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# Interpreting absolute value as distance

In this video, we work through a bunch of examples that stretch our thinking on absolute value.

## Want to join the conversation?

- What if the absolute value was 0(24 votes)
- The absolute value is the distance from zero. If the absolute value of a number is zero, that number itself is also zero.(30 votes)

- help! I don't get anything of this absolute value part . its very frustrating :((18 votes)
- I'm not sure what don't you understand, but absolute value is simply THE DISTANCE FROM THE NUMBER TO 0. For example, if you imagine two numbers on a number line: -4 to 0. That would be four blocks away right? Which means, the absolute value of -4 is 4. If you have any other questions you can reply to this. I will be checking them :)(34 votes)

- This is 11 minutes long... WHY!!(11 votes)
- At the7:11point, Sal quickly scribbles out the third option.

Although, I tried out this proposition with the values:

x = 5

and y = 9

Which gives the equation:

|11 - 9| = |-5 - (-3)|

and both sides of the equation equal 2.

So the third option, does in fact work.(6 votes) - Help! Is the equation |a-b|=|a|-|b| true all the time or just only for this special case?

at1:46, sal agreed that |a-b| is the same thing as |a|-|b|

also in a previous video sal said |a-b| = |b-a|

so, it seems like we can say |b-a| = |b|-|a|.

if this is logical then why the following equation is not true:

let's imagine a=-9 and b=-2

|a|-|b|=|b|-|a|

⇒ |-9|-|-2| = |-2|-|-9|

⇒ 9-2 = 2-9

⇒ 7 ≠ -7

thanks in advance 🙂(3 votes)- Good question. |a-b|=|b-a| is always true because the distance between the two are the same no matter which side you start from. However, |b-a|=|b|-|a| when a and b are the same signs. The second rule would be |b-a|=|b|+|a| if they are opposite signs. Your assumption that is special case is correct, but there are only two choices. I am not sure |a|-|b| ever equals |b|-|a| unless |b|=|a|.(6 votes)

- What is the distance between a and b?(4 votes)
- A and B are variables, so it depends what A and B are. if A=1 and B=(-2) then the answer is 3.(4 votes)

- At0:35to0:38he says the absolute value of a-b is the same thing as b-a. I don't understand how that is possible. It's making me nuts crazy and confused.(5 votes)
- what if the absolute value is x(3 votes)
- than it would be |x|(2 votes)

- Are there any more videos to explain just what absolute value is?(3 votes)
- Is there a easier way to do this(3 votes)
- search it up, google can tell you.(2 votes)

