Graphing two variable inequality

So what I would like to do in this video is graph the inequality negative 14x minus 7y is less than 4. And like always, I encourage you to pause this video and see if you can graph it on your own before we work through it together.

So the way that I like to do this is I like to isolate a y on one side of this inequality, and I'll do it on the left-hand side. So the first thing I want to do is let's get rid of this negative 14x here on the left. The best way to do that is to add 14x, but if I do that to the left-hand side, I also want to do that on the right-hand side as well.

So I'm adding 14x to both sides, and that leaves me with these two canceling out on the left-hand side. I have negative 7y, and that's going to be less than I have 14x plus 4, or 4 plus 14x, which is 14x plus 4.

Now, I want to divide both sides by negative 7 in order to isolate this y, but we have to be very careful. When you multiply or divide both sides of an inequality by a negative number, you need to swap the inequality sign. Instead of it being less than, it would have to become greater than.

So if you divide both sides by negative 7, you are going to be left with y greater than—let me do a different color just to highlight that I'm swapping this around—y is going to be greater than 14 divided by negative 7, which is negative 2x, and then 4 divided by negative 7, well, that's negative 4/7.

And now I'm ready to graph this. The way that I would graph this, I would graph y is equal to negative 2x minus 4/7. But since we're not greater than or equal to it, I would put a dotted line there and just shade everything above that.

Let me just do what I'm talking about, so let me draw some axes here. This is going to be an approximation; obviously, this is a hand-drawn graph that we're dealing with. So just like that, that's my x-axis.

Let's see, our y-intercept is negative 4/7, which is a little bit more than a half. So let's say that this right over here is, let's say this right over here is 1—or negative 1, I should say. Actually, I think I would have to put my axis a little bit higher, so let's do it like this.

So that is x; let's make this negative one, and then that's about negative two. So this would be one, and then this is going to be right about—this is about 1, and then we have 2 right over there.

So, looking at our y-intercept when x is 0, let's first just think about what we have if we're thinking about y is equal to negative 2x minus 4/7. Well, our y-intercept right over here would be negative 4/7, which is a little bit more than a half, so it will be roughly right around there.

Then we have a slope of negative 2. We have a slope of negative 2, so if you increase by 1, you are going to decrease by 2. So you're going to decrease; you're not just going to go there; you're going to go right around there.

So let me get my ruler tool out so I can actually draw that, so I'll connect these two dots with my ruler tool. Now remember, I'm not just trying to graph y equals—and in fact, this isn't greater than or equal; it's just strictly greater than.

So let me draw a dotted line there to show that I'm actually not going to include the graph where y equals negative 2x minus 4/7. So let me do that dotted line right over there to show that I'm not—it’s not going to be equal to that.

It's all the y’s for any x that are greater than that. My spacing is getting a little sloppy, so that does the job—there you go. So that would be the line y is equal to negative 2x minus 4/7.

But I dotted lined it because this isn't greater than or equal to; if it was greater than or equal to, I would fill it in, but it's greater than to show that we don't include the line, but we want all of the region, all of the area above the line.

So let me shade that in. That's going to be all of this. Actually, I can shade it in with a nice big juicy shading. Get the right tool out. All right, here we go.

So it would be all of this area right over here would be what I would actually shade in. So once again, I don't include the line; I include everything above the line, and I am done.