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How many values can the integer $$p=|x+3|-|x-3|$$ assume?

A)6
B)7
C)13
D)12
E)Infinite values

My own question, as always any feedback is appreciated

Originally posted by Zarrolou on 15 May 2013, 11:30.
Last edited by Zarrolou on 01 Jul 2013, 23:45, edited 2 times in total.
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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Good job arpanpatnaik, vinaymimani!

Official explanation

The function $$|x+3|-|x-3|$$ for values $$\geq{}3$$ equals $$6$$, and for values $$\leq{}-3$$ equals $$-6$$
For the middle values it follows the equation $$2x$$ (as the users above correctly say)

However there is a quicker way to get to the answer than counting the possible values.

Its upper limit is $$6$$, its lower limit is $$-6$$ and the function $$2x$$ is monotonic and increasing (and continuous), so will assume all values between 6 and -6 included.
(This is not theory necessary for the GMAT, but if notice the fact that $$2x$$ must pass for all values between 6 and -6, you can save time)

So the values that the integer p can assume are $$-6,-5,...,0,...,5,6$$ TOT=$$13$$

For for clarity, below there is the graph of $$|x+3|-|x-3|$$ that will make my explanation more clear.
Attachments Untitled.png [ 5.26 KiB | Viewed 31176 times ]

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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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Zarrolou wrote:
How many values can the integer $$p=|x+3|-|x-3|$$ assume?

A)0
B)7
C)13
D)14
E)Cannot be determined

My own question, as always any feedback is appreciated
Kudos to the first correct solution(s)!

I'd go with [C]!

For the expression $$p=|x+3|-|x-3|$$ , one can think of 3 possible ranges for x to lie in:
1. x > 3
2. x belongs to {-3,3}
3. x < -3

It can be calculated for x > 3 : P = 6
Also, for x < -3, P = -6 i.e. for the above ranges the value of x does not put an effect on P.

But when it comes to the region x belongs to {-3,3} P = 2x. Since P depends on x, and no definite rule has been defined for x, we have to assume x to be values so as to result 2x is a possible integer values. Hence x can be {-2,-1,0,1,2,-1/2,1/2,-3/2,3/2,-5/2,5/2} For each of the above values the expression P has an integral value and all the above lie between -3 and 3. Hence the total number would be 11+2 = 13.

Hope this is the right answer Regards,
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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Zarrolou wrote:
How many values can the integer $$p=|x+3|-|x-3|$$ assume?

A)0
B)7
C)13
D)14
E)Cannot be determined

My own question, as always any feedback is appreciated
Kudos to the first correct solution(s)!

For x=3/-3 we have the value of p = 6/-6.
For all x>3, we have the value of p = (x+3)-(x-3) = 6
For all -3<x<3, we have the value of p = (x+3)-(3-x) = 2x. We could have x = 1/2,1,3/2,2,5/2 ,0 and the same set of negatives--> 11 values
For all x<-3, we have the value of p = (-x-3)-(3-x) = -6.
Thus, the number of unique values that p can have are 13.
C.
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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Zarrolou wrote:
How many values can the integer $$p=|x+3|-|x-3|$$ assume?
A)0
B)7
C)13
D)14
E)Cannot be determined

My own question, as always any feedback is appreciated

Dear Zarrolou,
Here's some feedback. I like the question itself quite a bit. My only criticism are the answer choices. Obviously, the OA (C) need to be on the list, and (B) 7 is an excellent distractor (what folks get if they just plug in integer values of x). I don't know about (D) 14 --- who would pick that? If folks realize that the output is +6 for x > 3 and -6 for x < -3, I could see them saying, "Hmm, how many integers from -6 to 6? That must be 12." Forgetting inclusive counting in something like this --- that's a huge predictable error. I think both 6 & 12 would be good to have on the answer list for this reason. I have no idea how anyone could possibly pick (A) 0 ---- anyone who can plug in any single number and get an output will know there are now zero outputs. I see that as a wasted incorrect answer: one that no one will pick. Choice (E) is debatable. I would recommend 3, 6, 7, 12, 13 as the answer choices myself.
What do you think?
Mike _________________
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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WholeLottaLove wrote:
Hello, thanks for the response.

My reasoning came from this: find-the-value-of-135023.html post, which I thought was a similar situation. In it, a-1 can be a positive or negative value, thus causing the signs to flip accordingly.

I do not understand why if -3<x<3, P=(x+3)+(x-3)=2x and I am also not understanding the relation to the graph. Sorry...absolute value problems and concepts have proven to be very difficult to understand.

That question is similar and I am gonna apply the same method here, so:

How many values can the integer $$p=|x+3|-|x-3|$$ assume?

