still not be in the solution set for either because it's on a dashed green line if this was a solid green line that it would be part of the green solution set but since it's a dashed Green Line the line itself is not part of the solution set now at this point right over here this is a this is in this this point satisfies the green inequality it's part of its solution set but it does not satisfy the blue inequality so it's not a it's not a it's not in the solution set for the system now this point here this actually would satisfy both and the reason why it satisfies both it clearly is in the in the shaded area for the green inequality and it oh but it's it's on the line for the blue inequality but that's okay because we're including the line itself in the solution set it's a solid it's a solid blue line so this would be in the solution set for the system of inequalities this would be in the solution set for the system of inequalities all of these points because they're in the solution set of the blue inequality that we're seeing visually and Green when we assume that the green one just keeps going down that this is actually what we're seeing here is actually the overlap now let's now that we have a better understanding of things let's actually tackle the question so we're constraining ourselves to y equals negative 2 so I'll actually let me draw a line here that shows all the points where Y is equal to negative 2 so that shows at least on our graph where Y is equal to negative 2 so given y equals negative 2 what do we have what has to be true of X in order to satisfy the system of inequalities here well we're gonna have to deal with all of the X values to the including and to the right of this point I can say including I can say including because the blue inequality this is going to you can also be equal to the line you can actually be on the line so being on the line is part of the solution set for both or or anything to the right and so let me so all of this all of this is part of the solution set and so if we constrain y equals negative 2 we see that X has to be we see that X has to be greater than or equal to negative 3 and we see that this is that is this choice this choice right over there X is greater than or equal to negative 3 now let's do another one and instead of constraining y we're now going to constrain X which Y values make the ordered pair 4 comma Y a solution of the system of inequalities represented by the graph below and once again I encourage you to pause the video and see if you can work work through it on your own alright now let's work through it together we are in this scenario we are constraining X we're saying that X has to be equal to 4 so X equaling 4 that's all the points on that's all of the points on this line right over here so we are constraining ourselves to the points that sit on this line but we want to be part of the solution set so we want to be we want to be on this line that constrains us to x equals 4 but we want to be in the overlap of the solution sets of the two inequalities in order to satisfy the system so let's see we want to be we want to be in this area right over here that's the overlap of the solution sets of the two inequalities and so if we constrain ourselves to X equaling four we Y has to be let's see Y has to be greater than because we're not including the Green Line itself it's dashed so Y has to be greater than negative one or we could say negative one has to be less than Y and then Y can go all the way all the way up to and including three up to and including three because this blue line is actually filled in so anything that's on the blue line is is still going to be on the solution set of the blue inequality and this is and this point that I'm showing right here is clearly sitting in the overlap for both inequalities and so Y has to be less than or equal to three so if X is equal to four Y has to be in this front has to be greater than negative one and less than or equal to negative three or and less than or equal to three I should say so let's see which of these choices negative one is less than Y is less than or equal to three that's once again our first choice