Comparing absolute values

Let's do some examples comparing absolute values. So let's say we were to ask ourselves how the absolute value of negative 9, I should say, how that compares to the absolute value of-- let me think of a good number-- let's say the absolute value of negative 7. So let's think about this a little bit, and let's think about what negative 9 looks like, or where it is on the number line, where negative 7 is on the number line. Let's look at what the absolute values mean, and then we should probably be able to do this comparison. So there's a couple of ways to think about it. One is you could draw them on the number line. So if this is 0, if this is negative 7, and then this is negative 9 right over here. Now, when you take the absolute value of a number, you're really saying how far is that number from 0, whether it's to the left or to the right of 0. So, for example, negative 9 is 9 to the left of 0. So the absolute value of negative 9 is exactly 9. This evaluates to 9, Negative 7 is exactly 7 to the left of 0. So the absolute value of negative 7 is positive 7. And so if you were to compare 9 and 7, this is a little bit more straightforward. 9 is clearly greater than 7. And if you ever get confused with the greater than or less than symbols, just remember that the symbol is larger on the left-hand side. So that's the greater than side. If I were to write this-- and this is actually also a true statement. If you took these without the absolute value signs, it is also true that negative 9 is less than negative 7. Notice the smaller side is on the smaller number. And so that's the interesting thing. Negative 9 is less than negative 7, but their absolute value, since negative 9 is further to the left of 0, it is-- the absolute value of negative 9, which is 9, is greater than the absolute value of negative 7. Another way to think about it is if you take the absolute value of a number, it's really just going to be the positive version of that number. So if you took the absolute value of 9, that equals 9. Or the absolute value of negative 9, that is also equal to 9. Well, when you think of it visually, that's because both of these numbers are exactly 9 away from 0. This is 9 to the right of 0, and this is 9 to the left of 0. Let's do a few more of these. So let's say that we wanted to compare the absolute value of 2 to the absolute value of 3. Well, the absolute value of a positive number is just going to be that same value. 2 is two to the right of 0, so this is just going to evaluate to 2. And then the absolute value of 3, that's just going to evaluate to 3. It's actually pretty straightforward. So 2 is clearly the smaller number here. And so we clearly get 2 is less than 3, or the absolute value of 2 is less than 3. So we have a less than right over here. Let's say you wanted to compare-- I'm trying to find a suitable color-- the absolute value of negative 8 to the absolute value of 8. Well, one way to think about is that they're both 8 away from 0. This 8 to the left of 0. This is 8 to the right of 0. So both of these things evaluate to 8. Absolute value of negative 8 is 8. Absolute value of 8 is 8. And so, clearly, 8 is equal to 8. Let me do a couple more examples. Let's say I wanted to compare the absolute value of negative 1, and I want to compare that to positive 2. So the absolute value of negative 1 is just the positive version of 1, or it's just the positive version of negative 1, which is just 1. So 1 is clearly less than 2. Or the other way to think about it, the absolute value of negative 1 is clearly less than 2.