 So an extremely useful skill is to be able to graph an inequality. And this emerges because of the following. It's very often helpful to alternate between an algebraic representation of something, which is to say I might have a formula or some sort of equation, and a geometric representation, some sort of picture. As they say, a picture is worth a thousand words. Formulas are essentially words, pictures are essentially pictures. So it's nice to be able to make that transition, because very often some things that are extremely difficult algebraically turn out to be really simple geometrically. And likewise, things that are really difficult geometrically are sometimes easy to figure out algebraically. So for inequalities in particular, we want to be able to convert an algebraic statement like x less than or equal to 7, that's a formula, that's an equation, into some sort of geometric picture. And we'll do this in the following way. We'll set down a number line into our boundary numbers, what that means is any number that I can definitely indicate as a locate on the number line, that's 7. And then I'll shade the portion of the number line that corresponds to the inequality. So for example, let's take a look at that x less than or equal to 7. Now for future reference, because we will start to look at inequalities in other contexts, it's important to note that we're trying to graph on a number line. Very specifically, we're looking at the graph of x less than or equal to 7 on the number line as opposed to some other context. So we'll set down our number line, because that's where we want to do the graph. And I have a value of 7 here that seems to be part of the inequality. So I'll go ahead and locate that value, it's going to be someplace around there. And I want the graph to be x less than or equal to 7. And that inequality says that x is allowed to be 7. And so to indicate that, I'll put a filled-in circle at 7. And now I have to figure out which part of the number line to shade. And there's a couple of ways of doing this. But the easy one is that this 7 has divided the number line into two parts. There's this part to the right and there's this part to the left. And an easy way of deciding which part to shade is to pick a point in each region. So for example, if I take 0 and say 1 trillion, and I'll see if the point satisfies the inequality. And if it does, I'll shade that side. And if it doesn't, I won't. Now let's take a look at x at 0. Well, 0 is actually less than or equal to 7. So since 0 is less than or equal to 7, 0 satisfies the inequality. So I'll shade that point. And I'll also want to include everything that is part of the interval containing 0. And the only boundary I have is going to be right here at 7. So I'll shade everything on that side. Now we do have to check the other side. 1 trillion is not less than or equal to 7. So 1 trillion is not part of the solution. So I don't want to shade that region. Now there's another example. Negative 5 less than x, less than negative 1. So I'll graph my boundary points. Negative 5 and 1. So here's negative 5. Here's 1. I am working on a number line. So the inequality is strict. I'm not allowed to have x equal to negative 5. I'm not allowed to have x equal to negative 1. So I want to put open circles at those boundary points. And now I've partitioned the number line into 1, 2, 3 distinct intervals. And so I'll pick a point in each of those intervals. So for example, I might pick 0, I might take negative 2, and negative 10. So I'll pick a point in each of the intervals and I'll test each point. So let's see. So this negative 10, negative 10 less than negative 5, less than negative 1. This is not true. It is not true that negative 10 is greater than negative 5. So I don't want to shade this portion. I'll go on to the next boundary point. I'll go on to the next test point. That's at x equals negative 2. And it turns out that it is true. Negative 2 is greater than negative 5, is less than negative 1. So I had to shade this region that includes negative 2. So negative 2 definitely works. And any point in that interval will also work. And finally, I'll check x equals 0. And it's not true that while it is true that 0 is greater than negative 5, it must also be less than negative 1, which is not true. So I don't want to shade that region. Now once we have these regions shaded, we'd like to go, we may like to go back to an algebraic formulation of what those regions look like. So let's take a look at our region here. So first thing to notice is that I have boundary points at x equals 4 and x equals 10. So I might note those two boundary points. And I've shaded everything. So the thing to notice here is everything to the left of 4 has been shaded, but 4 itself has not been. So that says that I'm looking at everything that is less than 4, but not equal to 4. And I could write that this way, x less than 4. Now it turns out for convenience, what you want to do is you want to place the x in really the same location as the shading is. So I have 4, the shaded part is to the left, x is going to be less than 4. And this helps for a couple of reasons, which we'll see in a moment. On the other side, notice that I have 10 and everything to the right. So my x would actually be, I'll write my x on the other side of 10 and note that I'm actually looking at x values that are greater than 10. And I'm allowed to include 10. That's a filled in circle there. So I get to include 10. So I want x greater than or equal to 10, or 10 less than or equal to x. Now, one last thing to notice, a number can be either here or here, but it's never going to be in both. So I can put the two regions together by saying that one of these two things is the case. Either x is less than 4, in which case we're over here, or 10 is less than or equal to x, in which case we're over here. Well, let's take a look at the different inequality. So there's a different shading. And so again, take a look at our boundary points. That's negative 5 and 1. So I'll go ahead and set those down in the same order that they appear on the number line, negative 5 and 1. So now let's take a look at this. The shaded region is to the right of negative 5, but it doesn't include negative 5. So my x, the shaded region, is to the right. I'll write my x as here. And it's everything that is greater than negative 5, but I'm not allowed to have things equal to negative 5. On the other side, I have 1 and everything, and things to the left of it. So I'm going to put x here, and x is going to be less than 1, but it could actually be equal to 1. So I am going to say that x is less than or equal to 1. And here's the important thing. We want to put these two together. And the thing to notice here is that whatever our x values are, they have to satisfy both of them. They have to be greater than negative 5 and also less than 1. So I have to say that our inequality is going to look like this. And in such cases, I can rewrite this as a single three-sided inequality. It's not a great idea to do so, but it is common to do so. And just drop out this middle portion. Negative 5 less than x, less than or equal to 1, and my inequality gets written that way.