## Video transcript

- [Voiceover] What I
hope to do in this video is get a little bit more practice thinking about absolute value
of the difference of numbers as the distance between those two numbers. So in this first question, we are asked which of the following expressions are equivalent to the absolute value of A minus B. And just as a reminder, the absolute value of A minus B, this expression, this is going to give us the distance between A and B. So it's going to give us this
distance right over here. It's going to give us this
distance right over here. That is the absolute value of A minus B, which is of course the same thing as the absolute value of B minus A. So which of these
expressions are equivalent? So this first one has
the absolute value of A minus the absolute value of B. Well what is the absolute value of A? Well that's the distance
that A is from zero. So that's going to be this distance. This distance right over here
is the absolute value of A. That's the absolute value
of A right over there and then the absolute value of B is going to be this distance. That's the distance that B is from zero. So that right over there is going to be the absolute value of B. That's the absolute value of B. So if you take, if you say
the absolute value of A minus the absolute value of B, what are you going to be left with? Well, you're going to be left with, you're going to be left
with this distance. You're going to be left with
this distance right over here. This distance is the absolute value of A minus the absolute value of B. Absolute value of A minus this distance is gonna give you this green distance. Well that's exactly what we have up here. The absolute value of A minus B is the distance between A and B and that's what this
green distance is as well. So, this is going to be equivalent to the absolute value of A minus B. And if you wanna really verify it, you could try it with some numbers. I mean what they tell us about A and B is that both of them are
going to be negative. They're both to the left of zero. And we also see that B is greater than A, or it's less negative than A. So you could even try
it with some numbers. You could say well maybe B is negative one and A is negative five and then verify that this would be true. Now what about the absolute value of A plus the absolute value of B? Well that would be taking this distance, this magenta distance,
absolute value of A, and then adding it to this blue distance, that absolute value of B. So this would give you a larger distance than the distance
between those two points. Or, you could try it with numbers. I mean, imagine a world, just like I said, imagine a world where A
is equal to negative five and B is equal to negative one. Well in this world, the
distance between the two, the absolute value of A minus B would be equal to the absolute
value of negative five. Negative five minus negative one. Minus negative one. Which is the same thing as the absolute value of
negative five plus one. Which is equal to the absolute
value of negative four. Which is equal to four. So for these particular numbers, and I just picked them, I just picked two negative numbers where A
is more negative than B, the way it's drawn, this distance in green, or this distance right over here, would be four. Now the absolute value of A, absolute value of A plus
the absolute value of B, in this circumstance,
is going to be equal to five. It's going to be the absolute
value of negative five, which would just be five plus the absolute value of negative one, which would be one. This would be equal to six. So for these numbers,
once again I just picked two random numbers that
met the constraints that both are negative and that A is more negative than B is, it didn't hold up. So this is not going to be the case. And I'm not going to
select none of the above, because I found a choice that I know is going to be true. Let's do another one of these. Let's do several more of these. Which of the following expressions are equivalent to the
absolute value of A minus B? Once again, absolute value of A minus B, that is the distance between A and B. That is this distance that
I'm drawing right now. That is this distance right over here. That is the absolute value of A minus B. Well what is this first choice? Just A minus B without the absolute value. Well we see that A is less than B, it's more to the left. In fact, A is negative and B is positive. So if you take a negative number and then you subtract a
positive number from it, you're going to get a negative number. This thing right over here
is going to be negative. Or if you subtract a larger
number from a smaller number, you're going to get a negative value. But the distance between
these two numbers, we took the absolute value,
this is a positive value. This is just a distance. So this isn't going to be the case. Now let's look at this choice. The negative of B minus A. Well B minus A is going to be positive. How do we know that? Well B is larger than A. B is greater than A. So if B is greater than A, B minus A is going to be positive. But then we're taking the negative of it, so this whole expression
is going to be negative. It is going to be negative again. Another way to think about it, B is a positive number, you subtract a negative number from it, that's the same thing as adding the absolute value of
that negative number. This part is going to be positive. But then you have this
negative out front of it, it is going to be negative. And, like in the last example, you could try out numbers
that meet these constraints. Maybe B is positive three
and A is negative two. And I encourage you, try this out. Figure out what the absolute
value of A minus B is, it'll be five. And figure out which of these give you that same result. And neither of them will. So the answer here is none of the above. Let's keep going, this is a lot of fun. Alright, select the best interpretation of the following equation. So we have the absolute
value of 11 minus X. So this is the distance between 11 and X equals the absolute
value of Y minus three. So this is the distance
between Y and three. So this is telling us
that the distance between 11 and X is the same as the
distance between Y and three. So they say the distance between 11 and X is equal to the distance
between Y and three. Yeah, that's exactly, that's
exactly what I just said. So I would select that. But let's look at the other choices. The distance between 11 and a negative x is qual to the distance
between Y and negative three. Well the distance between, let me underline this, the distance between 11 and negative X, let me just in a different color. The distance, I'm having trouble changing colors. So, the distance between
11 and negative X, that's not going to be this over here, that's going to be, you could take the absolute value of
11 and then from that you would subtract negative X. That's this thing right over here. So this would actually simplify
to the absolute value of 11 plus X, which is not
what we have over here. And then the distance
between Y and negative three, same idea. That's gonna be Y minus negative three. Which is not what we have over here. So this is not, this is not what this equation represents. The distance between 11 and Y, okay so now they're really mixing. They're saying 11 and Y
is equal to the distance between negative X and negative three. So now they've just completely
mixed everything up. So that's not gonna be the case. Let's do one more of these. Let's do one more. So we are asked, which of
the following expressions is equal to the rectangle's area? Alright, so if we want to figure out the area of a rectangle, just multiply the width times the height. Or you could say the
length times the height. So let's see which of
these represent that. So the absolute value of J minus L. So, let me get a color here. So J minus L. The absolute value of J minus L. So J minus L, so J is this X coordinate, it's gonna be negative six, and L is this X coordinate, it's going to be, it's going to be positive six. So the absolute value of J minus L is going to be our, is going to the the difference
in the horizontal axis, or it's going to be the
distance on the horizontal axis between this point and that point. Or you say the horizontal
distance between those two points. So it would be the length
of this line segment. So that is the absolute
value of J minus L. Once again, the X coordinate
here is negative six, the X coordinate here is positive six, and you can even figure it out. It's going to be negative six minus six, which would be negative 12, and then you take the
absolute value of that, this is going to be 12. And you don't even have
to figure that out here, we just know that the length of this line is the absolute value of J minus L. So that's that. And then they have the
absolute value of M minus Q. So the absolute value, the absolute value of M minus Q. So they have M over here,
that's the Y coordinate here, and Q is the Y coordinate down here. So the absolute value of M minus Q is going to be the distance, the vertical distance
between these two points, which is really just, 'cause the X value isn't changing, this is actually going to be the length. This is going to be the
length of that side. That's going to be the
absolute value of M minus Q. So yeah, if you multiply this
length times this length, you're going to get the
area of the rectangle. So I didn't even have to
look at the other choices, I would definitely go with this one. But let's see where the other ones probably aren't correct. So this is the absolute
value of J minus M. So here you're taking the difference of the X coordinate here and
the Y coordinate over there. So that's kind of bizarre. This this already looks suspicious. Here you're saying the
absolute value of J minus N. Absolute value of J, absolute value of J minus N. Well, their X coordinates are the same, so this is actually going to be, this we actually know is going to be zero. J is equal to N, they're
both equal to negative six. That's not gonna give you
the length of this line, because we have no change along X here. All the change is along Y. If we wanted to figure out the length of this line right over here we would have to find
the absolute value of K, of the change in our Y coordinates. So absolute value of K minus O would give you the length of this line. And if you wanted the length
of this right over here, you'd want your change in X, so that would be the
absolute value of N minus P, or you could say the
absolute value of P minus N. But they didn't use those choices. So yeah, we feel good about that.