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Math Expert V
Joined: 02 Sep 2009
Posts: 59561
If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 28% (02:20) correct 72% (02:31) wrong based on 1519 sessions

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If $$y=|x+5|-|x-5|$$, then $$y$$ can take how many integer values?

A. 5
B. 10
C. 11
D. 20
E. 21

Kudos for a correct solution.

Math Expert V
Joined: 02 Sep 2009
Posts: 59561
If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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35
49
SOLUTION

If $$y=|x+5|-|x-5|$$, then $$y$$ can take how many integer values?

A. 5
B. 10
C. 11
D. 20
E. 21

When $$x\leq{-5}$$, then $$|x+5|=-(x+5)=-x-5$$ and $$|x-5|=-(x-5)=5-x$$.
Hence in this case $$y=|x+5|-|x-5|=-x-5-(5-x)=-10$$.
1 integer value of $$y$$ for this range.

When $$-5<x<5$$, then $$|x+5|=x+5$$ and $$|x-5|=-(x-5)=5-x$$.
Hence in this case $$y=|x+5|-|x-5|=x+5-(5-x)=2x$$.
Therefore for this range $$-10<(y=2x)<10$$.
19 integer values of $$y$$ for this range (from -9 to 9, inclusive).

When $$x\geq{5}$$, then $$|x+5|=x+5$$ and $$|x-5|=x-5$$.
Hence in this case $$y=|x+5|-|x-5|=x+5-(x-5)=10$$.
1 integer value of $$y$$ for this range.

Total = 1 + 19 + 1 = 21.

If anyone interested here is a graph of $$y=|x+5|-|x-5|$$: As you can see y is a continuous function from -10 to 10, inclusive.

Try NEW absolute value DS question.

Attachment: WolframAlpha--yx5-x-5_--2014-07-08_0728.png [ 12.01 KiB | Viewed 30141 times ]
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Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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9
3
y=|x+5| - |x-5|

Case 1 : x>5

y=x+5 -(x-5)
=10

Case 2: -5<= x <5

y=(x+5)+(x-5)
=2x

Now x could assume 10 values in this range
-5, -4, -3, -2, -1, 0, 1, 2, 3, 4

for which y could assume 10 values
-10, -8, -6, -4, -2, 0, 2, 4, 6, 8

However since x could be a fraction , x could also assume values like:
-4.5, -3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5, 4.5

for which y could take values like:
-9,-7,-5,-3,-1,1,3,5,7,9

Case 3: x< -5

y=-(x+5)+(x-5)
=-10

Therefore combining all the unique values of y we see there are 21 values of y(E)
##### General Discussion
Director  Joined: 25 Apr 2012
Posts: 651
Location: India
GPA: 3.21
Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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5
If $$y=|x+5|-|x-5|$$, then $$y$$ can take how many integer values?

A. 5
B. 10
C. 11
D. 20
E. 21

Sol :$$y=|x+5|-|x-5|$$, then $$y$$

We see that there is a following range of values

x<-5
-5< x<5
x>5

Now for any value of x<-5 the expression will give the value -10...
Consider x=-6 so we have |-1|-|-11|= 1-11=-10
Consider x=-10, we have |-5|-|-15|= 5-15=-10
Similarly when x>5 the expression will give value of 10

Consider x=7, |12|-|2|=10
X=100,|105|-|95|=10

So we have 2 values of y possible -10, 10

Consider the range -5<x<5...At x=-5 and 5 value is -10 and 10 respectively

Consider x=-4, |-1|-|-9|=-8
x=-3,|-2|-|-8|=-6
x=-2, |-3|-|-7|=-4
Similarly x=-1 will give you different value

Like wise x=0 will give you one value and value of x from 1 to 4 you get different values of y.

So total possible values -10,10,0, + 4 values -1 to-4 and +4 values for range of x 1 to 4= 11

Ans C
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Manager  Joined: 27 Jul 2012
Posts: 98
Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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1
If $$y=|x+5|-|x-5|$$, then $$y$$ can take how many integer values?

A. 5
B. 10
C. 11
D. 20
E. 21

X>=5 : Y= X+5-X+5=10;
X=<-5 : Y=-X-5+X-5=-10;
-5<X<5: Y=2X ; we have 9 values here
hence C, y can take 11 integer values.
Senior Manager  Joined: 13 Jun 2013
Posts: 257
Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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3
3
Bunuel wrote:

If $$y=|x+5|-|x-5|$$, then $$y$$ can take how many integer values?

A. 5
B. 10
C. 11
D. 20
E. 21

Kudos for a correct solution.

Attachments Capture.JPG [ 60.17 KiB | Viewed 27892 times ]

Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9848
Location: Pune, India
Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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15
9
Bunuel wrote:

If $$y=|x+5|-|x-5|$$, then $$y$$ can take how many integer values?

A. 5
B. 10
C. 11
D. 20
E. 21

Kudos for a correct solution.

Use the number line for mod questions:

___________-5 ______________0______________5___________

You want the values of y which is the difference between "distance from -5" and "distance from 5".

Anywhere on the left of -5,

<------------------------------------------------
<-------------
___________-5 ______________0______________5___________

The difference between the two distances will always be -10 (taking difference to mean what it does for GMAT)

In between -5 and 5, can y can the value of -9? Sure. Move 0.5 to the right of -5. At x = -4.5, distance from -5 will be .5 and distance from 5 will be 9.5. So y = 0.5 - 9.5 = -9
Note that x needn't be an integer. Only y needs to be an integer.

