 In this video I'm going to talk about graphing linear inequalities. This method of graphing is going to be very similar to what I've done in previous videos about graphing lines, about graphing linear functions. It's going to be basically, you'll just see a lot of the same vocabulary, a lot of the same words, but basically we're just adding more to it. We're adding inequalities to it. So what we're going to be doing is we're going to be shading, we're going to be doing solid lines and dotted lines for boundaries, things like that. Now in your extensive studies of inequalities, inequalities are actually going to be much, much more helpful in real life than regular equations are, simply because in real life you have numbers that don't quite fit a model or when you're selling or buying, your numbers of selling and buying are going to be up and down. So actually inequalities could be much, much more useful in the real world than actual regular equations. But anyway, we're just here just to talk about graphing linear inequalities, just the very basics. So here we go. The first thing you want to do, I'm going to concentrate here on the left, I have my first equation, my first inequality I should say, y is greater than negative one-third x plus two. So you can see that this inequality is in slope-intercept form. So let me, since I'm referring to it, slope-intercept y equals mx plus b. So with slope-intercept form you can graph an equation. We're going to use that to our advantage here. In the previous videos I've talked about many, many different ways of graphing equations. We're going to stick with slope-intercept form because it is the most efficient, most effective form of graphing equations. So we're going to stick with that. Okay, the first thing we're going to do is we're going to draw, not just a line, when we're graphing inequalities, we don't call it a line, we're going to call it the boundary. And you'll see why we call it a boundary here in a minute when we start doing some shading. So what I'm going to do first is I'm going to treat this inequality, I'm going to treat it like an equal sign. So I'm going to graph the equation y equals negative one-third x plus two. Okay, I'm going to graph that first and then I'm going to kind of shade my boundaries. Shade it either above or below my boundaries. Okay, now the first thing I need to decide is where to start. So on slope-intercept form you always start with a y-intercept. I have a y-intercept of two so that's going to be right here. Okay, so that's the first thing I start with. Now from there I'm going to use my slope. I have a negative slope so my line is going to look something like that. Okay, but that's not what the actual line is but I know that's what it's going to look like. So I have a slope of negative one-third. Get rid of that. Slope of negative one-third. I'm going to drop one and then go one, two, three. Rise one, run three. There's one point. I can't run three anymore so I'm actually going to go back up here. I'm going to rise one, run one, two, three. So there we go right there. Okay, so now I have three points to use to graph this. When you're graphing inequalities I suggest that you put a bunch of points, three, four, five, whatever it takes. Make sure that when you create this boundary that you go from the edge of your graph to the edge of your graph. You'll see why here in a minute again with the shading that we have to do. It's much, much easier to understand if you make a very large line, a very large boundary. It makes a lot more sense. Okay, so now I mentioned earlier about dotted lines and solid lines. When I'm drawing boundaries for my inequalities, when I'm drawing boundaries, I either need to use a solid line or a dashed line. Okay, now the easy way to remember this is if I have greater than or less than I use a dotted line. Alright, if I have greater than or equal to, less than or equal to, I use a solid line. That's a short, easy version for it. Now, technically what that means is dotted lines mean you do not include points that are on that line. Okay, dotted lines mean you do not include points on that line. They're not included in your solution. Okay, so we want numbers wise that are bigger than this line. So the thing is we want to be bigger than this line. So anything that's actually on this line is not going to count. If we're on the line, we're not bigger than the line. We want to be bigger than it. So that's kind of a logical rationale to this. So what that basically means is I want to make dotted lines. So a couple dotted lines here. It doesn't have to be quite perfect. I'm a little slanted there, but that's okay. That right there, that is my boundary. That is my boundary. With that boundary, I'm either going to shade up or I'm going to shade down. And that sometimes can be confusing depending on what your line looks like. A lot of students will think, oh, should I shade left or right? No, no, no, no. With inequalities, you shade up or you shade down. That is it. You shade up or you shade down. There is one specific time where you might shade left and right, but I might go over that in a later video. Okay. Now from here, I need to shade. This creates a boundary and now I need an area to shade. Wys are greater than this line. That tells you where to shade. I want to shade greater than this line. This is how I read this. The whys that I want are bigger than this line. So as I look at this line, where are the whys greater? The whys are greater up. Up. Up here, the whys are bigger. They are greater. That is the area that I want. That is the area that I want. So this is my shaded area and that is it. Okay. That is how you graph inequalities. Now let's go into what this actually means. Why do we shade? Why do we have boundaries? Why do we have these things? Now when we shade