I think that we both agree that if $$x>3$$, both $$|x+3|-|x-3|$$ will be positive so we can eliminate the || signs and keep $$x+3-(x-3)=6$$.
Same reasoning for $$x<-3$$: if that's the case then both $$|x+3|-|x-3|$$ will be negative so we must "flip" the sign according to the rule
$$|-value|=+value$$. We obtain $$-(x+3)-(-)(x-3)=-6$$.

I hope that this so far makes sense.

Now, what if -3<x<3? In this case the term $$|x+3|$$ will be positive (you can try with values -2+3>0 for example).
The other term will be negative $$|x-3|$$ (-2-3<0 for example). The first one will be positive=> the abs value will not affect the sign; The second one will be negative=> the abs value will turn it positive. Lets write this down:
$$p=(x+3)-(-)(x-3)=2x$$ - if -3<x<3

Now we have
for x>3 we have P=(x+3)-(x-3)=6, for -3<x<3 P=(x+3)+(x-3)=2x, for x<-3 P=-(x-3)+(x-3)=-6
the graph just represents those equations. For example if x>3 the graph shows you a line P=6
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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WholeLottaLove wrote:
Can you tell me if my explanation is correct? I have a good deal of difficultly with absolute value!

For problems like this, we get a range of numbers to test by seeing what values with make x=0. For example, -3 makes x+3 =0 and 3 makes x-3 = 0. So we test those cases and get two results. Then, we test cases where X is greater and less than 3, which get's us the same result for every integer greater than + x and less than - x. Then, we test integers between -3 and 3 getting us the remainder of the possible answers.

Out of curiosity, why is P = (x+3)-(x-3) when X>0 and p = (x+3)-(x-3) when x<0? Is it because a negative value of x will get a negative value for |x-3| and for all |-x| = -(-x)?

Thanks!

The explanation above is correct, but I don't think that is doable. I mean that testing each value is not a good way to find the answer, you could miss something; and more important if the answers involved bigger numbers as 150 for example, I dare you to count all the possibilities...

"Out of curiosity, why is P = (x+3)-(x-3) when X>0 and p = (x+3)-(x-3) when x<0? Is it because a negative value of x will get a negative value for |x-3| and for all |-x| = -(-x)?"

This is not correct.

if x>3 we have P=(x+3)-(x-3)=6: a straight line
if -3<x<3 we have P=(x+3)+(x-3)=2x
if x<-3 we have P=-(x-3)+(x-3)=-6: a straight line

We have three functions in P for the intervals above described. The quickest way to get an asnwer is to see that P will assume all values form -6 to 6 inclusive.

Let me know if you have more doubts
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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WholeLottaLove wrote:
But how does 2x translate into an additional 11 answers? If you plug in values for 2x of -2≤x≤2 then I see only 5!

If this is your question, I assume that the abs value problem now is clear. Let me know if I assume correctly

First of all it's "values for 2x of -3≤x≤3 not -2≤x≤2 ".

for $$-3<x<3$$ $$2x$$ can have all values between 2*(-3) and 2*(3) or in the range $$-6,6$$.
How many integers are there between -6 and 6? -5,-4,...,0,...4,5 so 11
To those we sum 6 and -6 and the total reaches 11+2=13

You see only 5 values I think because you are not considering cases as
$$x=\frac{1}{2}$$(fractions), for this $$p=2x$$ will be an integer for example. Same case for $$x=-\frac{1}{2}$$ and so on
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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WholeLottaLove wrote:
Oooh. Because P must be an integer. So x can be any integer or fraction between -3 and 3 that yields an integer result for x...i.e. 2, -2.....-1/2, 1/2?

Yes, almost correct! Remeber that we are considering the range -3<x<3 so the list of values is
-5/2 (-2.5), -2,-3/2,-1,-1/2,0,1/2,1,3/2,2,5/2 => 11 values

I said "almost correct" because in "x...i.e. 2, -2.....-1/2, 1/2" you missed -2.5 at the beginning, but everything else is fine
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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WholeLottaLove wrote:
How many values can the integer p=|x+3|-|x-3| assume?

A)6
B)7
C)13
D)12
E)Cannot be determined

Hi all.

I thought I understood this problem but upon further review, I am still a bit uncertain.

You say that the range is between -6 and 6, and for any x value that holds true. For example, if x=20 then p=|x+3|-|x-3| ==> p=|20+3|-|20-3| = 23-17 = 6.

One explanation that I liked (because it was simple haha) was that P could equal any integer between -6 and 6 inclusive. That seems too simple however so I want to see if I can understand the problem further.

P has to = to an integer, so that limits values (I suppose it would be nearly impossible to solve in a few minutes if P could = any number) but how can we figure out exactly which integers it can equal. For example, isn't it possible that a problem similar to this might not be valid for every number in the range? (i.e. between -6 and 6)

Thanks!

Good question!

In this question every integer between -6 and 6 is a valid result. How can we be sure of that?

