 So we have this nice algebraic definition of absolute value, the absolute value of x written this way is the absolute value of x is either equal to x if x is greater than or equal to 0 or minus x if x is less than 0. But it's much easier to work with absolute value using the following idea. The absolute value of a minus b is the distance between a and b on the number line. Now if b is 0, then since a minus 0 is just a, this leads to a related idea. The absolute value of x is the distance between x and 0 on the number line. And the important thing to remember here is that absolute value only changes the sign, never the magnitude. So let's say we want to find the absolute value of negative 5. So let's plot negative 5 on the number line. So remember the absolute value of x is the distance between x and 0 on the number line. And also on the number line, negative numbers are to the left of 0. So negative 5 will be 5 units to the left of 0. And since the distance of negative 5 from 0 is 5, so the absolute value of negative 5 is 5. What if we throw absolute value into an arithmetic expression? Find the absolute value of 3 minus 8 plus 7. Now the important thing to remember here is the absolute value bars are a grouping symbol so that operations within them must be done first. And so first we have to find absolute value of 3 minus 8. And we can do this in two ways. We can do it the easy way. And the easy way is to remember that the absolute value of a minus b is the distance between a and b on the number line. So we'll plot 3 and 8 on the number line. And we find the distance between them, which will be 5. And so the absolute value of 3 minus 8 is 5. Now if you don't want to do things the easy way, we can also do things the hard way. And so we have to figure out what 3 minus 8 is. So remember that a minus b is the same as minus b minus a. So 3 minus 8, well I can't subtract a bigger number from a smaller number, but that's the same as minus 8 minus 3. I can find 8 minus 3, that's 5. The negative is still there, so that's negative 5. And I have the absolute value of negative 5, which is going to be 5. And I still have the plus 7, so I'll add 7 and get our final answer, 12. So again, we can find this the easy way or the hard way, and just for variety, we'll find absolute value of negative 12 minus 7 the hard way, and 7 minus 15 absolute value the easy way. Let's see, a minus b is the same as negative b minus a, so I'll switch this order. That gives me negative 7 minus negative 12. Now I know that if I subtract a negative, I get a sum, so that becomes negative quantity 7 plus 12. I add to get 19, but the negative is still there, so that's negative 19, and the absolute value gives me 19. You know what? Let's do this the easy way as well. If we'd done negative 12 minus 7 the easy way, we'd plot minus 12, we'd plot 7, and then we'd find the distance between them, which is going to be 12 plus 7, or 19. Well, let's find the absolute value of 7 minus 15 the easy way, so this is the distance between 7 and 15. So we plot 7, we plot 15, and we find the distance between them, which is going to be 15 minus 7 or 8. And so this absolute value expression becomes 27.