 Again, part of the point of introducing basic arithmetic is so that you can do algebra. And let's take a deeper look at absolute value, and we will note the following thing. Absolute value often appears in the context of the concept of a difference. In particular, we like to claim there is a relationship between subtraction and difference. And there is, but it's a rather peculiar one, and the peculiarity can be observed as follows. If I take a look at the value of 8 minus 5, that is the subtraction, 8 minus 5, the value is 3, and I would also say that the difference between 8 and 5 is 3. So far so good. On the other hand, if I take a look at the subtraction 5 minus 8, the actual value is negative 3. On the other hand, if I take a look at the difference between 5 and 8, it's going to be, well, it's still 3. And what makes things interesting here is that the subtraction has two different values, depending on the order that we perform the subtraction. 8 minus 5 is 3, 5 minus 8 is negative 3, yet when we speak of the difference between the two numbers, we don't actually distinguish between whether it's the difference between 8 and 5 or the difference between 5 and 8. Essentially, what we've done is we've automatically taken the absolute value of the one minus the other. So 8 minus 5 is 3, 5 minus 8, take the absolute value of that, you also get 3. And what that suggests is that if I take a look at the absolute value of A minus B, I can interpret this as the difference between A and B. Now an easy way to represent this difference is to use a number line. And this also allows us to do some algebra, some very complicated algebra, if you don't understand the basic concepts, but very easy algebra if you do. So let's consider this algebraic problem, absolute value of x minus 5 equals 7. Now if you've taken an algebra class before, you know there's some particular algorithm that we use to solve this, but for the most part that algorithm gets a little bit complicated. But if we understand what we're talking about, then this problem is actually very easy to solve. So again what I have here, x minus 5, well I can interpret A minus B absolute value as the difference between A and B. So what this equation is claiming here is that the difference between x and 5 is 7. Well since there is a difference between x and 5, what that says is that x is not equal to 5, and I can use a number line to show the situation. Since x is not equal to 5, it might be larger. So here's my number line, here's 5 on the number line, here's x, which is larger. And what do I know? Well the difference between x and 5 is 7, which says that the difference between x and 5, this amount here, is going to be 7. So if I take a look at my number line, what that says is that if I start at 5 and take a hop up 7, then I get 2x, there's my difference between x and 5 equal to 7, and I can just look at the diagram, no algebra here. Now I'm not using any sort of algebra here, but I am algebraically thinking. And so from 5 to x plus 7, that takes me up to 12. And so one possibility is that x is equal to 12. Now the only tricky part about absolute value is that we have to consider all possible cases. So here I assume that x was larger than 5, it's possible that x might be smaller than 5. So x is, again, not equal to 5, because there is a difference between x and 5. I've taken the case x is larger, now let's take a look at if x is smaller. And again the difference between x and 5 is 7. So if I hop down a bit, 7 units in particular from 5, I'll get to x. And again I can see from here from 5 back 7 to x, and that tells me that my second possibility is that x is equal to negative 2, and I can summarize my solutions here. My solutions are either going to be x equals negative 2 or x is equal to 12. Well let's take a look at a different problem, 5 minus x, absolute value equal to 7. And again same sort of problem. So again I want to interpret the absolute value of a minus b as the difference between a and b. And so I can read this as claiming the difference between the two is 7. And again I'll use the number line to model the situation. And because there is a difference, 5 and x are different. So again x might be larger than 5, and I can hop up 7 to get to x, and that tells me that x must be 12, or x might be smaller than 5. I can hop back 7 to get to x, and that tells me x is equal to negative 2, and I have my solutions x equals negative 2 or 12. And it's worth asking the question. Notice anything interesting here. This solution is exactly the same as our solution to absolute value of x minus 5 equals 7. There is no fundamental difference between the two equations. They look different, but they are essentially the same equation, solved it exactly the same way.