 In this video I'm going to talk about solving systems of linear inequalities. Basically how you solve a system of linear inequalities is you have to graph the system and then your solution is going to be a shaded region. I'll show you what that is here in a minute. So here are my two inequalities. I graph the system of inequalities. Here are my two inequalities. y is less than one half x minus three and y is greater than or equal to negative x plus two. So what I'm going to do is I'm going to take these two equations. The first thing I need to do is I need to graph them. Graph them as if they are normal equations. Graph it as if it's y equals one half x minus three and also y equals negative x plus two. Graph it almost as if it's a regular equation. Now one thing you got to remember though is that when you graph inequalities they're either going to be solid lines or dashed lines. In this case the first one is going to be a dashed line. The first one is going to be a dashed line. The second one is going to be a solid line. I'm just trying to keep track of that. So I will graph the first one. So negative three for my y intercepts right here. And then one half for my slope. So one, one, two, one, one, two, one, one, two, one, one, two, one, one, two. Now notice that this is a dotted line. So I'm just going to put dashes on all of my points right here. So that way it looks more like a, there we go, there we go. So basically I'm not including any points that are on that line. So any part of my solution is not going to be on that line. That's kind of what that dotted line means. Alright, next one. Y intercept of two and a slope of negative one. So one, one, one, one, one, one, one, one, one, one. Lots of points on here so we can see where our shaded regions are going to be at. And here's my solid line for this one. Now I'm also going to write the equations next to them. So this one is y equals negative x plus two. And then this one down here, this one down here is y, that's not an equals up there, I'll change it here in a second, y is greater than or equal to negative x plus two. So let me go back and change this one up here. It's not an equals, it is a, whoa. Looks like I wrote down the wrong ones. Let me change that one more time. Alright, gotta make sure I write down the right equations. Alright, so this is the negative slope. So this is a greater than or equal to. And then this one down here is y is less than one half x minus three. There we go. So we always do our mathematics with pencils. That way if we make mistakes like that we can always erase them. Okay anyway, moving on. Now what we've created here is we have four different regions. I'm going to number these regions. This one right here is region one. This one over here is region two. This one down here is region three. And this one over here is region four. Okay now one of those four regions is going to be the solution to this system of inequalities. So basically what's gonna happen is I'm going to shade in one of those four regions and all of my points that are in that region are going to be solutions to this system of inequalities. Okay, so now I just gotta figure out where to shade. Alright, so my version of doing this. Now you can plug in points if you want to. My version of doing this is simply just reading this to myself. The y's that are smaller than this line. So I'm gonna go back over here. So the y's that are smaller than this line. Okay, so as I look at this line, the smaller y's are going to be down. So that means region three down here or region four over here. So notice I put dots in those regions. Okay, what that does, that tells me that I should shade in one of those two regions. Okay, now you'll see why I do the dots here in a moment when I do the other equations. So let's look at this next one. The dots are greater than this line. So where are the y's greater? So as I look at this line, the greater y's are going to be above this line. They're gonna be up here. So that includes this region here, number one, and this region over here, number four. Okay, now notice that I have two dots right here in region four. That is where both of my equations are going to graph. So what that means is that area is going to be my common solution. So what I'm going to do is I am going to shade in, I'm gonna shade in this region right over here by the edge of my screen here, right there. That is my shaded area. There's my common solution already. And again, what that basically means is all of the points that are in that shaded area are going to be solutions to the system. Any point in here is going to be a solution to this system, is going to make both of those equations true. So that is how you represent the answer to this system. It's not gonna be a number, it's not gonna be a coordinate, it's not gonna be anything like that. It has to be a picture in a shaded region because there's many, many, many solutions. You can also argue that there's an infinite number of solutions within that shaded area right there. Anyway, that's my first example of working with this. Now I'll do one more example, give you something else to look at. So I'm doing another example, but this one here looks a little bit different. The first one just looks like a normal equation. The second one there, this one is kind of a reminder on how to graph horizontal and vertical lines. This one here is actually going to be a horizontal line and that's gonna be at negative one. So actually I'm gonna graph that one first since I'm talking about it. Okay, so over here, right here is where my horizontal line is at. It's right there at negative one. Oh, crud, is that supposed to be a solid line? The or equal to tells you that we include points on that line and so we use a solid line for this one. Now this top one here is going to be a dotted line. The graph of that is going to be plus two for my y-intercept, negative three for my slopes, one, two, three, one. Right there, one, two, three, one. One, two, three, one. That one's gonna go off, so I go dotted line here, do a dotted line in the middle. There we go. So this one is y is less than negative three x plus two and then this line here is y's are greater than or equal to negative one. Okay, so now I gotta figure out where to shade. So I got my four regions, we'll call this region one, call this region two, call this one over here region three, call this one region four. So what we'll do here is figure out where to shade. Now this first one, y's that are smaller than the line, y's that are less than, so if I look at this line, the less thans, the y's that are smaller are going to be down, which means down from this line is gonna be region one or down from this line is also gonna be region four. Okay, so either one of those regions is going to be in my solution and now over to this one here, y's are greater than or equal to negative one. So the y's that are greater, so as I look at this line, the y's that are greater are going to be up. So that's gonna include region one here and it's gonna include region two over here. Now notice that region one has two solutions to it. Okay, region one has two dots that go with it. So what I'm going to do is I'm going to shade in this entire region. I'm gonna shade in this entire region right here. There we go. Now make sure you get real close to both of the lines right there. And again, remember to use pencil while doing this because if you make a mistake, you want to be able to erase it. If you use pen, it doesn't really work out that well. Alright, and there's my solution region. And again, any point, any point in this region right here is going to be a solution to both of those inequalities. Alright, that's just a couple of examples of how to solve systems of linear inequalities.