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Identify and order absolute values
We're told to list in order from smallest to largest. So each of these quantities, it looks like we have expressions inside of absolute value signs. And just as a bit of review, absolute value just means your distance from 0. Or another way to think about it is, if it's a negative number inside of the absolute value sign, it becomes positive. If it's already positive, it stays positive. So let's think about these numbers. So the first one is the absolute value of 5. How far is 5 away from 0? Well, it's 5 away from 0. So it's equal to 5. So I were to actually draw a number line, you would see that. 0 is here, 5 is over here, this distance right there is 5. So the absolute value of 5 is 5. Now, the next quantity they want us to figure out is the absolute value of 9 minus 7. Well, this is the same thing as the absolute value of 2. 9 minus 7 is 2. And once again, 2 is just 2 units away from 0, so it's just 2. If you have a positive value in the absolute value sign, it just is itself. The absolute value of 2 is 2. Then we have the absolute value of 5 minus 15. Well, that's going to be the same thing as the absolute value. 5 minus 15 is negative 10, so it's the same thing as the absolute value of negative 10. Now, there's two ways you can think about it. If it's a negative number inside of the absolute value sign, it just becomes the positive version of it, so it just becomes 10. Another way to think about it is, if you had negative 10, you would plot it-- it would be out here someplace, negative 10, we'd have to extend the number line-- it is 10 to the left of 0. That's what the absolute value is telling us. Then we have the absolute value of 0. Well, 0 is just 0 away from the number line. Absolute value of 0 is just 0. That is just 0. That's just right there. It has no distance from the origin. And then finally, we have the absolute value of negative 3. That's 3 to the left of 0, or you can just think of it as getting rid of this negative sign, so it is equal to 3. So, now that we've expressed them all as just simple integers, let's list them in order from smallest to largest. So of all of these values, which is the smallest? This one is the smallest, the absolute value of 0. So let me write that down, absolute value of 0. What's the next smallest? What's the next one? Since we have a 2 there, this is the next smallest, right there. And that original expression was the absolute value of 9 minus 7, so the absolute value of 9 minus 7. And then what's the next smallest? We have this 3, and then we have a 5 and a 10. So the next smallest is this 3, right here. That 3, and that original expression was the absolute value of negative 3. Then we have this 5 over here, which is just the absolute value of 5. And then finally, we have this 10 over here, which was the absolute value of 5 minus 15. And we are done.