Modeling inequalities with negative coefficients

0:00- [Instructor] The goal of this video is see if we can simplify this inequality or think about the solution set of this inequality, but also do it in a visual way. So we have negative 3n plus 7 is greater than one. Or you can visually think about it with these algebra tiles as you have three negative ns, which is negative 3n and then you have seven positive ones, that's greater than this one, this one positive one right over here.

0:25Now, the first thing that I wanma do is I want to get rid of the seven on the left-hand side. I just wanna have the things involving the N on one side of this inequality. So the way I would do that is I would subtract seven from both sides. Or another way to think about it is I could add seven negative ones.

0:43So one, two, three, four, five, six and seven. And each of these are a negative one. I know the writing is getting a little small, but I think you know what I'm talking about. But we can't just do it on the left-hand side; we have to do it on the right-hand side too.

1:01So one, two, three, four, five, six and seven. Each of these are negative ones, just like that. And what happens when you do that? Well, each of the positive ones cancel out with a negative one. So you are left with none of that constant on the left-hand side.

1:28They have all canceled out nicely. And on the right-hand side, that positive one cancels out with one of the negative ones. And so what we're left with is negative 3n. Negative 3n is greater than, on the right-hand side, we have one, two, three, four, five, six negative ones is greater than negative six.

1:56And let me clean this up a little bit just so it becomes pretty clear what just happened. So let me get rid of all of this messiness and let me get rid of some of that. Now, the next step, you might say, "Well, what if I just divide things into three equal parts?"

2:12But that's a little bit difficult because you might say, "Well, what happens to the inequality sign here? Can I divide both sides by negative three and keep it as a greater than? Or does it need to flip somehow?" But to make it make sense to us, why don't we try to get the n onto the other side of the equation?

2:28And the one way to do that is we could add 3n to both sides. Let's add 3n to both sides. And what will that look like? Well, on the left-hand side here, we're gonna have one, two, and three ns n, n, n. And on the right-hand side, we're also gonna have three ns.

2:51I'll do it over here since I'm running out of space horizontally. One, two, and three ns. Now what does our inequality look like? Well, on the left-hand side, we have zero. All of this stuff cancels out. So actually, what I could put here is nothing. I could cancel that out, I could put a zero, I could put a blank.

3:15However we wanna describe it. Nothing is going to be greater than what we have on the right-hand side. Well, the right-hand side, we have 3n and then we have a minus six. So zero is greater than 3n minus six, minus six. Now, the next thing we might say is, "All right, well, it's nice that we have a nice positive three on this right-hand side.

3:41How can we get this negative six onto the other side?" And this wasn't the fastest way of simplifying it, but you'll see there is a payoff here. We're going to see what to do with the inequality sign when we're simplifying these types of inequalities. Well, we could add six to both sides.

3:55Let's do that. We're going to add six to both sides. If we do that visually, we're goin to have one, two, three, four, five and six. So six ones on the right-hand side. And then we're also going to have one, two, three, four, five, six ones on the left-hand side.

4:24And so we're going to have six is greater than, the negative six and the positive six, those cancel out, is going to be greater than 3n. And we see that over here, all of these two cancel with these two. These two cancel with these two. These two cancel with these two.

4:44So we have 3n. Well, we have six is greater than 3n. But what's interesting here is we remember a time where we had negative 3n is less... Sorry, we had negative 3n is greater than negative six. And now we have positive six is greater than positive 3n. But this thing down here could also be written as, it could be written as positive 3n is less than six, is less than six.

5:20So the difference between the inequality on the left here, negative 3n is greater than negative six. And positive 3n is less than six. And these are the same. We've just been doing the same algebra. So these are saying the same thing, is we've essentially divided both sides of this left inequality by negative one, or we've multiplied both sides by negative one to get this inequality on the right-hand side.

5:40But when we do that, what happened? The inequality switched and that's always going to be the case. If you multiply or divide by a negative, the direction of your inequality is going to switch. But this right over here, or maybe I'll focus on this one that we had originally written, six is greater than 3n.

5:58This is straightforward to solve. We can divide both sides by three here. So we could divide both sides by three, and we clearly get that n, or maybe I should say it this way, two, and let me scroll down a little bit. We'd get that two is greater than n, two is greater than n.

6:22Each of these equal groups of three have two in them. They also have an N in them. So two is greater than n, and we're done.