 In this video, I'm going to talk about solving compound inequalities. So your first question might be, what is a compound inequality? Basically what it is is it's two inequalities in one. So we got an equality here on the left, an inequality here on the right, and it's connected with his or word. So that's a compound inequality. And this other example over here, again, two. You got one over here on the right, one on the left. These are two inequalities connected with an and sign. Or excuse me, an word. So now, what's the difference between the two? Well, what I'm going to do first, I'm going to solve both of these, and then I'll kind of show the difference between the two. It's quite obvious once you look at the graphs of these two inequalities. It's quite obvious to see the difference between the two. So that's what I'm going to do. I'm going to solve each of these compound inequalities and then graph the solution set. So not only are we solving it, but we're graphing it. We're doing the pictures. So the first one here, I'm just going to divide by six on both sides. Y is less than negative four. Or now I'm going to subtract five from both sides. Y is going to be greater than or equal to negative two after subtracting five on both sides. All right, so those are my two inequalities. Now, basically what I'm going to do is I'm going to graph both of them on the same number line. So draw my number line here. Now again, depending on what your teacher wants, you're going to put different numbers here. What I like to do is I like to do a negative four and a negative two. Those are the numbers I'm concentrating on. And then just a couple of numbers that are around it. So for example, in between we have negative three on the outside here. We have a negative five. And then on the outside here we have a negative one. Okay, this just shows that you know the numbers, the priority numbers, where they get smaller, where they get bigger. It shows a lot of information. Okay, so now I'm going to figure out what to do. Now remember, when we have greater than or less than, we have open circle. So I'm going to have an open circle on negative four. Now greater than or less than or equal to, what I do when I get this line underneath it, those are going to be closed circles. So actually number two, or excuse me, negative two is going to be a closed circle. Okay, now from here I'm going to figure out which direction to go with these. I'm either going to go left or going to go right. It just kind of depends on what the problem is. Okay, so y is less than negative four. So the y's that I want are going to be less than negative four. So as I look here, the numbers that are less than negative four are going to be this way. Okay, and then for the other one, the y's, the ones that I want to shade, the ones that I want are going to be greater than negative two. Okay, so if I look here, the numbers that are greater than are going to be that way. Okay, so that right there is my compound inequality. And again, you can visually see why we call it a compound inequality. There's multiple. So we've got an interval over here. We've got an interval over there. We've got a lot of stuff going on. Okay, that's why we call it a compound inequality. Okay, moving on to the next one, moving on to this one over here. We're going to do the exact same thing. Now notice we have this AND word here. Over here we had OR, and I've got an AND word here. Okay, it's a little bit different. The solving and the graphing is the same, but it's just going to look a little bit different. We'll see what that looks like here in a moment. Okay, this side for this equation, this is supposed to be one half, if we can see that, one half. One half C is greater than or equal to negative two. So what I'm going to do is I'm going to multiply both sides by two. C is greater than or equal to negative four. And I'm going to solve this one. This one's going to take a couple of steps. So I have to subtract one from both sides. Negative two C is less than zero. Okay, now this one gets a little tricky. This negative two, I have to divide both sides by negative two. Now the reason I put this example in here is to remind us that whenever you multiply or divide by a negative, you always have to switch your inequality. So right here I'm going to multiply by negative two and switch my inequality. I divide by negative two, I think I said multiply. So I want to divide by negative two, excuse me. So now the thing is when I divide by a negative number, I have to flip my inequality symbol. Now notice that when I actually divide by negative two, zero's just going to stay zero. Love work with zero, nothing really happens. When you add subtract or multiply with zero, it just kind of stays zero. When you multiply or divide, when you multiply by zero, everything stays zero. I can't divide by zero, can't do that. But when you add or subtract with zero, zero's great to work with. Anyway, enough about zero. Now I need to graph this. So I'll put my number line out here. Now again, the numbers I want to concentrate on are zero and negative four. So I'm going to put those on there. Negative four over here, zero here. And just a couple of numbers around it. So negative five over here. We'll put a one over here. And then right between a negative two, that's right about halfway between. That's exactly halfway between, in fact. And now, open and closed circles. So it looks like for the negative four, I'm going to have a closed circle. Closed circle because of the less than or equal to, or equal to part, closed circle. And now here, I'm going to have an open circle on zero, open circle on zero. And now I've got to figure out which way to shade. Now I've got to figure out which way to shade. All right, so let's go back to this first one. The C's that I want, the C's that I want to shade, are greater than or equal to negative four. The C's I want are greater than or equal to negative four. So the greater than numbers, greater than numbers. The greater than numbers are going to be this way. Now notice I didn't put an arrow on the end of that because what I'm anticipating is that if I'm going, if I go towards my other circle over here, if I go towards it, then I think these are going to come together. We'll see. Let's go to the next one and actually see if that works. All right, C is less than zero. So the numbers that are less than zero are this way. So it looks like they join up right there. They join up right there. Okay, now if you look at the difference between these two, these are solved, and one has an OR word over here and one has an AND word, there's a difference between the two. Now notice that it's very blatantly apparent once you look at the drawings. These ones are going away from each other and these ones are going towards each other. Okay, this, what this does, this gives us two types of compound inequalities. This one over here on the left is called a disjunction. Disjunction. Okay, disjunction. They're going away from each other. That makes sense to call it a disjunction. And then this one over here, since everything is coming together, we're going to call this a conjunction. Conjunction, the ones that come together. There we go. All righty. Those are just a couple of different examples of solving compounded inequalities. Now remember, you not only have to solve them, you got to solve them, you got to solve all of the different solutions that you have for these type of inequalities. And again, if we, just to go back over this vocabulary here, it's a disjunction when they go away from each other or when we have this OR word, for the most part, or it's a conjunction if we have the inequalities coming together with this AND word. Okay, anyway, that is just a couple of examples of solving compound inequalities.