Absolute Value

[doclkk]

I guess whenever I see absolute value - I get confused.

I know the basics of absolute value. |5| = 5 or -5.

Here's my question

If you are redefining the question (or whatever the right expression is) and your equation becomes |-5| = -5 or 5, what does that mean?

In class, there was a DS questino that had |-5| = 5. I said - Yes, that's correct, and it ended up coming up as incorrect. Can someone go through each of the nuances of absolute value with me.

|X| + |Y| = |X+Y|

So what does this equation imply?

I don't have my EIV book with me because I can only fit 4 books with me a week in my suitcase and am doing word translations, CR, RC and SC instead. I was reviewing my MGMAT cats and am still confused with absolute value concept when it comes down to an absolute value of a neg number.

[mike22629]

First of all, l5l can NOT equal -5

Remember absolute value must ALWAYS be positive. I believe that most people find absolute value difficult because they do not truly understand what it means. Absolute value measures distance. Distance can NOT be negative.

So what does lxl actually mean? Well the best way to visualize absolute value is to use a number line. The lxl actually means the distance from zero.

So:

------(x=-4)----------0-----------------

This means that lxl = 4 because -4 is 4 spots away from zero.


As far as the equation lxl + lyl = lx+yl

Lets break it down into both sides of the equation...

lxl + lyl simply means that you must add up the distance between x and 0 and the distance of y and 0.

lx+yl = lx-(-y)l or ly-(-x)l

So what does lx-(-y)l actually mean? It is the distance between x and y.

Note: you have to make one variable negative because when determing the distance between two variables inside of an absolute value sign, the variable must be subtracted from one anotherl
For example lz - yl means the distance between z and y

But lz+yl does not mean the distance between z and y

So what does the equation lxl + lyl = lx+yl actually imply? To put it simply it implies that the two variables must be on opposite sides of zero on the number line, or more clearly, that one is negative and the other is positive.

Let me know if this helps.

[Svedankae]

lx+yl = lx-(-y)l or ly-(-x)l

So what does lx-(-y)l actually mean? It is the distance between x and y.

Note: you have to make one variable negative because when determing the distance between two variables inside of an absolute value sign, the variable must be subtracted from one anotherl
For example lz - yl means the distance between z and y

But lz+yl does not mean the distance between z and y

Hey man, can you explain that again? I dont really get why you have to subtract x from y or vice versa

[Osirus@VeritasPrep]

What he was illustrating was the how to solve | x + y|. He was showing that even if the numbers are negative the values are positive so you would add and not subtract. lets say x = 4 and y = -5. With the expression x + y alone without the absolute value symbols then the answer would be -1. With absolute value however the value of any number is the distance from 0 on the number line so all the positive or negative sign does is tells you which direction the number is from zero but the distance from 0 is always a positive number. So for |x + y| = |x - (-y)| you do this to change the sign of the y value to a positive so you can add it to x.

Example:

x = 8, y = -9

x is 8 units from 0

y is 9 units from zero

in order to add the absolute values of x and y you would never want to subtract anything. You would want to make any number with a negative value positive and to do this you would write it as |8 - (-9)| because the answer is 17, but if you leave the sign the same then the answer would be -1 and an absolute value could never be negative.

[maihuna]

So what does the equation lxl + lyl = lx+yl actually imply? To put it simply it implies that the two variables must be on opposite sides of zero on the number line, or more clearly, that one is negative and the other is positive.

Let me know if this helps.

Nopes...they both are of same sign either both positive or both negative...

[mike22629]

Well remember that I said that the easiest way to deal with absolute value is to use a number line.

Now instead of using variables, use actual numbers to make it clearer.

x=2 and z = 1
so
lx+zl = 3, but 3 is not the distance between x and z

However,
lx-zl = 1, and 1 is indeed the distance between 2 and 1

Now to take it a step further,
2+1 = 2 -(-1)

So if you ever recieve a question on the GMAT with lx+yl it would be in your best interest to convert it to lx-(-yl (if the problem requires it of course)

PS. According to Ian Stewart, difficult GMAT absolute value questions test your understanding of absolute value as a measure of distance which is what makes this technique valuable.

