Hi, and welcome. This is Anthony Varela. And today, we're going to talk about a system of inequalities. So first, what we're going to do is compare a system of equations to a system of inequalities. We'll talk about solutions to a system of inequalities. And we'll also see what those solutions look like on a graph.
So let's first do a bit of comparison between a system of equations and a system of inequalities. So here, we have a system of equations. This is two or more equations that we're considering at the same time. And a solution to this system would be a specific point xy that satisfies all equations that make up your system.
And taking a look at a solution on a graph, so here I have two lines that make up a system. That point of intersection is going to be the solution of those equations that we see on the graph. So how does this then compare to a system of inequalities?
Well, this is what a system of inequalities looks like. We can see it's almost identical to our equations. But we, of course, have inequality symbols here instead of equal signs. Now the solution is not a specific point, but it's a region of points xy on the coordinate plane that satisfy all of the inequalities.
So what this might look like then on a graph. Here, I have two lines on a graph. And the solution region is part of that coordinate plane. And any point xy that's in this highlighted region is a solution to the system because it satisfies all of the inequalities that make up that system.
So the solution region, when we're talking about systems of inequalities, that's all x and y values in a defined solution region that satisfies every inequality in the system. So in summace, system of inequalities is two or more inequalities with the same variables considered at the same time. Very similar to systems of equations, we're just dealing with inequalities. And our solution isn't a specific point; it's a region of points.
So let's take a look at a solution region on a graph. So I have a couple of different inequalities that make up a system. And what we're going to do is graph each of them individually, and then we'll take a look at a solution region.
So I'm going to start out with 2x plus y is less than 7. Now I'm going to graph this as a line. So I'm going to isolate y. So I can identify a y-intercept and a slope. So I can graph a line.
So I'm going to take away 2x from both sides. And this will give me then y is less than negative 2x plus 7. I just like to write my x term first. This is negative, and this is positive. So now I can identify then a y-intercept occurring at x equals 0, y equals 7 and a slope then of negative 2.
And when I'm graphing inequalities, remember this inequality symbol tells me to use a dotted line instead of a solid line because it's a strict inequality symbol. So here's my y-intercept here at y equals 7 and my slope of negative 2. And now I need to shade half of this coordinate plane. And I have y is less than negative 2x plus 7, so I'm going to shade underneath.
Well, that's just one of my inequalities in my system. Let's go ahead and plot x minus 3y is less than negative 6. So once again, I'm going to rewrite this to isolate y so I can plot this on my graph.
So I'm going to take away x from both sides of this inequality here. And I have negative 3y is less than negative 6 minus 6. So now once again, to isolate y, I'm going to divide. And I'm going to divide by negative 3.
And remember, when I divide or multiply an inequality by a negative number, that inequality sign must change. So I'm noting that here. And we have y is greater than 1/3 x plus 2. So now I know that I have a y-intercept at y equals 2, and my slope is 1/3. And once again, I'm going to be using a dotted line.
So here is the line y equals 1/3 x plus 2. And to show this as an inequality, I'm shading everything above. Now I have one more inequality to put on this graph. x is greater than or equal to 0.
So here, I can use a solid line because my inequality includes or equal to And I can draw on my graph x equals 0. That is just the y-axis itself. And now I need to shade in x being greater than or equal to 0. So that's going to be everything on the right side of that line x equals 0.
So now that I have plotted all of my inequalities, I'm looking for a solution region that is the overlap of all of my individual solution regions. So I see lots of different overlaps here, but I'm looking for an overlap of all of the inequalities in the system. That would be this area right here.
Now while other areas might represent some overlaps of two of my inequalities in the system, it needs to be an overlap of all of them. So what this means then is any xy of-- x and y of value that fits within here, this region right here on my graph, is going to be a solution to my system of inequalities.
I'd like to show you one more graph of a system of inequalities. And then not actually going to show the equations. I'm just going to show us some solution regions on the graph. So when plotting one of my inequalities in my system, I get this solution region here-- some line and everything underneath it.
And when I graph a second inequality that makes up my system, here I have another line and the solution region to that inequality. And then I'm plotting one more. And now what I'm looking for is an overlap of all of my inequalities that make up the system.
And here, I noticed that I have a couple of overlaps. This green region represents an overlap. This blue region represents an overlap. This red region represents an overlap. But those all represent an overlap of only two out of the three inequalities. So this actually has no solution because it must overlap all individual regions, not just some in order to be a solution to our system.
So let's review an introduction to a system of inequalities. We talked about the solution region of a system of inequalities, being all x and y values that fit within a defined solution region that satisfies every inequality in the system. So here's a graphical representation of that. We saw several different overlaps, but this black area here represents the overlap of all of the inequalities in the system. So that's the solution region.
We also talked about how a system of inequalities could have no solution because it must overlap all individual regions, not just some. So thanks for watching this tutorial and an introduction to a system of inequalities. Hope to see you next time.
