Step by step
Think distance, not direction
Two elevators leave the lobby, floor 0. One goes up to floor 3 and the other goes down to floor −3. Both rides are 3 floors long, because up or down doesn't change how far you went. That's the idea behind absolute value.
▶ Watch this step 0:00Count hops from zero on a number line
Tip the building on its side and the floors become a number line. The bars | | mean absolute value. For |−4|, start at 0 and count left: 1, 2, 3, 4, so |−4| = 4, the same as |4|. Distance is never negative, and |0| is just 0.
▶ Watch this step 0:55Compare distances, not positions
Without the bars, −8 is less than 5. With the bars, you compare distances: −8 is 8 hops from zero and 5 is 5 hops. Since 8 > 5, |−8| > |5|. Smaller number, but bigger distance.
▶ Watch this step 3:37Work inside the bars first
For |2 − 9|, solve 2 − 9 first. Start at 2 and hop 9 to the left, since it's subtraction, and you land on −7. Now finish the job: |−7| = 7. The bars don't just go away when the inside is done.
▶ Watch this step 5:08Keep signs that are outside the bars
In −|−7|, the negative sits outside the bars. Solve the inside first: |−7| = 7. The outside negative just comes along, so −|−7| = −7.
▶ Watch this step 6:45Work backward for |x| = 5
Ask which numbers are 5 away from zero. Count 5 hops right and you land on 5; count 5 hops left and you land on −5. So x = 5 or x = −5, and you need both answers.
▶ Watch this step 7:38
“Absolute value just means the distance away from zero.”
Worked example · with Digit the Fox
Try one more, step by step
Problem: Find |6 − 15|.
- Work inside the bars first: 6 − 15.
- Start at 6 on the number line and hop 15 to the left. You land on −9.
- Now the problem is |−9|.
- Count the hops from −9 back to zero: 9.
Answer: |6 − 15| = 9
Check: Count the hops again: −9 sits 9 steps left of zero, and a distance is never negative.
Common mistakes
- Stopping too early: working out |−3 − 5| = |−8|, then writing −8 as the final answer. Take one more step and find the absolute value: |−8| = 8. Mr DJ warns about this
- Comparing the numbers instead of the distances, and deciding |−8| < |5| because −8 < 5. Find each absolute value first, 8 and 5, then compare: |−8| > |5|. Mr DJ warns about this
- Turning the outside negative positive, and writing −|−7| = 7. Only the inside becomes a distance. The minus outside the bars stays, so the answer is −7. Mr DJ warns about this
- Giving only x = 5 as the answer to |x| = 5. −5 is also 5 hops from zero, so write x = 5 or x = −5. Mr DJ warns about this
Questions
Can absolute value be negative?
No. Absolute value is a distance from zero, and a distance is never negative. A negative answer can still show up when there's a minus sign outside the bars, like −|−7| = −7.
Why does |x| = 5 have two answers?
Because 5 and −5 are both 5 hops away from zero on the number line, so both make the equation true.
Read the full transcript
0:00 Hi, I'm Mr. DJ and today absolute value. Everything you need and it starts with a riddle about elevators. Here's a building with two elevators. They both start at the lobby, but we're going to call this floor zero right here. Elevator A goes up one two three floors. It stops on floor three. Elevator B goes down one two three. It's like going down into the basement somewhere. Right? We're going to call that -3. That floor is going to be -3. So, which ride was longer? Was it up three or was it down three floors? Neither.
0:45 Both rides are three floors long. You know, up or down, it doesn't change how far you went. That's the principle behind absolute value. Absolute value is all about distance, all about how far from zero. Now, pretend like we tip that building on its side and we make this number line. All right, the floors become the number line. So, remember like the lobby was zero right here and the negative floors like going down were are going to be the negative numbers going left and the floors going up are going to be the ones going to the right. Remember, absolute value is the distance from zero. You always start counting from zero. If we were going to floor three again, right? The distance from zero is one two three.