If you are still having difficulty understanding what I am saying, search for posts on absolute value where Ian Stewart replied. He explains it very well.

[Osirus@VeritasPrep]

lx+yl = lx-(-y)l or ly-(-x)l

So what does lx-(-y)l actually mean? It is the distance between x and y.

Note: you have to make one variable negative because when determing the distance between two variables inside of an absolute value sign, the variable must be subtracted from one anotherl
For example lz - yl means the distance between z and y

But lz+yl does not mean the distance between z and y

Hey man, can you explain that again? I dont really get why you have to subtract x from y or vice versa

just thought of another way to think about it that may help you.

Think about calculating how far two people, Jim and Mike, are from one another and we use some point in between them as the point of reference, Mike's house. If Jim is 8 miles east of the house and Mike is 9 miles west of the house, then they would be 17 miles from each other. With a number line though Mike's location would be -9 to illustrate direction. So to find |Jim + Mike| you would never subtract the distance Jim is from mike, that wouldn't make sense, so to account for this you have to make the negative value positive so you can add the two totals. Hence the |Jim - (-Mike)| Hopefully the example made it somewhat clearer.

[mike22629]

By the way,
Mahiuna is right. They are both the same sign, my mistake. But the principles still hold true, I just did not think it out entirely. Plugging in numbers will prove this.

x=-2, y=2

lxl + lyl =4
lx+yl = 0

x=-2, y=-2
lxl + lyl = 4
lx+yl = 4

Thanks for clearing that up mahiuna.

[doclkk]

lx+yl = lx-(-y)l or ly-(-x)l

So what does lx-(-y)l actually mean? It is the distance between x and y.

Note: you have to make one variable negative because when determing the distance between two variables inside of an absolute value sign, the variable must be subtracted from one anotherl
For example lz - yl means the distance between z and y

But lz+yl does not mean the distance between z and y

Hey man, can you explain that again? I dont really get why you have to subtract x from y or vice versa

just thought of another way to think about it that may help you.

Think about calculating how far two people, Jim and Mike, are from one another and we use some point in between them as the point of reference, Mike's house. If Jim is 8 miles east of the house and Mike is 9 miles west of the house, then they would be 17 miles from each other. With a number line though Mike's location would be -9 to illustrate direction. So to find |Jim + Mike| you would never subtract the distance Jim is from mike, that wouldn't make sense, so to account for this you have to make the negative value positive so you can add the two totals. Hence the |Jim - (-Mike)| Hopefully the example made it somewhat clearer.

I'm starting to get it - and btw - thanks for everyone's post.

So I just want to confirm

|X+Y| => means that X and Y are in different places because adding them together would equal a greater distance from each other. But I'm not understanding why you do not just ADD the two. Why is it necessary to do X-(-Y) because adding has the same effect?

Can you also more thoroughly explain why when |X| + |Y| = |X+Y|, the signs are the same?

Why does |Z+X| not equal the total distance between Z and X. If Z is 4 and X is 3, does this suggest that Z is positive 4 and Z is -3 and therefore the distance is 7 between the two?

I think I'm almost there in terms of understanding, maybe one last example will do it.

[Osirus@VeritasPrep]

lx+yl = lx-(-y)l or ly-(-x)l

So what does lx-(-y)l actually mean? It is the distance between x and y.

Note: you have to make one variable negative because when determing the distance between two variables inside of an absolute value sign, the variable must be subtracted from one anotherl
For example lz - yl means the distance between z and y

But lz+yl does not mean the distance between z and y

Hey man, can you explain that again? I dont really get why you have to subtract x from y or vice versa

just thought of another way to think about it that may help you.

