 In this video, I'm going to talk about solving absolute value inequalities. Now the first example that I'm going to do is solving absolute value inequalities with disjunctions. So if you remember the vocabulary, a disjunction is an inequality that's going to go away from each other. Now disjunctions, you first heard that when we talked about compound inequalities. Now when we talked about compound inequalities, we had two inequalities. As you look at this example, there's only one inequality here. But now remember, when we solve absolute value equations, we had to split them up and it became two equations. Well, we're kind of going to do the same thing here. We need to split this up and solve two different inequalities. And so that's going to create our disjunction. That's going to create our two separate inequalities. So I'm going to solve this inequality, solve this inequality, and then graph the solution set. And we'll see that it's a disjunction that's going to go away from each other. Okay. So the first thing that I'm going to do is I'm going to split this up. Now notice that I have the absolute value on the left by itself and then the number is on the right. So that is in perfect position for me to kind of split this up. So the first split is I'm going to rewrite this as I see it without the absolute values. Negative 4q plus 2 is greater than or equal to 10. And then I'm going to split this up again. I'm going to write it as I see it, except for this right side here I'm going to change. So negative 4q plus 2, and now here's where the difference is. When we solved absolute value equations, we just changed the sign on this number over here. So it became a negative 10. But now remember we're dealing with inequalities when we've changed a sign, divide by a negative 1, that kind of stuff. Remember we had to flip the sign. It's the same rule. So this symbol right here, I need to switch that around from a greater than or equal to to a less than or equal to. All right. So now we have our two inequalities. So now we have our compound inequality. That's what we need. We need two different inequalities so that we can write a disjunction. Okay. So that's how you split it up. So now what I'm going to do is I'm going to solve it. Now notice that over here I'm going to subtract 2 and then divide by negative 4. Same thing over here. I'm going to subtract 2 and divide by negative 4. So it's actually really nice because everything's going to kind of, I'm going to do the same steps on the left and on the right. Now I'm going to get different answers in the end, but the steps themselves are going to be the same. So that makes your solving a little bit faster, a little bit easier. All right. So subtract 2, so negative 4q is greater than or equal to 8, and then divide by negative 4. Now remember, when you divide by a negative, multiply or divide by a negative, you switch that symbol. So divide by negative 4, I get a negative 2. 8 divided by negative 4 is a negative 2. Okay. So there's that first one. Next, what I'm going to do is solve this one. So subtract 2 from both sides. Negative 4q is less than or equal to negative 12. Two negative numbers, they get bigger, bigger negative to negative 12. All right. Then I'm going to divide by negative 4. Same deal on this side. I got to switch this inequality. 2 is now greater than or equal to 3. Negative 12 divided by negative 4 is a positive 3. Okay. So now what I have here is I have my compound inequality. Now what I need to do is I simply just need to graph this. So I'm going to graph a number. I'm going to grab my number line. Okay. A little number line over here. And again, my important numbers, the ones I really want to pay attention to are 3 and negative 2. So I'm going to put those on the number line. So here's a negative 2 over here. We'll put a 3 here. Now of course, depending on who your teacher is, they want you to put more numbers on here. I just like to have numbers in between, so I know that 0 is in between these numbers. Not halfway between, but it is between them. I like to have a negative 3 on this side and then a 4 on this side. So on the right side, I got bigger numbers. On the left side, I got smaller numbers. And in the middle, I got numbers like 2 and 1 and 0 and negative 1. I got numbers like that. Okay. So now I'm going to put my circles on. All right. So with the negative 2, it's a less than or equal to. So the or equal to part tells me to have a shaded circle. And over here, it looks like we have the same. Now notice when I split this up, it's actually going to be the same symbol. It's either going to be less than or greater than, or it's going to be less than or equal to, greater than or equal to. So notice that you're always going to have either a line under it or not a line under it. So basically what that equates to is that you're always going to have either an open circle or a closed circle for both of these numbers. So on the 3, that means I'm going to have an open circle, which is kind of nice. So when you solve these, they're either going to be both open or both closed. Kind of nice when you solve these. Okay. So now which direction they're supposed to go? Let's go back to the negative 2. The ones that I want to shade are less than negative 2. The ones I want to shade are less than negative 2. So the less than numbers are that way. The less than numbers are that way. All right. So now let's go to this next one. 2, the numbers that I want, the numbers that I want are bigger than 3. So the numbers that are bigger than 3, well, they're that way. And there we go. There's my disjunction. We can kind of see that. If you remember the vocabulary, a disjunction is an inequality, is a compound inequality that graphs away from each other. So these are pointing away from each other. There's my disjunction. There we go. Again, one thing to remember about this, when we split things up, when we solve absolute value inequalities, when we split this up, the first one, we write as we see it. So we just basically take away the absolute value sign, symbols, and just rewrite it as we see it. And then the second time we write it, the left side, we just get rid of the absolute value symbols. But the 10 becomes a negative 10, and