1:32 The absolute value of three is just three. But, if we were going to floor -3, it's one two three in this direction. The absolute value of -3 is still three. We still went three hops. It doesn't matter that we were going negative. The absolute value of four. Count the blue box from zero. We have one, two, three, four of them. So, the absolute value of four is four. And remember, this these lines right here, these mean absolute value. I'm kind of making them messy, but that's what they mean. The absolute value of -4. You want to start at zero, right here,
2:18 and you want to count to the left. One, two, three, four. Yeah, the absolute value of -4 is four. So, we said the absolute value of -4 is four, and we said the absolute value of four is four. Yeah, let's take a look again. Ready? The distance going in this direction is one, two, three, four. And the distance going in this direction is one, two, three, four. It doesn't matter whether it's positive four or negative four. Absolute value just means the distance away from zero. Remember, the distance is never negative. You you can't like walk negative steps, right? It just It just distance is always a positive number. There's one
3:03 special case, zero. The absolute value of zero, though, is just zero. All right, example one. Let's try this out. What is the absolute value of -6? And remember, when we're saying the absolute value, we're talking about the distance away from zero. All right, so what you do is you find -6 on the number line, and you count the steps. It's one, two, three, four, five, six away from zero. So, the absolute value of -6 is just six. Let's compare now, cuz this one trips up a lot of people. Let's compare the absolute value of -8 to the absolute value of five. Now, without the absolute value signs, -8 is further to the left than
3:49 positive five. -8 is less than positive five. But when you include the absolute value part of it, now we're just talking about the distance away from zero. So five is 1 2 3 4 5 hops away from zero. -8 is 1 2 3 4 5 6 7 8 hops away from zero. So if we're comparing -8, the absolute value of -8 to the absolute value of five, remember, the absolute value of -8 is eight. And the absolute value of five
4:35 is five. So when you're comparing those, you're really just comparing eight to five. And guess what? Eight is greater than five. So that means the absolute value of -8 is greater than the absolute value of five. Take a look right here. See? The absolute value of -8 is eight. And the absolute value of five is five. So absolute value of -8 is greater than the absolute value of five. Smaller number, but bigger distance. What if we need to work inside the absolute value bars? So for example, we have the absolute value of 2 - 9. So yes, you're going to want to solve everything inside of the absolute value bars first. 2 - 9.
5:20 This means start at two on the number line, right here. This says we're going to go to the left cuz it's subtraction. And we're going to go to the left by nine hops. So we're starting at two on the number line, and we go 1 2 3 4 5 6 7 8 9, and you land at -7. Right there. So now, once you solved 2 - 9 inside the absolute value, you now have the absolute value of -7. And you just solve for that. Well, what's the absolute value of -7? The absolute value of -7 is just positive 7 or just 7. It's seven hops away from zero. Remember, absolute value is the distance from zero. Take a look at it
6:05 right there. Watch out, here is a common mistake that students make. -3 and -5 inside the absolute value bars. Yes, -3 -5 gets you -8, but you have to remember to take that one more step. You have to remember that it's still inside the absolute value bars. And the absolute value of -8 is 8. It is not -8 as a final answer. See, you take a look. The bars aren't acting like brackets, right? You have to finish that. They don't just go away. You have to do that next step and find the absolute value. What if there's signs outside of the absolute value bar? Right, this example, we have negative on the outside here. We have the
6:50 absolute value of -7. All right, let's take a look how we handle this. So, what you're going to do is you're just going to solve everything on the inside of the absolute value bars. So, the absolute value of -7 is 7. Then, the outside minus or negative sign just kind of gets tacked on. So, this just comes right there. Simple as that. So, one more time real quickly, we had this, something like this. Right? You would say, you would solve, you would say the absolute value of -7 is just 7. And then this just comes straight down like that. That's how you get -7. See, the minus is outside the bars, so it just stays there. Backwards. Find the
7:36 number. Hmm. Example five. The absolute value of x is equal to 5. Well, which numbers are five away from zero? Well, let's take a look. We have 1 2 3 4 5. So, we know five can be X. X is going to be equal to five. But, what about -5? 1 2 3 4 5. Hey, X can be equal to five or -5. See, there's five. It's five blocks across, but can also be -5. This one has two answers. They're both correct. You have to have both of them. X = 5 or X can be equal to -5. Number one, we're talking about the distance
8:21 from zero. In number two, you do inside the bars first. Three, the outside signs stay. Four, the absolute value of X = 5 has two answers. Now, it's your turn. I'm going to give you a mild problem, and then a medium problem, and then a spicy problem. I want you to do the mild problem, and then we'll go over it. We'll do the medium problem, and then we'll go over it. The spicy problem, I'm going to leave the answer in the comments. So, take a look. I'll pin it. All right, here's the mild one. The absolute value of -15. Pause the video now at this point if you want to think about it. And don't forget you don't you don't want to just the answer. Just also think
9:06 about why. Why is the absolute value of -15 to your answer? Like, what does that mean? The absolute value of 3 8. Don't forget to pause the video. Take a look. And here's the spicy one. Then, you get the negative absolute value of -6 plus the absolute value of 4 - 9. Multi-step problem here, but you can do it. All right, let's go over the mild and medium problems. Again, the spicy answer will be in the comments below. All right, the absolute value of -15 is 15 because -15 is 15 hops away from zero. The absolute value of 3 - 8, this is going to get us the absolute value of -5, right? And the absolute value of -5 is just
9:51 five. Did you get that? All right, take a look at those comments for the answer to the spicy one. See if you got it right. If you didn't, that's okay. Remember, mistakes in math are okay if you're learning from them. So, take a moment, think about where you made your mistake. If you found value in this video, please take a moment to like the video, subscribe to the channel cuz I don't want you to miss any of my lessons. And if you know somebody else who is working on this stuff, share it with them. Maybe they'll find some value in it as well. And I'll see you in the next lesson. Take care. Bye.
From the video’s captions.
