 In this video, I'm going to talk about graphing linear inequalities using intercepts. This is a little bit different than my previous video where we graped linear inequalities using slope intercept form. In this case, if you look at our equation here, one of our examples, on the left side we have the x's and y's. This is the standard form of a linear equation. We've got the x's and y's on this side, and we've got the numbers on this other side. This is the standard form of an equation. I want to graph each one of these inequalities. What we have to do first is we have to find the intercepts, the x-intercept, where the line is going to intercept on the x-axis, and we want to find the y-intercept, which is where the line is going to intercept the y-axis. We've got to find those two things first. I'm going to very quickly do those. If you don't know how to find the intercepts of an equation, I do have a previous video. It's called Graphing Lines Using Intercepts. I would search for that, look for that if you don't know how to graph using intercepts, because I'm going to very quickly go through that. What I'm going to do is I'm going to treat this like an equal sign. I'm going to treat this like an equal sign. First thing I'm going to do is I'm going to find the x-intercept first. To do that, I need to plug in 0 for y, so this is 3x plus 4 times 0 equals 8. Notice again, I'm treating that like an equal sign. 3x is equal to 8, x is equal to 8 thirds, x is equal to 2 and 2 thirds. We're making a mixed number, so we understand a little bit better. My x-intercept is right here at 2 and 2 thirds, right there. Now be as close as you can with this point. It's 2 thirds of the way to 3, so that's about 2 thirds. Be as accurate as you can with those points. Sometimes that's kind of hard to see. OK, second thing, I want to find the y-intercept. I want to find the y-intercept. So in this case, I'm going to actually make the x is 0, so this is 3 times 0 plus 4y equals 8. Again, using equals right here. So this is 4y equals 8 divided by 4, so y is equal to 2. This one's a little bit easier, no fractions here. But y is equal to 2, that makes my y-intercept right there. OK, so now I have found the intercepts. And now what I'm going to do is I'm going to draw my boundary line. In this case, my boundary line is going to be a solid line. In my previous video, when we talked about graphing in slope-intercept form, graphing inequalities in slope-intercept form, I explained why it's a solid or a dashed line. If you want to re-watch that, you can. OK, so I have a solid boundary this time. Now I need to decide, do I shade up or do I shade down? Where is my solution going to be? Now this one's a little tough, a little different. And on the previous one, I just did a kind of a logical explanation of looking at the equation, saying it to yourself and figure out, OK, do I need to shade up or shade down? In this case, we can't do that. We have a little bit more of an arithmetic way of trying to figure out if you shade up or shade down. So in this case, the arithmetic that we use, what we're going to do is we're going to test the point 0, 0. Now notice that in this boundary, we have, OK, here's one boundary, here's one shaded portion up here, and then we have a shaded portion down here. Notice where 0, 0 is that. 0, 0 is down here. What we're going to do is we're going to use a point and we're going to test to see if it works. If it works, then that is the region that we shade. If it does not work, then that's the region that we do not shade, and then we would choose the other one. So let's test this point, plug it in, see what happens. So here's my equation. I'm going to plug it in. 3 times 0 plus 4 times 0, both x and y are 0 in this case. Is that going to be a little question mark there? Is that going to be less than or equal to 8? Now the reason that we choose 0 is because it's very easy to work with. You plug it in, you multiply, everything goes away, everything goes to 0. Is 0 going to be less than or equal to 8? Yes, yes, this is true. We plugged in 0, 0. Yes, it is true. 0 is less than or equal to 8. So it is true. That means this point, 0, 0, that point is in the solution area, which means that this has to be the solution area. So this is the part that gets shaded in. That is the part that gets shaded in. Let me redo that shading real quick. I want to make sure and get every part that I can. Get as close to the line as you can. OK, there we go. Notice a little bit of the first quadrant here, almost all of the second quadrant, all of the third quadrant, and most of the fourth here. We want to try to get everything that we can. This 0, 0 point, that tells us we test it, and that told us that 0, 0 was part of the solution set, and so that we should shade there. That is how you determine where you're going to shade. OK. All right, now for the second example. Second example over here. Give me a moment. I need just a, actually, you know what? Let's move this over here, since I'm not going to need that over there anymore. Give myself a little bit more room to work with. OK, so now the second example over here, 5x minus 2y, is greater than negative 4. So I'm going to do the same thing. I'm going to go through the same process. What I'm going to do is I'm going to find out what the intercepts are. I'm going to plot them. I'm going to draw my boundary. In this case, my boundary is dotted. And then I'm going to test the point 0, 0 to see if that is the point, if that's in the solution set or not. OK, so in this case, I'm going to quickly go through this. So my first is my x-intercept, which the y is 0. So all I'm left with is 5x equals negative 4. So x, in this case, is equal to negative 4 fifths. Now, again, we're dealing with fractions here. That's OK. Divide by 5. It doesn't divide evenly. That's OK. Just leave it as negative 4 fifths. That is totally OK. x-intercept of negative 4 fifths, that's over here. That's not quite bigger than 1, but it's right about there. Put a big dot right there. That's where negative 4 fifths is at. Now, for the other intercept, now we have the y-intercept. We have the y-intercept. Now I want to make the x0. So that leaves me with negative 2y equals negative 4. Divide by negative 2. That gives me 2. Negative 4 divided by negative 2 is a positive 2. So I have a y of positive 2. So there it is, right about there. All right, now I'm going to create my line here, bear with. Oh, wait, it's a dotted line. Dotted line. So dotted line here, dotted line here, and just kind of keep going. Make sure it's obvious that it's going to be dotted. There we go. There's my dotted boundary. Now I'm going to test the 0.00. So I'm either going to shade up, which is this region here, or I'm going to shade down, which is this region here. So I need to test this 0.00 to see if that's a solution or not. So let's take the 0.00, and we're going to test this out. So 5 times 0 minus 2 times 0 is that bigger than negative 4. Let's see, 0 is lovely to work with. But is 0 bigger than negative 4? Yes, it is. 0 is bigger than negative 4. So that actually tells us that 0.00 is in the solution set. So once again, this here, all of this, that boundary, sorry, that area on the bottom side of this boundary, on the bottom side of this boundary here, all of this area, all those points are solutions to my inequality. All right, that is solving, or excuse me, graphing linear inequalities using intercepts. Just a couple of examples of each. Again, I referred you back to graphing lines using intercepts. If you don't know how to graph using intercepts, that's a good video to go watch. Also, a couple of things here. Make sure you find the y-intercept first, or excuse me, x-intercept, then you find the y-intercept. And once you do that, you create your boundary, and then you need to see where should you shade. Test the 0.00, plug it into your inequality. If it works, then that point is included in your solution area. Then that point is included in your solution area. OK? All right, those are a couple of examples of that. So hopefully this video was helpful. Thank you for watching.