Think about calculating how far two people, Jim and Mike, are from one another and we use some point in between them as the point of reference, Mike's house. If Jim is 8 miles east of the house and Mike is 9 miles west of the house, then they would be 17 miles from each other. With a number line though Mike's location would be -9 to illustrate direction. So to find |Jim + Mike| you would never subtract the distance Jim is from mike, that wouldn't make sense, so to account for this you have to make the negative value positive so you can add the two totals. Hence the |Jim - (-Mike)| Hopefully the example made it somewhat clearer.

I'm starting to get it - and btw - thanks for everyone's post.

So I just want to confirm

|X+Y| => means that X and Y are in different places because adding them together would equal a greater distance from each other. But I'm not understanding why you do not just ADD the two. Why is it necessary to do X-(-Y) because adding has the same effect?

Can you also more thoroughly explain why when |X| + |Y| = |X+Y|, the signs are the same?

Why does |Z+X| not equal the total distance between Z and X. If Z is 4 and X is 3, does this suggest that Z is positive 4 and Z is -3 and therefore the distance is 7 between the two?

I think I'm almost there in terms of understanding, maybe one last example will do it.

You are just adding the two values. The reason for the X -(-Y) is to make it X + Y. If you have a positive number and "add" a negative number that is subtraction. Example: 5 + (-10) = -5. Since an absolute value can't be negative, or to use the earlier example. Mike or John can't be negative miles from one another, it is necessary to change the negative sign, so in these cases when you want to add absolute values you must make the negative sign positive. You do this buy creating a double negative which is always positive. The take away point is that if you have a problem where | x + y | if why is negative you would not subtract y from x. In the example Mike was 9 miles west (-9) from the house (zero on a number line). If you wanted to add the two values without changing the sign you would have 8 + (-9) which would be negative 9, but we know that you can't have a negative value, so it is necessary to convert the negative number into a positive one and that is what the -(-y) is doing.

[Svedankae]

x=-2, y=-2
lxl + lyl = 4
lx+yl = 4

So thats like saying:

|-2 + -2| = |-2 - (--2)|?

I totally dont understand how this fits to what mike said earlier:

"lx+yl = lx-(-y)l or ly-(-x)l

So what does lx-(-y)l actually mean? It is the distance between x and y."

The distance between -2 and -2 is equal to zero and not equal to 4 as it is suggested abvoe.


I am hella confused. Can somebody maybe put it in simple words what this question is really asking?

[Osirus@VeritasPrep]

x=-2, y=-2
lxl + lyl = 4
lx+yl = 4

So thats like saying:

|-2 + -2| = |-2 - (--2)|?

I totally dont understand how this fits to what mike said earlier:

"lx+yl = lx-(-y)l or ly-(-x)l

So what does lx-(-y)l actually mean? It is the distance between x and y."

The distance between -2 and -2 is equal to zero and not equal to 4 as it is suggested abvoe.


I am hella confused. Can somebody maybe put it in simple words what this question is really asking?

I think I'm confusing more people than I'm helping so I apologize and hope that someone can explain it in a clearer manner.

[mike22629]

Ok picture with a number line:

x = 4, y =2

---------------------0-------y-------x-------

Now, lx+yl = 6

HOWEVER, to look at it another way
x = 4, y =-2

------------y--------0---------------x
Now, the distance between x and y is equal to lx+yl, when they are both positive this is not the case (distance between x and y is 2 when they are both positive but lx+yl = 6). So in effect, you are manipulating the equation, by adding a double negative, to find lx+yl.

[mike22629]

So basically what I am getting at:

lets say lx + yl = 6

the distance between x and y will not equal 6.

BUT, if you make one of them negative, then the distance between the two variables will = 6

[doclkk]

So basically what I am getting at:

lets say lx + yl = 6

the distance between x and y will not equal 6.

BUT, if you make one of them negative, then the distance between the two variables will = 6

I thought what you were saying the entire time is that |X+Y| = 6 therefore the DISTANCE will equal six but the equation does not equal 6?

My understanding of the take away is that

If the question is X = |4+(-2)| then the distance would be 6 and not 2.

But again, right now - i'm way more confused and it seems that another northern california person is too.