Welcome back everyone. Previously, in other videos, we've seen how to graph a one-dimensional inequality, something like x≥1. It was pretty straightforward. You just find the point at x=1 on a number line, and then you would just figure out all of the numbers that satisfy this inequality. It would be everything to the right of that point. But we're going to do the exact same thing here for a 2-dimensional inequality. It's just that now our equations that we've been dealing with have multiple variables like ys and x. So the whole idea is that we're going to have to plot inequalities now on a 2-dimensional graph instead of just a 1-dimensional number line. But I'm going to show you that it's pretty straightforward, and I'm going to show you a step-by-step way to do this.

So let's go ahead and take a look here. So, if I had to graph the inequality x≥1 on a two-dimensional number line, let's see how this would work here. We've already seen what the equation x=1 looks like. Remember, on a two-dimensional graph, this will actually be a vertical line. It's kind of weird because on a one-dimensional number line, it's just a point. But remember, now we have the sort of a y-axis, so you almost could extend this up and down. You'll see that you'll get a vertical line. So, the key idea here is that to graph an inequality; first, you actually have to graph what the corresponding line is. But now we have to look at how to shade the parts of the graph that will satisfy this inequality.

So what you're going to do here is you're going to shade the side of the graph with points that make the inequality true. You remember, you're just going to figure out the points that make this inequality a true statement. And what I can do here is just pick random points. Like, let's say, for example, I pick this point, 2,0. If I take these x and y coordinates and I plug them into this inequality, we'll see that we get true statements for this. For example, the x coordinate for this is 2, so 2≥1. That is a true statement. But now we also have an infinite number of y values to consider. We can pick a point over here. This would be 4,3. When you plug it in this inequality, you'll also get a true statement. So the key difference here is that we're not just going to shade the axis like one line; we're actually going to have to shade everywhere on this graph with points that satisfy this inequality. And for this particular line, it's actually going to be all of these points that are to the right of this inequality. So it kind of makes sense. For a one-dimensional number line, you just had a line. For a two-dimensional graph, we actually have an area, a region of points that satisfy this inequality.

That's all there is to it. All the points on this side of the graph are where x>1, so that satisfies this inequality. All the points over here are less than 1. So if you had something like y<x, for example, we could graph this pretty simply. All you'd have to do is just graph the corresponding line, y=x. Now one thing I want to point out here is the symbols are different. Notice we have a less than symbol, whereas we had a greater than or equal to symbol. So, if you have equations with the less than or equal to or greater than equal to symbols, you draw a solid line. The way I like to think about this is if you see a solid bar underneath the symbol, then you draw a solid line. But if you have something like less than or greater than alone, then you draw a dashed line. So y<x is going to look like y=x, but we're going to have to draw it with a dashed line. So it's basically going to look something like this.

These examples are pretty straightforward, but sometimes you're going to get more complicated equations, something like y>2x-4. So I actually want to show you a step-by-step process of how to graph these inequalities. Let's get started here. So the first thing you want to do is you're going to graph the solid or dashed line depending on the symbol by switching the inequality symbol with an equal sign. So remember that this symbol over here means that we're going to be drawing a dashed line. And the way we graph this is by graphing y=2x-4. Well, it just looks like a line that goes through the y-intercept of negative 4 and has a slope of 2. So it's going to look something like this. Remember, we have to use a dashed line for this. So this is going to be what that graph looks like. It's kind of just sketched out. So that's the first step.

Right? So the second thing we have to do is we have to figure out which points will satisfy our inequality. With x≥1, it was pretty simple because we just highlighted everything to the right of this graph. But for this, it's going to be a little bit tricky. Right? Is it going to be this side over here? Is it going to be this side? How does it actually work out for different angles and different steepnesses? So the second step is you're actually going to basically test a point on either side of the line. And the way that you test it is you just basically Plug the x, y coordinates into the inequality, which is exactly what we did with this 2,0 over here. So the idea is that we're just going to pick a point randomly. You can just pick any point that you want. It won't matter. But it's always just best to use a point on the x or y axis because it makes math a little bit easier. So for example, if I pick this point at random, this is going to be 0,1. I'm going to test it by plugging the x and y values into this inequality. So 0,1, if I plug it into this inequality over here, what this says is that is 1 greater than 2 times 0 minus 4. Right? Because that's what this inequality says over here. So let's work this out. Is 1, in fact, greater than this inequality? Well, 2 times 0 is 0. And again, that's why using a point on the axis is easier because one of the variables will sort of cancel out. So 1, we'll see, is greater than negative 4. Is that a true statement? It is.1 is greater than negative 4, and so this is going to be a true statement. So that's the second step, sort of testing out a point. Now the third one says that if the point makes the inequality true, like exactly what we had in this case over here, then you're going to shade the side that includes that point. So shade the side with that point. So on this graph here, the line 2x-4 is like a barrier, and we're going to shade everything that includes this point over here that makes our statement true. So really what happens is the region that makes this inequality true is actually going to be all of this area over here. If you were to take a point at random anywhere inside this region, the x and y values would make this inequality a true statement. Versus if you pick something in this region over here, you'll get something that doesn't make sense. Like, 5 is greater than 7. Right? That does not make any sense. Alright.

So if your point makes the inequality false, which is not what happened here, then you're going to shade the side without that point. And that's really all there is to it. Now there's actually a cool shortcut here that I'll teach you, which is that if you can ever isolate y to one side of your inequality and get it in slope-intercept form, which is basically what we have here, then if you have something like a greater than symbol or greater than or equal to, then you can just shade everything that is above the line. So that's exactly what we had here. y>2x-4. This is the line. You would shade everything that is above this line. Whereas if you had something like a less than symbol, then you would shade everything below that line. Alright? So that's how to graph linear inequalities. Thanks for watching, and I'll see you in the next one.