Everyone, welcome back. So previously in other videos, we've seen how to graph a one-dimensional inequality, something like \(x \geq 1\). It was pretty straightforward. You just find the point \(x = 1\) on a number line, and then you would just sort of figure out all of the numbers that satisfy this inequality. And 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 \(y\) 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 \(y\), or sorry, \(x \geq 1\) on a 2-dimensional number line, let's see how this would work here. Now 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 we have a \(y\)-axis, so you almost could kind of, like, extend this up and down. You'll see that you'll get a vertical line. So the key whole 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 sort of how to shade the parts of the graph that will satisfy this inequality. And so what you're gonna do here is you're gonna 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 sort of, like, 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 \(2 \geq 1\). That is a true statement. But now we also have, like, an infinite number of \(y\) values to consider. We can pick, like, a point over here. This would be \(4, 3\). When you plug it into this inequality, you'll also get a true statement. So the key difference here is that we're not just gonna shade the axis like one line. We're actually going to have to shade everywhere in 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 kinda makes sense. For a one-dimensional number line, you just had a line. For a two-dimensional graph, we actually have, like, an area, a region of points that satisfy this inequality. Alright? So 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 or 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 or less than, 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. Alright. Now, 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 those inequalities. Let's get started here. So the first thing you want to do is you're gonna graph the solid or dashed line depending on the symbol by switching the inequality symbol with an equal sign. So remember that this graph this symbol over here means that we're gonna be drawing a dashed line. And the way we graph this is by graphing the corresponding line. What does \(y = 2x - 4\) look like? 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, like, 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 \(x \geq 1\), it was pretty simple because we just highlighted everything to the right of this graph. But for this, it's gonna be a little bit tricky. Right? So is it gonna be this side over here? Is it gonna 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 or by plugging the \(x\), \(y\) coordinates into the inequality, which is exactly like what we did with this \(2, 0\) over here. So the idea is that we'