Today we're going to talk about absolute value. Absolute value is defined as the distance that a number is away from 0 on a number line. So we'll talk a little bit more about what exactly absolute value is, and then we'll do some examples.
So let's look a little bit more closely about the definition of absolute value. So if I want to find the absolute value of 3, we use two straight lines as our absolute value sign, and the way that I say this is the absolute value of 3.
Using my definition, I want to find the distance that 3 is away from 0 on this number line. So I can simply count 1, 2, 3, and I see that 3 is 3 away from 0 on my number line. So the absolute value of 3 is just 3.
Now if I want to find the absolute value of negative 3, again using my definition, I need to find the distance between negative 3 and 0 on my number line. Counting again, 1, 2, 3, I see that negative 3 is 3 away from 0 on my number line.
So the important thing to see here is that whether your number inside the absolute value sign is positive or negative, absolute value will always evaluate to be a non-negative number. And that's because, again, absolute value is the distance from 0. So it's always going to be non-negative, which basically just means either 0 or a positive number.
So here's my first example. I've got the absolute value of 8 plus the absolute value of negative 4 plus the absolute value of negative 2 minus the absolute value of 1. So to evaluate this expression, I'm going to find the absolute value of each of these numbers and then either add or subtract.
So the absolute value of 8-- how far away is 8 away from 0 on a number line? We know that that's going to be 8. Absolute value of negative 4-- the distance between 0 and negative 4 is still positive 4. So this will evaluate to be positive 4. Absolute value of negative 2-- same thing here. Negative 2 is still two away from 0 on our number line, so absolute value of negative 2 is just positive 2. And the absolute value of 1 is going to be 1.
So now that I've evaluated all of my absolute values, I can go ahead and just add and subtract from left to right. And that's going to give me 8 plus 4 is 12, plus 2 is 14, minus 1 will give me 13 for my final answer.
So for my second example, I've got one absolute value sign, and on the inside I've got 10 minus 3 plus negative 5 plus 2. So thinking of order of operations, I know that I need to use my absolute value sign as a grouping symbol, meaning that I need to evaluate everything inside of the absolute value sign before I would evaluate anything that's on the outside.
So inside of my absolute value sign, I've got addition and subtraction, and I'm just going to do that from left to right. So 10 minus 3 is going to give me 7. 7 plus a negative 5 is the same as 7 minus 5, so that's going to give me 2. And if I add 2 tp that, that's going to give me 4. So this simplifies to the absolute value of 4, and the absolute value of 4 is just going to give me 4.
So here's my third example. I've got negative 2 times the absolute value of 4 plus negative 6. I'm going to start by evaluating what's in my absolute value sign. So 4 plus negative 6 is going to give me a negative 2. Bring down my absolute value sign, and bring down the negative 2 being multiplied in front.
Now the absolute value of negative 2 is just going to give us 2. And I'll bring down my negative 2, again, that's being multiplied in front. Negative 2 times 2 is going to give me a negative 4.
So it's important to see here that even though we have an absolute value sign in our expression, we can still get a negative number depending on what you have outside of your absolute value sign. So even though absolute value always evaluates to be a non-negative number, your answer can still be negative.
The last thing I want to talk about involving absolute value are the product and quotient properties of absolute value. So let's look at that product property first. So that tells me if I have two numbers being multiplied, a and b, inside of their own absolute value sign, then I can separate that out and do the absolute value of a times the absolute value of b. So let's put some numbers with it and see how that works.
Let's say I've got the absolute value of 3 times negative 4, and they're both in the same absolute value sign. So to evaluate this I'm going to start by multiplying 3 times negative 4, and that's going to give me negative 12 inside my absolute value. And the absolute value of negative 12 is just positive 12.
So let's see if we separate them out if we get the same answer. So my a number was 3, so the absolute value of 3. My b number was negative 4, absolute value of negative 4. The absolute value of 3 is going to give me 3 times the absolute value of negative 4 is going to give me positive 4. And 3 times positive 4 is also going to give me 12. So we see that this property does hold true.
Let's look at it for the quotient property. So let's put some numbers with this. The property tells us that if we have two numbers a and b that are being divided in an absolute value sign, you can again separate them out. So that's the absolute value of a divided by the absolute value of b. So if my a number is 9 and my b number is 3, inside absolute value sign, 9 divided by 3 is going to give me 3. And the absolute value of 3 is just 3.
Let's see if it works on the other side. So here my a number again was 9. Absolute value of 9 over the absolute value of 3 was my b number. So I'm going to simplify the absolute values first. Absolute value of 9 is just 9, and the absolute value of 3 is just 3. And 9 divided by 3 will also give me an answer of 3. So again, we can see that the quotient property of absolute value does work.
So let's go over our key points from today. As usual, make sure you get them in your notes so you can refer to them later.
Absolute value is defined as the distance a number is away from 0 on a number line. Because it's a distance, it will always be non-negative number. In other words, it will either be zero or a positive number. The absolute value sign acts as a grouping symbol. In other words, when you're simplifying you would do things in the absolute value symbol first, just like parentheses. And finally the product quotient properties of absolute value will allow us to simplify absolute value expressions.
So I hope that these key points and examples helped you understand a little bit more about absolute value. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.
The distance between a number and zero on the number line; it is always non-negative.