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# Interpreting absolute value as distance

## Thuyết minh video

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 this is a reminder the absolute value of a minus B this 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 going to 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 want to 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 can 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 then the then 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 5 minus negative 1 minus negative 1 which is the same thing as the absolute value of negative 5 plus 1 which is equal to the absolute value of negative 4 which is equal to 4 so for these particular numbers and I just picked them I just pick 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 4 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 5 it's going to be the absolute value of negative 5 which just be 5 plus the absolute value of negative 1 which would be 1 this would be equal to 6 so for these numbers once again I just pick 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 go to 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 for but 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 but 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 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 then you have this negative out front of it it is going to be negative and like the last example you could try out numbers that meet these constraints maybe B is positive three and a is negative two and encourage you to 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 there's a lot of fun all right 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 3 so this is the distance between y + 3 so this is telling us that the distance between 11 and X is the same as the distance between y + 3 so they say the distance between 11 and X is equal to the distance between y + 3 yeah that's exactly that's exactly what I just said so I would select that let's look at the other choices the distance between 11 and at negative x is equal to the distance between y and negative 3 well the distance between let me let me underline this the distance between 11 and negative x I mean just in a different color the distance 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 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 3 same idea that's going to be y minus negative 3 which is not what we have over here so this is not this is not what this would this eat 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 3 so now they've just completely mixed everything up so that's not going to 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 rectangles area all right so if we want to figure out the area of a rectangle just multiply the width times the height or you could say the 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 going to be negative 6 and L is this x-coordinate it's going to be it's going to be positive 6 so the absolute value of J minus L is going to be it's going to be R it's going to be the difference in the horizontal axis on the or it's going to be the distance on the other horizontal axis between this point and that point if we're going or you say the horizontal distance between those two points so it would be the length of this of this line segment so that is the absolute value of J minus L once again the x-coordinate here is negative 6 the x-coordinate here is positive 6 and you can even figure it out it's going to be negative 6 minus 6 which would be 12 negative 12 and then you take the absolute value of that this is going to be 12 you don't even have to figure that out here this we just know that the length of this line is the absolute value of J - 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 because this 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 if you multiply this length times this length you are going to get the area of the rectangle so I didn't have to look at the other choices I would definitely go with this one but let's see where the other ones probably are not aren't correct so this is the absolute 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 so this already looks looks suspicious here you're taking the absolute value of J minus n absolute value of J absolute value of J minus n well they're x coordinates are the same so this is actually going to be this we actually know is going to be 0 J is equal to n they're both equal to negative 6 that's not going to give you the length of this line because we have no change along the along X here all the changes along Y if we wanted to figure out the length of this line right over here we would have to find the the absolute value of K of the change in our 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