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## Algebra (all content)

### Course: Algebra (all content)>Unit 6

Lesson 2: Constraining solutions of two-variable inequalities

# Constraining solutions of systems of inequalities

Given the graph of a system of inequalities, Sal finds the x-values that make the ordered pair (x,-2) a solution of the system.

## Video transcript

- [Voiceover] Which x-values make the ordered pair, x comma negative two, 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 two which is another way of saying we're going to constrain ourselves to y equaling negative two. If we constrain ourselves to y equaling negative two, 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 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 of what's going on here. Let me pick some points. So this point right over here, this is a solution to neither of the inequalities in our system. You can think of this as the green system and this as the blue system. In order to be a solution set, you have to be in the shaded area for that system. So this point right over here, it's in the solution set for neither of the inequalities of the system. This point right over here, 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, then 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 this point right over here, 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 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 shaded area for the green inequality but it sits 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 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 in the green one. We assume that the green one just keeps going down. And this is actually what we're seeing here is actually the overlap. Now that we have a better understanding of things, let's actually tackle the questions. We're constraining ourselves to y equals negative two. So actually, let me draw a line here that shows all of the points where y is equal to negative two. So that shows, at least on our graph where y is equal to negative two. So given y equals negative two, what has to be true of x in order to satisfy the system of inequalities here? Well, we're going to have to deal with all of the x-values including and to the right of this point. I can say including because the blue inequality, 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 anything to the right. So all of this is part of the solution set. And so, if we constrain y equals negative two, we see that x has to be greater than or equal to negative three. And we see that that is this choice, this choice right over there, x is greater than or equal to negative three. 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, four 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 through it on your own. All right now, let's work through it together. In this scenario, we are constraining x. We're saying that x has to be equal to four. So x equaling four, 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 on this line that constrains us to x equals four, 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 in this area right over here, that's the overlap of the solutions sets of the two inequalities. And so, if we constrain ourselves to x equaling four, 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 can say negative one has to be less than y, and then y can go 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 still going to be on the solution set of the blue inequality. 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 greater than negative one and less than or equal to negative three... 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.