when is |x-4| equal to 4-x? MGMAT # Properties Page 43

[farukh_a]

Hello Folks,

Below is the question on Page 43 of MGMAT Number Properties Strategy Guide:

5. when is |x-4| equal to 4-x?

The answer description of this question is rather vague and doesn't clarify the concept of Modulus. In fact, the book doesn't contain too much information about Modulus.

Can anyone give further comments regarding the answer of this question.

Best Regards
Farukh

[rohinivt]

I do not have MGMAT Number Prop Guide.
Anyway, I hope the following explanation would be reasonable.

|x-4| means two cases: +(x-4) and -(x-4)

i)
(x-4) = 4-x, solve for x, and you will get x = 4

ii)
-(x-4) = 4-x. If you bring the x terms to one side and '4's on other side, you will end up in -x+x = 4-4; means 0 = 0.

So |x-4| becomes equal to 4-x when x = 4 or x = 0.

Let me know whether this explanation is correct.
Thank you.

[farukh_a]

Yes, you actually did reach close. the OA to this questions is:

x<=4

But the way you solved the question did give a new way of looking at the questions involving absolute values. So it indeed was helpful.

Thanks for replying!

[tim]

totally wrong, rohinivt. notice your second solution was 0=0, not x=0. this means any number would work, but only subject to the constraint that x-4 is negative (that's the only reason you would use the opposite of x-4 as the absolute value)..

to answer the original question about modulus, this is a very powerful tool for dealing with absolute value inequalities. keep in mind that |a-b| literally means "the distance between a and b" and you can use that to help you. in this case, the equation turns into:

the distance between x and 4 equals 4-x

now use logic to finish this out: the distance between x and 4 is either x-4 or 4-x, depending on which is the larger number. in this case, if the distance is 4-x that means that x is smaller than 4 (or equal)..

another thing to keep in mind is that if you have something like |m+n| you need to turn it into a subtraction problem before talking about a distance. in this case we can turn |m+n| into |m-(-n)| which means it is the distance between m and -n..

[philip.traumatic]

Correct me if I'm wrong but the formal definition for absolute value states the following:

|a| = a if a >= 0 and -a if a < 0

That definition makes both the response above AND the explanation in the book wrong. The answer should be x < 4 not x =< 4.

[tim]

Depends on whose "formal definition" you're using. Most mathematicians would agree that the definition using distance is a more formally correct definition. Regardless, your definition is correct on the real number line. Can you shed some more light on why you think that makes the answer wrong though?

[philip.traumatic]

I have actually never seen a formal definition that strayed from what I posted above and having done 2 tours in graduate level math departments (pure math & quantitative finance) I can assure you that your definition using distance is more of a heuristic explanation than any sort of mathematically rigorous definition.

The only time you multiply the interior quantity of an absolute value by -1 is when that quantity is strictly less than zero. When the interior quantity is non-negative it is left alone. That's why the answer is x < 4.

If you did it your way you would have a piecewise defined operator with non-disjoint domains (they would overlap at zero) and that can't happen.

If you have a legit resource, however, that states otherwise please let me know.

[dkl322]

I had a question regarding this answer as well. The answer guide states that x must be between 0 and 4. Why are answers 0 through (-4) not valid as well?

[tim]

dkl, just plug in values in the range you specify and you'll see that they don't work..

[tim]

Philip, i'm sorry to hear that your two tours of graduate level mathematics did not introduce you to complex numbers or multi-dimensional euclidean vector spaces, but c'est la vie. As i said before, your definition does in fact work ON THE REAL NUMBER LINE, so you and i are both using acceptable definitions in the context of this problem..

As to your observation, the fact that the intervals on a piecewise function overlap is not a problem as long as the function evaluates to the same number in both cases. As long as the function is well-defined, the overlap is okay..

As to your continued insistence that the solution is x<4 rather than x<=4, i invite you to demonstrate how x=4 fails to satisfy the equation..

[philip.traumatic]

Philip, i'm sorry to hear that your two tours of graduate level mathematics did not introduce you to complex numbers or multi-dimensional euclidean vector spaces, but c'est la vie.

You learn about complex numbers in jr. high and Euclidean vector spaces in calc III/linear algebra, both high school/undergraduate courses. So yeah my graduate level classes didn't introduce me to things I would have taken many years before.

In any event that was a very poor attempt to smear someone who is trying to help you clean up a simple mistake. It's bad enough to resort to an ad hominem but you look really childish when the content has no contact with reality. How could one even glean that information from anything I posted?

