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## Constraining solutions of two-variable inequalities

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# Constraining solutions of systems of inequalities

## 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.