 Alright, so let's take a look at another example of an absolute value equation, and here's one that is absolutely horrific if you just know an algorithmic procedure for solving absolute value equations. On the other hand, if you actually understand what the absolute value represents, and what you're doing when you are implementing basic arithmetic operations, this is something that's actually pretty easy to solve, and you can solve this at a very, very early stage in mathematics education. So let's take this equation apart. We see that we're adding, we're putting together two different things, the difference between X and 9, and the difference between X and 5, and when we put these two things together, what we get is 10. So let's see if we can draw a picture of that, so we'll model our expression. We have some unknown value, X, and the first part of our sum is the difference between X and 9. Now I don't know where 9 is, but I will imagine for now that X is someplace beyond 9. So 9 is over here someplace. So I have the difference between X and 9, and I'll represent that using this bar this time because I want to do something with these later on. And I also have the difference between X and 5. Now once I've placed 9, 5 has to be over here someplace. So there's 5, and there's my difference between X and 9. And so there's my representation of the sum, or there's my representation of the difference is X minus 9 and X plus 5. Now what I want to do is my equation tells me that if I put those two differences together, what I get is 10. So let's go ahead and put those two together. So I have red plus green gives me a 10. And now I have a nice simple equation that I can solve using a tape diagram. So let's take a look at that. If I take this longer piece apart here, I see it consists of a portion here plus something that's exactly the same shape as the green piece there. And well, how big is that piece? Well, that is the difference between 5 and 9. So I actually know how big that is. That's going to be size 4. And so what that means is this 10 bar, this green and the red, it's the green plus this thing here, which is green and 4. So if I put those things together, I end up with something like this. And at this point, I can solve this system, this equation here. 10 is 4 plus 2 things I don't know. Well the things I don't know must be 3. So I can recover what those values 3 are. And then what that tells me, x is, well here's 9. The difference up to x is 3. That means that x must be 12. Now when we're dealing with absolute values, we actually do have to remember to consider all possibilities. Here we assume that x was greater than 9, but the other things I have fixed here are 5 and 9. So x could be greater than 9 over here someplace. X could also be less than 5, or x might be right here in the middle. So let's consider the situation where x is smaller than 5. And I'll redraw my diagram. So here x is smaller than 5, still smaller than 9. I still have my difference between x and 5. I still have my difference between x and 9. And as before, I'm going to put these two things together. When I put these two things together, I get 10. And so there's my tape diagram, and my reasoning is the same as before. Actually, it's the same equation as before. It's the same problem as before, but I'll go through the reasoning one more time. This 9 bar here is the same as the green, plus 4 more. And so this, together with green, gives me 10, so I'll fill those in. That tells me that bit has to be 3. And so that tells me that here's 5. The difference to x is 3. That tells me x has to be 2. And so there's another solution. And my last case, my last possibility, is maybe x is somewhere between 5 and 9. So I'll draw that. So here's my x, here's the difference to 9, here's the difference to 5. If I put these together, I want to make sure that I have 10. So I glue those two things together, except there's a problem. If I look at my pieces carefully. If I put these two pieces together, what I get is the difference between 5 and 9. And the problem is, the difference between 5 and 9 is 4, which means that a 4 bar has to be equal to a 10 bar. Well, that's not possible. I can't have 4 equal to 10. It's not true. And since 4 is not equal to 10, that also tells me that this particular situation doesn't correspond to a solution. So there's no solution in this case. And the only two solutions I have, x equals 12, x equals 2.