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# Graphing systems of inequalities

Sal graphs the solution set of the system "y≥2x+1 and y<2x-5 and x>1.". Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

We're asked to determine the
solution set of this system, and we actually have three
inequalities right here. A good place to start is just to
graph the solution sets for each of these inequalities and
then see where they overlap. And that's the region of the
x, y coordinate plane that will satisfy all of them. So let's first graph y is equal
to 2x plus 1, and that includes this line, and then
it's all the points greater than that as well. So the y-intercept
right here is 1. If x is 0, y is 1, and
the slope is 2. If we move forward in the
x-direction 1, we move up 2. If we move forward 2, we'll
move up 4, just like that. So this graph is going to look
something like this. Let me graph a couple more
points here just so that I make sure that I'm drawing
it reasonably accurately. So it would look something
like this. That's the graph of y is
equal to 2x plus 1. Now, for y is greater than or
equal, or if it's equal or greater than, so we have to put
all the region above this. For any x, 2x plus 1 will be
right on the line, but all the y's greater than that
are also valid. So the solution set of that
first equation is all of this area up here, all of the area
above the line, including the line, because it's greater
than or equal to. So that's the first inequality
right there. Now let's do the second
inequality. The second inequality is y
is less than 2x minus 5. So if we were to graph 2x minus
5, and something already might jump out at you
that these two are parallel to each other. They have the same slope. So 2x minus 5, the y-intercept
is negative 5. x is 0, y is negative 1, negative 2, negative
3, negative 4, negative 5. Slope is 2 again. And this is only less than,
strictly less than, so we're not going to actually
include the line. The slope is 2, so it will
look something like that. It has the exact same slope
as this other line. So I could draw a bit of a
dotted line here if you like, and we're not going to include
the dotted line because we're strictly less than. So the solution set for this
second inequality is going to be all of the area
below the line. For any x, this is 2x minus 5,
and we care about the y's that are less than that. So let me shade that in. So before we even get to this
last inequality, in order for there to be something that
satisfies both of these inequalities, it has to be in
both of their solution sets. But as you can see, their
solutions sets are completely non-overlapping. There's no point on the x, y
plane that is in both of these solution sets. They're separated by this kind
of no-man's land between these two parallel lines. So there is actually
no solution set. It's actually the null set. There's the empty set. Maybe we could put an empty set
like that, two brackets with nothing in it. There's no solution set
or the solution set of the system is empty. We could do the x is
greater than 1. This is x is equal to 1, so
we put a dotted line there because we don't want
include that. So it would be all
of this stuff. But once again, there's nothing
that satisfies all three of these. This area right here satisfies
the bottom two. This area up here satisfies
the last one and the first one. But there's nothing that
satisfies both these top two. Empty set.