these, we are looking for solutions. Inequalities have many, many solutions. Remember your basic inequalities. If x is greater than 3, if you have a number line, if x is greater than 3, if 3 is right here, all the numbers that are that way are bigger than 3. And so you have a lot of solutions. You have an infinite number of solutions and we use a number line to represent this. Okay. So that is kind of a blast from the past there. Now when we have multiple variables, when we have multiple variables like we have over here, we have x's and y's, we are looking for x, y coordinates that are solutions to this equation. We are looking for x's and y's that satisfy this equation, that work for this equation. So what I am actually going to do to show you this is I am actually going to choose a few points to show you what I mean by solutions. So actually in this area right here, big dot right here, okay, that right there, that's a point right there at that intersection, kind of a big point so we can all see it. Okay, that point is 3, 3. That's okay. 3, 3, that's that point right there. That is within my boundary, that is in the shaded area, okay, that point is a solution to my equation. Any point in this shaded area is a solution to my equation, is a solution to my inequality. Okay, that means any one of these points I can plug in to my inequality and it will be satisfied. Okay, so let's actually try that. Let's try the point 3, 3. So I have 3 for a y and I have a 3 for an x. Okay, a little bit of math here, 3 is greater than, 3 is actually canceled, times by 3, divided by 3, they'll cancel to negative 1 and 3 is greater than 1. Negative 1 plus 2 is 1. Is that true? Is 3 greater than 1? Yes it is. So this point, 3, 3 that we plugged in satisfies the inequality. It is true. You can think of it this way. If you plug it in, the inequality is true. Fantastic. Okay, so now the thing is, what if you go outside the boundary? What about all this stuff down here? What does this mean? Okay, so let's try a point down here. Let's try one negative, let's try this point down here. Actually, let me use a different color here. Let's use black. Okay, so that point right there, 1, negative 2. So I'm going to do the same thing. I'm going to take, I'm going to do the work, but I'm going to do the work with a different point. I'm going to use 1, negative 2. Okay, so negative 2 for the y's and 1 for the x's. Of course I made it difficult for myself because now this is going to multiply and give me a fraction. That's okay. Negative 2 is greater than negative 1 third plus 2. Okay, negative 2 is greater than, okay, so 2 and then take away 1 third is going to be 1 and 2 thirds. Okay, 1 and 2 thirds. All right, now take a look at our end result. Negative 2 is greater than 1 and 2 thirds. That's not true. That's not even close to true. This negative 2, that's a lot smaller than 1 and 2 thirds. This is not true. Not true. Okay, so this point, which is outside of the shaded area, outside the boundary, this point is not true. Okay, that's why this shaded area, these are all the solutions that satisfy the inequality. These numbers out here, they're not going to. They're going to create stuff like this that's not true. Okay, so that tells you, that gives you an idea of why we shade, why we have boundaries, and what the shading and what the not shading, what all of that means. That's all the stuff that we have for inequality. Okay? All right, so basically we're looking for points that work and for points that don't work. Okay. All right, so now I'm on to my second example. That was a lot there. There's a lot of explaining to go through there. You might have to rewind and check that out again, but there's a lot of explaining to go through right there. We get rid of this. We get rid of this to concentrate on my second problem, my second graph that I have over here on the right. Okay, here we go. All right, different color. So now why is, that's my equation, excuse me, inequality. That's my inequality. Why is less than or equal to negative 1? Hmm. If I look back over here, I had x's, so if I look back over here, I don't have x's. So that actually tells me this is a vertical or horizontal line. Okay, now remember back to when you studied those, if I have y equals negative 1, the y's, those are horizontal lines. Okay, so keep that in mind as we're making this. So okay, it's gonna be a horizontal line at negative 1. Now, there's my starting point. Now I gotta decide, is it a dotted line or solid line? y is less than or equal to negative 1. It is going to be a solid line. It's gonna be a solid horizontal line. It's gonna be a solid boundary. Okay, now in this case, we only have two choices. We can either shade up or shade down. All right, so let's read our problem and see what direction we're supposed to shade. The y's, the shaded portion that I want are less than negative 1. y's are less than negative 1. So as I look at this point, where are the y's less than? The y's are less than down here. So this is where I want to shade. This is where I want to shade. There we go. And that's it. So this top part up here, this top part up here, all of these points will not work for this inequality. Down here, on the other hand, all of these points will work for this inequality. That's what the shaded areas mean. Okay, that's it. That's all I have for graphing inequalities. Those are my two examples. I know the first one was a little bit long winded, but I had to go over the meaning behind the shading. Why do we shade? Why do we have dotted or solid lines? Those type of questions, once answered, give you a better understanding of why we do all this stuff. What do shaded areas actually mean? Okay, hopefully that was helpful. That is graphing later inequalities. I hope this video was of some help to you, and thank you for watching.