 In this video, I'm going to talk about solving absolute value equations. So basically what I'm going to do is look at these two equations, unless they have an absolute value in both of those. Basically what I'm going to do is I'm going to solve both these equations, show you how to solve equations that have absolute values in them. It's a little bit different here, a little bit different. Now, if you have an absolute value that's just on a number, it's not really that big of a deal. It doesn't really change very much. But if you have an absolute value on a variable here, that actually changes a lot. Changes a lot. So I'm just going to show you. I'm not going to go into a lot of description of why we solve it this way. But I'm just going to show you the way that we solve this. A little bit different from what you usually do. So now with absolute values, we know that an absolute value is the distance from that number to 0 on the number line. So now you can actually have two distances. You can either go the positive direction or the negative direction, ending up getting the same distance. Now, in that kind of aspect, we're going to have to solve this two different ways, because we have to think about the positive way to solve this and the negative way to solve this. So we're going to solve this twice, basically, is what it equates to. So what we're going to do, we're going to take this inequality, we're going to split it up. We're going to kind of split it up. It's kind of a split, not really. Basically, what we're going to do is we're going to rewrite this thing twice. The first time that we're going to rewrite it, we're going to rewrite it as we see it, except for we're not going to put the absolute value sign as symbols. So 9, or excuse me, x plus 9 is equal to 13. So that's the first one I'm going to go right, this first split that I'm going to make. The second one over here is actually, when I split this up, when I rewrite this, I'm going to rewrite the left side without the absolute values, but then the right side, I'm actually going to change. The right side, I'm going to change to a negative 13. So take a moment and look at those two. x plus 9 is equal to 13, and x plus 9 is equal to negative 13. Now again, remember, absolute values are distances on a number line. So basically what we're looking for is a distance that's going to be 13. Well, from 0 to 13, that's a distance of 13, but then also a distance of 0 to negative 13 is also a distance of 13. So you got to take both into account. That's kind of why we split both of these up. Now again, that's a very vague and very quick generalization of why we split this up twice, but it gives you an idea of why we have to do this twice. Okay, so now what we do is we just solve this, just like you would any other equations. So what I'm going to do is I'm going to subtract 9 from both sides, x equals, what would that be? x equals 4, okay, subtract 9 from both sides, x equals two negative numbers, that would be negative 22. And that's it. Okay, so now if you're still a little confused on this, what you can always do is just plug these numbers back in to figure out if they still work. So let's take that, let's take that 4. Okay, let's take a little bit of time. Take this 4 here and we're going to plug it back in. Okay, so 4 plus 9, absolute value of that, is that going to be equal to 13? Okay, that's kind of the question we have here. Is that going to be equal to 13? Okay, well 4 plus 9 is 13, and the absolute value of 13 is 13. So this does work. Okay, a little check mark there, this does work. All right, so let's do this with the next one. Let's try this with negative 22, okay? So negative 22 plus 9, we're going to take the absolute value of that, and we're going to see if that still equals 13. Okay, well negative 22 and positive 9, that actually makes a negative 13, and the absolute value of negative 13 is 13. So look at the left and the right, it does work. So basically what we have here is two numbers, two numbers that both work in this equation. Two numbers that both work in this equation. Again, because we're working with absolute values, absolute value is a distance, so the absolute value of 13, and the absolute value of negative 13, it's basically the same thing. All right, so that's kind of a quick explanation of what that is. So I'm going to work on a second example over here, work on the second example. Now notice it's a little bit different. Over here, everything was, I got the absolute values on the left, numbers on the right, and this one's a little different. We got absolute value and a number on the left, and then numbers on the right. So the first thing you actually have to do with this before you split it up, is you have to kind of move everything across or away from one another. This absolute value needs to be by itself, and the numbers need to be on the other side. So the first thing I need to do, the first thing I need to do is actually add eight to both sides. Now I have the absolute value of 6x, I got that by itself, and now I got 30 by itself. I got the absolute values on one side, numbers on the other. Now I can actually start to split this apart. Now I can actually split this apart. The left side is gonna be 6x equals 30. I'm just gonna basically write it as I see it without the absolute values, and the other split is going to be 6x equals negative 30. Equals negative 30, that's a bad three there. That's a bad three, that's okay, we'll just leave that. Okay, so now what I'm going to do is I'm gonna solve both of these. So now actually, interestingly enough, and you can kind of catch it from the first one, the first one we just subtracted nine both times. This one over here, we're gonna divide by six both times. So actually, when you split this apart and you solve it, you're gonna do the same exact steps on this left one over here and on this right one. You're gonna do the same exact steps. Now you're doing them to a different number, but again, the steps are gonna be the same. So it's kinda nice, kinda nice. X is equal to divide by six, this is gonna be five. Divide by six, x is equal to negative five. All right, now again, what I'm gonna do is real quickly, just plug this in and to see if this actually works. I'm gonna plug this back in to see if this actually works. All right, so six times five in the absolute value and then subtract eight from it, is that going to be equal to 22? Okay, well six times five is 30. Okay, the absolute value of 30. All right, the rest of us down. The absolute value of 30 is 30. So I have 30 minus eight is equal to 22. Well that, in fact, 22 is equal to 22. That does work. That does work. On the other hand, getting down here in the corner of my work here, on the other hand, what happens if I plug in negative five? So I have six times, well parentheses here, negative five, absolute values, minus eight equals 22, is that going to be true? Is that going to be true? So six times negative five is a negative 30. So I'm taking the absolute value of negative 30, subtracting eight from it, is that going to be equal to 22? Okay, so the absolute value of negative 30 is in fact 30 minus eight is equal to 22. Well, in fact, yes, 30 minus eight is 22. Okay, so notice that actually down here, these two were relatively similar. Okay, we actually had the kind of same steps here. Alrighty, and in fact it did work, 22 is equal to 22, so these are my solutions. Five and negative five are both solutions that work for this. Okay, so there's a quick couple of examples of solving absolute value equations. The thing you gotta remember here, two things you gotta remember, is that first of all, you gotta switch things up, or excuse me, you gotta split things up. Your first split is gonna be the equation as you see it without the absolute values. The second split is going to be the left side, the absolute value is gonna stay the same, but the right side, you're gonna change the sign of it. So it's gonna go from positive to negative or negative to positive, depending on your problem. And also, one thing to remember is that you have to get the, before you split everything apart, you have to get the absolute value by itself and the numbers by itself. So notice we had to add eight to both sides here before we could do anything, before we could split it apart. Okay, so that's a nice quick short video on how to solve absolute value equations.