we also switch the inequality symbol. Notice here, we switch the inequality symbol. All right. So that's an example of a disjunction. Now I'm going to do an example of a conjunction. So here's another example. Here's another example. I'm going to go a little bit faster through this, since we already have the gist of what we're supposed to do. OK. So now as I look at this, the first thing that we're supposed to know that we're supposed to have is that the absolute values need to be on one side, and the numbers need to be on the other. Now as I look at this, I have the absolute values up here, but then I got this divide by 2. I have this divide by 2. It's really messing things up. OK. So let's get rid of that. I don't want to, if I want to get rid of divide by 2, what I'm going to do is I'm going to multiply both sides by 2. OK. So after I multiply both sides by 2, what I'm going to get is x minus 5, the absolute value of that, less than or equal to 8. OK. If I multiply 2 on both sides, that gets the absolute value by itself, and then the 8 there on this side. So I've got the numbers on one side, and the absolute values on the other. So there we go. That's what it's supposed to look like. Now I can start splitting everything up. Now I can write my compound inequality. I can write my 2 inequality. So here we go. The first split, I'm going to write it as I see it. X minus 5 is greater than, or excuse me, less than, less than or equal to 8. And split it up one more time, and it's going to be x minus 5. Change this from a positive 8 to a negative 8. Now one side. Oops. There we go. A little clicker problem. OK. So once I change that from a positive 8 to a negative 8, I also have to change this symbol to a greater than or equal to. So when you change that sign, you also got to change the symbol around. OK. So now I'm going to graph this. Now I'm going to graph this. So actually it's pretty easy to, excuse me, not graph it. I'm going to solve this. I've got to solve the rest of it before I graph it. Anyway, so what I'm going to do is I've got to add 5 to both sides. That's the only step I've got to do. This is really nice. The only thing I've got to do is add 5 to both sides. So here we go. So I'm going to take add 5 here to get 13. And I'm going to add 5 here to get a negative 3. So there we go. Now this is going to be a conjunction. So everything should shade together. OK. Everything should shade together. All right. Now there's one additional thing that I need to show you on the conjunctions is that you can actually write this inequality. You can actually write this as a single inequality. A single inequality. OK. I'm going to kind of show you how here in just a moment. I'm going to graph it first. I'm going to graph it first. And then I'm going to show you how to kind of combine things together. OK. I'm going to graph it. Put my number line over here on the right side. My important numbers are negative 3 and 13. Now make sure I know negative 3 is on the right side here when I solve. 13 is on the left. But when you put them on the number line, put them in the proper position. OK. So negative 3 is on the left and 13 is on the right. So again, when I put numbers in here, 0, 0 is there in the middle. It's not exactly in the middle. That's one of the numbers in the middle. I like to put numbers there. See I got a negative 4 on this side. And then over here I'm going to have a 14. Those numbers there, to give you an idea of what other numbers are around the negative 3 and the 13. So here we go. Both of them are going to be shaded. I'm going to have closed circles on both of them. Because of the or equal twos. Because of the or equal twos. OK. So I got shaded circles, shaded circles. And now you know it's a conjunction. So they should shade together. So let's figure out if they do. The numbers I want to shade are smaller than 13. Right there. The numbers I want to shade are bigger than negative 3. Right there. So they do in fact shade together. Awesome. So I did get a conjunction like I was supposed to. All right. Now, again, I'm going to show you how to bring these together as a single inequality. And it's not as hard as what you might think. But I'm going to switch things around just a little bit. I'm going to switch things around just a little bit. So what I'm going to do is I'm going to take this inequality. Since it's the smaller one, since I have the negative 3, I'm going to switch everything around. Negative 3 is less than or equal to x. Now I switch everything around. x goes to the right, negative 3 goes to the left. And I also switch around the inequality. Now I'm going to compare these two right here. I'm going to compare these two right here. And I'm going to combine them together. I'm going to combine them together. OK. So negative 3. OK. Negative 3 and 13. So what I'm going to do, actually, you know what? Let me do it this way. Since I have the great technology, let me use the great old copy paste. OK. That's one thing that's great about technology that I can use copy paste. Or maybe I can't. It doesn't look like I'll be able to because these things are too tightly grouped together. If they're too tightly grouped together, they think it's one annotation. So I can't do it that way. That's OK. We'll move on with the good old fashioned writing. Move on with the good old fashioned writing. OK. So anyway, I want to bring this together as one inequality. So I'm going to rewrite this as negative 3 less than or equal to X. And now I'm going to write this one. X is greater, less than or equal to 13. So notice how I brought that together. OK. X is in the middle. X is in the middle. And we still have less than or equal to 13. And we have negative 3 is less than or equal to X. We have it on both sides. OK. And so that's one way of writing this as a single inequality. Now you can do that with conjunctions. You can do that with conjunctions, but you can't do that with disjunctions. Disjunctions don't really work that way. I mean, you could, but we have to differentiate the difference between a conjunction and a disjunction. It doesn't quite work. But anyway, there's two examples there. One of a conjunction and one of a disjunction of how to solve absolute value inequalities.