As i said before, your definition does in fact work ON THE REAL NUMBER LINE, so you and i are both using acceptable definitions in the context of this problem..

The only definition I have ever seen for absolute value on the real number line is the following:

1) |a| = {a if a >= 0} or {-a if a < 0}

The first is the definition given by every textbook in my office as well as the GMAT OG 12th Edition on page 126. This definition makes the answer x < 4 correct. GMAC says I'm right. They write the GMAT. I'm going to go with them.

If you don't like it, however, be sure to contact GMAC and suggest they learn about complex numbers and multi-dimensional euclidean vector spaces. I'm sure your unwarranted pretension will be well received just as it was here.

As to your observation, the fact that the intervals on a piecewise function overlap is not a problem as long as the function evaluates to the same number in both cases. As long as the function is well-defined, the overlap is okay..

You realize the definition of a piecewise defined function requires disjoint subdomains right? No. No I guess you don't.

As to your continued insistence that the solution is x<4 rather than x<=4, i invite you to demonstrate how x=4 fails to satisfy the equation..

Irrelevant. It violates the definition of absolute value and that's the entire point of my first post here.

Given that the definition supplied by GMAC contradicts the MGMAT answer to this question it's pretty safe to say that the MGMAT approach could potentially yield an incorrect answer on the GMAT. That's all I'm saying.

[tim]

Philip, I'm not trying to smear you. I am aware that many graduate programs don't require the type of multi-dimensional analysis that would require an alternate definition of absolute value, and it's not a slam on you if this didn't happen in your program. Just try to keep in mind that there can be multiple valid perspectives on a particular mathematical definition, and whether you've taken them or not, there are classes out there that present different perspectives on absolute value. The main message here is this: Relax; the GMAT does not require this analysis either, so you're safe! You just need to focus on how the GMAT approaches this type of problem..

Bringing things back on topic, as I've mentioned before, your definitions work fine. My definitions work fine. Everything's cool on the definition side of things; you just seem to have misapplied your definition. To summarize your approach, you disagree with the answer that x<=4 and say instead that the answer should be x<4. That means you are claiming that x=4 is not a solution to this equation. However, if you plug x=4 into the equation it says |4-4| = 4-4, in other words |0| = 0, which is clearly true meaning x=4 is a valid solution..

I hope this helps. Something got misapplied when you tried to use your definition on this problem, but I'm not entirely sure where things went wrong for you. Again, your definition is absolutely fine. If you can talk me through some more of your steps on this problem I'll be happy to help you figure out what went wrong for you..

[philip.traumatic]

Philip, I'm not trying to smear you. I am aware that many graduate programs don't require the type of multi-dimensional analysis that would require an alternate definition of absolute value, and it's not a slam on you if this didn't happen in your program. Just try to keep in mind that there can be multiple valid perspectives on a particular mathematical definition, and whether you've taken them or not, there are classes out there that present different perspectives on absolute value. The main message here is this: Relax; the GMAT does not require this analysis either, so you're safe!

lol my indignation stems from your assumption that because I'm questioning your explanation I somehow didn't take certain classes.

I've seen other definitions for absolute value in different contexts but not once have I seen absolute value defined another way for a point on a line.

You just need to focus on how the GMAT approaches this type of problem..

Actually this is what YOU need to do.

One more time:

In The Official Guide 12th Edition GMAT Review halfway down page 126 there is this verbatim definition for absolute value:

"The absolute value of x, denoted |x|, is defined to be x if x >= 0 and -x if x < 0."

That is exactly the definition I gave you above and it is that definition which makes the answer to this threads original problem x < 4. As I explained earlier, according to GMAC the only time you multiply the argument of the absolute value by -1 is when the argument is strictly less than zero. If x = 4, the argument IS zero and you do not multiply by -1...the argument stays the same...x-4.

Given GMAC's definition, if you were to answer the original problem with x <= 4 you would be wrong because it violates the definition provided by GMAC.

[tim]

Hi Philip,

That's totally cool if you want to stick with the definition you've been using. I hope I've managed to get across the point that the GMAT definition is totally correct and that regardless of any other definitions that exist you're safe to use that one. What I am really trying to help you see is that the reason you've gotten the answer incorrect is because you're using the correct definition but you are somehow misapplying it.

I would love to help you identify where this problem is occurring so you don't end up making the same mistake on the GMAT, but you're going to have to let me help. I've demonstrated why it makes logical sense that x=4 is a solution (check that number specifically against the GMAT definition if you haven't already). As I said awhile back, if you were to walk me through your version of the solution I could help you analyze where it went wrong.