Because in the range -6,6 the function equals $$2x$$ [(x+3)+(x-3)=2x], we can be sure that it will assume every integer value between -6 and 6.
I can say so because the funcion is defined in every point (it's a line) so does not have any invalid point, you can draw it and control yourself: it is defined in every point.

To give you an example of a function that is NOT defined in every point I could use $$p=\frac{3}{3+x}$$.
This is beyond the question itself, but that function is NOT defined for $$x=-3$$ : $$p=\frac{3}{0}$$ is not a denfined value.

But since 2x is a line, it will assume every value between -6 and 6.

Hope it's clear
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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WholeLottaLove wrote:
So,

This equation will always yield a value between -6 and 6 for any value of x.
Any value of x (positive or negative) will result in
(x+3) - (x-3)

But then how does that = 2x?

The original function is $$p= |x+3|-|x-3|$$.
For values of x>3 it will equal 6
$$p=x+3-x+3=6$$
For values of x<3 it will equal -6
$$p=-x-3+x-3=-6$$

For values $$-6\leq{x}\leq{6}$$ it equals $$p = (x+3)+(x-3)=2x$$, any value of x between $$-6\leq{x}\leq{6}$$ will result in p=2x.

Some posts back we enstablished that
"Yes, almost correct! Remeber that we are considering the range -3<x<3 so the list of values is
-5/2 (-2.5), -2,-3/2,-1,-1/2,0,1/2,1,3/2,2,5/2 => 11 values

I said "almost correct" because in "x...i.e. 2, -2.....-1/2, 1/2" you missed -2.5 at the beginning, but everything else is fine "

what I am trying to say is that p=2x is a line, so I am asking you: "How many integer values does the function p=2x assume in the interval -6,6?"
Answer: if X=-5/2 P=-5, if X=-2 P= -4,...
Plus 6 and -6 that are the "edge" values: total 13.
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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WholeLottaLove wrote:
Ok,

So 2x must be an integer and the result must lie within -6 and 6 when the values of x lie within -3 and 3?

And (I know I asked this before but I'm not 100% sure) how do I know if it's a trap and not all values between -6 and 6 are valid?

Again, thank you for your patience and help.

Yes, correct.

You know that there are no trap values because $$2x$$ is a straight line, so is defined for every $$x$$.
Lines in general are defined for every value of x. With this I mean that you can draw a line, and for whatever value of x you pick, you'll always find a corresponding value on the line.
You can ask yourself: is there any value of $$x$$ for which $$2x$$ is not defined? The answer is no.

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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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WholeLottaLove wrote:
How many values can the integer p=|x+3|-|x-3| assume?

A)6
B)7
C)13
D)12
E)Cannot be determined

We're not looking for how many valid solutions of "x" there are...we are looking for how many integers "p" there are (that's what was tripping me up before!!!)

We can do this by finding the range of values of x (i.e. what numbers, if any, does x lie between)

Find the check points: -3, 3

We have three ranges to test for: -3< x, -3<x<3, 3>x

For x<-3: |x+3|-|x-3| -(x+3)- -(x-3) -x-3 - (-x+3) -x-3 + x-3 P=-6
For -3<x<3 |x+3|-|x-3| (x+3) - -(x-3) (x+3) - (-x+3) (x+3) + x -3 P=2x
For x>3: |x+3|-|x-3| (x+3) - (x-3) (x+3) -x+3 P=6

So, the range of P is from -6 ≤ P ≤ 6. There are 13 integers between -6 and 6 inclusive.

Just one question - how do I know the values are inclusive (-6 ≤ p ≤ 6) as opposed to not (-6 < p < 6)?

Thanks!

Well, if you are not sure, you can plug in a value greater than 3 or less than -3 and see what you find.

$$p=|10+3|-|10-3|=|13|-|7|=6$$ so 6 is a possible value, same thing for x=-10

from a more methodical point of view, if x is greater than 3, the whole expression becomes

$$p=(x+3)-(x-3)=6$$ so 6 is a possible value

same thing for values less than -3.
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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nancerella wrote:
thanks to everyone's explanations, i think i've finally understood the solution to this problem.

now my question is, what's the best way to be solving this and similar type (multiple mods) questions to keep within the 2 mins mark?
would it be first, identifying the key ref points (e.g. 3 and -3, in this case) and then plugging in numbers within the ranges?
or is this just a concept that you need to get really good at and be able to quickly recognize the +/- setups of each mod for each of the respective scenarios? (e.g. if x<3, then setup equation with neg (x+3) and pos (x-3) cases).

I think the quickest way is the one explained here: how-many-values-can-the-integer-p-x-3-x-3-assume-152859.html#p1225488

You find that the function $$p=|x+3|-|x-3|$$ ranges from 6 to -6 : will assume every value in that range. The question asks for the number of INTEGER values p can have, so just count the integers between -6 and 6 included.

