 Another important type of equation in mathematics is called an absolute value equation, and this is an equation that involves an absolute value. Now there's a number of ways of solving these equations, but here's one way that's a very nice approach that relies on a connection between algebra and geometry, and it's really helpful to make the following identification. The absolute difference between two numbers A and B, which we're going to write as the absolute value of A minus B, is the distance between A and B on the number line. Now you might have been told that the absolute value represents the distance from zero, and that's true, and it's part of a more general concept that the absolute value in general tells you the distance between two things. So the absolute difference is the absolute value of A minus B is the distance between A and B. So let's take a look at that. So here's the number line, and I've marked a couple of points on that, and let's see if we can find which of these quantities is going to be greater. So let's take a look at that individually. So again, the key here is we want to identify the absolute difference. The absolute value of A minus B is the distance between A and B on the number line. And if we can interpret it this way, if we can locate where A and B are, we can see geometrically what that distance is going to be. So in that first case, this absolute value of Y minus Z, well, that's the distance between Y and Z on the number line. And if only I knew where Y and Z were located, I could plot them and I can represent what that distance is. Oh, wait, here it is. Here we know where Y is because we've been given that information. We know where Z is because we've been given that information. So we can actually identify what that distance is. So I'll go ahead and put my number line down. Here's Y, here's Z. The distance between them, the absolute value of Y minus Z, well, I'm going to represent that using this little blue bar here. And likewise, this absolute value minus 3, minus 5, that's the distance between negative 3 and 5. Well, here's negative 3, here's 5. And I can represent that distance as this bar that runs from negative 3 up to 5. And now I have the very difficult task of figuring out which two of these is bigger. Oh, wait, I can use the geometry. Here's an important lesson. A lot of problems in algebra can be reduced to a problem in geometry. And in general, one of those two representations will be far easier to work with. With absolute value, it's very frequent with the geometric representation that's easier to work with. And I can look at these and say this absolute difference Y minus Z is clearly less than the absolute difference negative 3, minus 5. So I know that Y minus Z absolute value is going to be less than absolute value minus 3, minus 5. Now, that doesn't mean that sometimes we need a little bit of help to see that absolute value of X, absolute value of Y plus 3. Well, the important thing here is to view any absolute value expression as a difference, as a distance between two points. And so I can view X as, well, I can always subtract 0. This is X minus 0. This is the distance between X and 0. And I know where X is. I know where 0 is. And so I can graph what that X minus 0 is going to be here. Absolute value of X minus 0, distance between X and 0. It's this about here. And again, you might remember from your definitions of absolute value way back when that we defined absolute values the distance of a point from 0. And again, it's part of a more general concept. So Y plus 3, we have to do a little bit of algebraic manipulation. But here, I need a subtraction Y plus 3. Well, I can change that to Y minus negative 3. And again, I can change in addition to minus the additive inverse. So this is Y minus negative 3. And again, as an absolute value, as an absolute difference, this is the distance between Y and negative 3. Well, I know where Y is. I know where negative 3 is. So I can represent this distance geometrically by this bar here. And I now have that task of deciding which of these two is bigger. This distance here, representing the absolute value of X, or this distance here, representing the absolute value of Y plus 3. And again, the geometry is much easier to work with. The absolute value of X is definitely greater than the absolute value of Y plus 3. Well, what if I want to solve some equations? For example, let's take a look at absolute value of 2 minus X equal to 3. Well, again, I can view the absolute value of 2 minus X. Well, this is the distance between 2 and X. So let's see. Well, the equation says that the distance between 2 and X is going to be 3. Well, let's go ahead and plot that on the number line. Now, I know where 2 is. I don't know where X is, but I know where 2 is. And I can place that on the number line. So there's 2. And we want to place X the distance between 2 and X is going to be 3. So I want to place X. So it's 3 away from 2. And here's the important thing to remember when dealing with an absolute value equation. In general, my location can be in one of two places. So it's possible that X is 3 more than 2. So here, the distance between X and 2 is 3. The other possibility is that X might be less than 2. So, again, here, the distance between X and 2, this distance is going to be 3 as well. So I know that X is in one of these two places. And now I have the very difficult task of figuring out what X is. Well, if X is over here, X is 3 more than 2, well, it must be that this location here must be 5. And so X and 5 are the same thing. So I could say X equals 5. Or if I'm back here, if I'm 3 less than 2, I have the very difficult task of figuring out where this is. If X is 3 less than 2, well, that's going to put me at negative 1. And so 3 back from 2, that's at negative 1. And there, X is. And X and negative 1 have to be the same thing. Well, how about a more complicated absolute value expression? The key here is that what we have is an interpretation for what the absolute value is. So what we'd like to do is we'd like to say the absolute value is equal to something. So let's arrange our equation so we can conclude something about a distance. So let's see. I have absolute value 2X plus 5 minus 4. And so I'll add forward both sides to make this an absolute value on the left and just 4 on the right. And what this tells me is 2X plus 5 equals 4. And again, I'll change that addition into the subtraction of a negative. So 2 plus 2X plus 5, 2X minus negative 5. And again, what this tells me is that the absolute difference between 2X and negative 5, that distance is equal to 4. And so, well, I don't know X, so I don't know where 2X is. But I do know where negative 5 is. So I can put 5 negative 5 on the number line. And I know that wherever 2X is, the distance to negative 5 is going to be equal to 4. So I want to put 2X so that it's 4 away from negative 5. So 2X is either up here or above negative 5, and it's down here below negative 5. And so let's see. If I go from negative 5 up 4, I'll get to 2X. Well, that's going to take me to negative 1. Or if I go down 4, that's going to take me to 2X, but that will also take me down to negative 9. So I have 2X is at the same place that negative 9 is. 2X is at the same place negative 1 is. There's another equation. And so that tells me that X is either equal to negative 1 half, which is what I'll get by solving that first equation. Or X2X is negative 9, so that tells me that X is going to be equal to negative 9 halves, which is what I get from solving that second equation.