Absolute value and number lines
We're told to plot these values on a number line. And you see every one of these values have an absolute value sign. So let's take a little bit of a review of what absolute value even is. The way I think about it, there's two ways to think about it. The first way to think about it is, how far is something from 0? So let me plot this negative 3 here. So let me do a number line. This isn't the number line for our actual answer, or to this command-- plot these values on a number line. I'm just first going to plot the numbers inside the absolute value sign, and then we're going to take the absolute value and plot those, just like they're asking us to do. So on this number line, if this is 0, if we go to the negative, we're going to go to the left of 0. So this is negative 1, negative 2, negative 3. Negative 3 sits right over there, so this is negative 3 right there. The absolute value of negative 3 is essentially saying, how far are you away from 0? How far is negative 3 from 0? And you say, well, it's 1, 2, 3 away from 0. So you'd say that the absolute value of negative 3 is equal to positive 3. Now that's really the conceptual way to imagine absolute value. How far are you away from 0? But the easy way to calculate absolute value signs, if you don't care too much about the concept, is whether it's negative or positive, the absolute value of it's always going to be positive. Absolute value of negative 3 is positive 3. Absolute value of positive 3 is still positive 3. So you're always going to get the positive version of the number, so to speak. But conceptualy, you're just saying how far away are you from 0. So let's do what they asked. So that first value, on this number line, so all of these are absolute values. So they're all going to be positive values. So they're all going to be greater than 0. So let me draw my number line, like this. I can do a straighter number line than that. Let's see. Well, that's a little bit straighter. And let's say, if this is 0, this would be negative 1, then you'd have 1, 2, 3, 4, 5, 6, 7. I think that'll do the trick. So this first quantity here-- I'll do it in orange --the absolute value of negative 3, we just figured out, that is positive 3. So I'll plot it right over there, positive 3. Then this next value, right here, the absolute value of 7. If we look over here, 1, 2, 3, 4, 5, 6, 7. 7 is how far away from 0? It is 7 away from 0. So the absolute value of 7 is equal to 7. So you already see the pattern there. If it's negative, it just becomes positive. If it's already positive, it just equals itself. So plotting this value, I'll just place it right over there. So the absolute value of 7 is 7. Absolute value of negative 3 is positive 3. Let me mark out the 0 a little bit better, so you see relative to 0. Now we have the absolute value of 8 minus 12. Well, first of all, let's figure out what 8 minus 12 is. So if you take 12 away from 8, you're at negative 4. 12 less than 8 is negative 4. And you can do that on a number line if you don't quite remember how to do this. But if you, you know, if you take 8 away from 8, you're at 0, and then you take another one, you're at negative 1, negative 2, negative 3, all the way to negative 4. So this is equal to the absolute value of negative 4. If we just plot negative 4, we go 1, 2, 3, negative 4 is right over there. But if we're taking its absolute value, we're saying how far is negative 4 from 0? Well it's 4 away from 0. 1, 2, 3, 4. So this is equal to positive 4. So we'll plot it right here. This number line is the answer to this command up here. So the absolute value of 8 minus 12, which is negative 4, is positive 4. Then we have the absolute value of 0. So how far is 0 from 0? Well, it's 0 away from 0. The absolute value of 0 is 0, so you can just plot it right over there. And we have one left. Let me pick a suitable color here. The absolute value of 7 minus 2. Well, 7 minus 2 is 5, so this is the same thing as the absolute value of 5. How far is 5 away from 0? Well, it's just 5 away. It's almost, you know, too easy. That's what makes it confusing. If I were to plot 5, it's 1, 2, 3, 4, 5. It is 1, 2, 3, 4, 5 spaces from 0. So the absolute value of 5 is 5. So you plot it just like that. So conceptually, it's how far you are away from 0. But if you think about it in kind of just very simple terms, if it's a negative number, it becomes a positive version of it. If it's a positive number already, it just equals itself when you take the absolute value.