Hope it makes sense Veritas Prep GMAT Instructor V
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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Zarrolou wrote:
How many values can the integer $$p=|x+3|-|x-3|$$ assume?

A)6
B)7
C)13
D)12
E)Cannot be determined

My own question, as always any feedback is appreciated

Also, it is not a 600-700 level question. It is certainly 700+ level.
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How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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sehosayho wrote:
How many values can the integer p=|x+3|-|x-3| assume?

A)6
B)7
C)13
D)12
E)Infinite values

Many folks used folloing approach
1. x >or= 3
2. x belongs to {-3,3}
3. x < or = -3
leaving 3 possible valuse for P ( 6,-6, and 2x)

I understand this approach, but I am confused with another approach of soloving inequalities questions like this one
the other approach is following
|x+3|>0 and |x-3|>0
|x+3|>0 and |x-3|<0
|x+3|<0 and |x-3|>0
|x+3|<0 and |x-3|<0
this case there are 4 possible valuse for P
P= 6, -6, 2x, or -2x
what am I missing here??
If I cannot use the second approach could you explain why please.
Thank you

The first approach is the logical conclusion of the second approach (with modification - the second approach as given by you in incorrect). The absolute value can never be negative. The four cases are
x+3 >0 and x-3 >0
x+3 >0 and x-3 <0
x+3 <0 and x-3 >0
x+3 <0 and x-3<0

Case 1:
x+3 >0 and x-3 >0
x > -3 and x > 3
This is the case of x > 3

Case 2:
x+3 >0 and x-3 <0
x > -3 and x < 3
This is the case of -3 < x< 3

Case 3:
x+3 <0 and x-3 >0
x < -3 and x > 3
Is this possible? Can x be less than -3 and greater than 3 simultaneously? No.

Case 4:
x+3 <0 and x-3<0
x<-3 and x < 3
This is the case of x < -3

Note that the cases come from the definition of absolute value:
|x| = x when x > 0 and -x when x < 0

You will find this post helpful: http://www.veritasprep.com/blog/2014/06 ... -the-gmat/
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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mikemcgarry wrote:
Dear Zarrolou,
Here's some feedback. I like the question itself quite a bit. My only criticism are the answer choices. Obviously, the OA (C) need to be on the list, and (B) 7 is an excellent distractor (what folks get if they just plug in integer values of x). I don't know about (D) 14 --- who would pick that? If folks realize that the output is +6 for x > 3 and -6 for x < -3, I could see them saying, "Hmm, how many integers from -6 to 6? That must be 12." Forgetting inclusive counting in something like this --- that's a huge predictable error. I think both 6 & 12 would be good to have on the answer list for this reason. I have no idea how anyone could possibly pick (A) 0 ---- anyone who can plug in any single number and get an output will know there are now zero outputs. I see that as a wasted incorrect answer: one that no one will pick. Choice (E) is debatable. I would recommend 3, 6, 7, 12, 13 as the answer choices myself.
What do you think?
Mike Many thanks for the feedback first of all, it's always good to hear your opinion.

I think that your advise is great, I will update the original question.
Just I will not use that order, otherwise folks who got it right and pressed C will have a wrong answer on their workbook (I think...)

Just one thing before I proceed, my option E "cannot be determined" sounds good as possible answer.
Some users wrote to me saying that is the first thing that came into their mind when they saw the question. What do you think?

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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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Can you tell me if my explanation is correct? I have a good deal of difficultly with absolute value!

For problems like this, we get a range of numbers to test by seeing what values with make x=0. For example, -3 makes x+3 =0 and 3 makes x-3 = 0. So we test those cases and get two results. Then, we test cases where X is greater and less than 3, which get's us the same result for every integer greater than + x and less than - x. Then, we test integers between -3 and 3 getting us the remainder of the possible answers.

Out of curiosity, why is P = (x+3)-(x-3) when X>0 and p = (x+3)-(x-3) when x<0? Is it because a negative value of x will get a negative value for |x-3| and for all |-x| = -(-x)?

Thanks!
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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Hello, thanks for the response.

My reasoning came from this: find-the-value-of-135023.html post, which I thought was a similar situation. In it, a-1 can be a positive or negative value, thus causing the signs to flip accordingly.

I do not understand why if -3<x<3, P=(x+3)+(x-3)=2x and I am also not understanding the relation to the graph. Sorry...absolute value problems and concepts have proven to be very difficult to understand.
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Re: How many values can the integer p=|x+3|-|x-3| assume?  [#permalink]

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But how does 2x translate into an additional 11 answers? If you plug in values for 2x of -2≤x≤2 then I see only 5! Re: How many values can the integer p=|x+3|-|x-3| assume?   [#permalink] 30 May 2013, 12:24

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