Similarly, at various points between -5 and 5, y will take all integer values from -9 to 9.

To the right of 5, the difference between the two distances will always be 10.

So values that y can take: -10, -9, ... 0 ... 9, 10 i.e. a total of 21 values.
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Location: India
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Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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Plastelin wrote:
proposed solution was wrong.
del.

You solution i presume was correct...No need to delete your post just because someone else did it differently.
Your post will be helpful for other members in the forum as well as one can learn from different ways of attempting a question...

Good luck
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Manager  Status: Please do not forget to give kudos if you like my post
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Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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Just test the number. Alway start with one of the integer given in the question. Let start with -5 we get -10. Now test 5 gives us 10. Now test 0 gives us 0. Now test 6 gives us 10 or -6 gives us 9. So the pattern is as we increase beyond 6 or -6 value is constant at 10. So the total is 11 including 0 between -5 and 5.
Math Expert V
Joined: 02 Sep 2009
Posts: 59561
Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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3
9
SOLUTION

If $$y=|x+5|-|x-5|$$, then $$y$$ can take how many integer values?

A. 5
B. 10
C. 11
D. 20
E. 21

When $$x\leq{-5}$$, then $$|x+5|=-(x+5)=-x-5$$ and $$|x-5|=-(x-5)=5-x$$.
Hence in this case $$y=|x+5|-|x-5|=-x-5-(5-x)=-10$$.
1 integer value of $$y$$ for this range.

When $$-5<x<5$$, then $$|x+5|=x+5$$ and $$|x-5|=-(x-5)=5-x$$.
Hence in this case $$y=|x+5|-|x-5|=x+5-(5-x)=2x$$.
Therefore for this range $$-10<(y=2x)<10$$.
19 integer values of $$y$$ for this range (from -9 to 9, inclusive).

When $$x\geq{5}$$, then $$|x+5|=x+5$$ and $$|x-5|=x-5$$.
Hence in this case $$y=|x+5|-|x-5|=x+5-(x-5)=10$$.
1 integer value of $$y$$ for this range.

Total = 1 + 19 + 1 = 21.

If anyone interested here is a graph of $$y=|x+5|-|x-5|$$: As you can see y is a continuous function from -10 to 10, inclusive.

Kudos points given to correct solutions above.

Try NEW absolute value DS question.
Manager  Joined: 29 May 2013
Posts: 53
GMAT 1: 710 Q49 V38 GPA: 4
Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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Bunuel wrote:
SOLUTION

If $$y=|x+5|-|x-5|$$, then $$y$$ can take how many integer values?

A. 5
B. 10
C. 11
D. 20
E. 21

When $$x\leq{-5}$$, then $$|x+5|=-(x+5)=-x-5$$ and $$|x-5|=-(x-5)=5-x$$.
Hence in this case $$y=|x+5|-|x-5|=-x-5-(5-x)=-10$$.
1 integer value of $$y$$ for this range.

When $$-5<x<5$$, then $$|x+5|=x+5$$ and $$|x-5|=-(x-5)=5-x$$.
Hence in this case $$y=|x+5|-|x-5|=x+5-(5-x)=2x$$.
Therefore for this range $$-10<(y=2x)<10$$.
19 integer values of $$y$$ for this range (from -9 to 9, inclusive).

When $$x\geq{5}$$, then $$|x+5|=x+5$$ and $$|x-5|=x-5$$.
Hence in this case $$y=|x+5|-|x-5|=x+5-(x-5)=10$$.
1 integer value of $$y$$ for this range.

Total = 1 + 19 + 1 = 21.

If anyone interested here is a graph of $$y=|x+5|-|x-5|$$:
Attachment:
WolframAlpha--yx5-x-5_--2014-07-08_0728.png
As you can see y is a continuous function from -10 to 10, inclusive.

Try NEW absolute value DS question.

I think the answer is 11. Options C.
The highlighted section should have 9 values instead of 19.
Math Expert V
Joined: 02 Sep 2009
Posts: 59561
Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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dransa wrote:
Bunuel wrote:
SOLUTION

If $$y=|x+5|-|x-5|$$, then $$y$$ can take how many integer values?

A. 5
B. 10
C. 11
D. 20
E. 21

When $$x\leq{-5}$$, then $$|x+5|=-(x+5)=-x-5$$ and $$|x-5|=-(x-5)=5-x$$.
Hence in this case $$y=|x+5|-|x-5|=-x-5-(5-x)=-10$$.
1 integer value of $$y$$ for this range.

When $$-5<x<5$$, then $$|x+5|=x+5$$ and $$|x-5|=-(x-5)=5-x$$.
Hence in this case $$y=|x+5|-|x-5|=x+5-(5-x)=2x$$.
Therefore for this range $$-10<(y=2x)<10$$.
19 integer values of $$y$$ for this range (from -9 to 9, inclusive).

When $$x\geq{5}$$, then $$|x+5|=x+5$$ and $$|x-5|=x-5$$.
Hence in this case $$y=|x+5|-|x-5|=x+5-(x-5)=10$$.
1 integer value of $$y$$ for this range.

Total = 1 + 19 + 1 = 21.

If anyone interested here is a graph of $$y=|x+5|-|x-5|$$:
Attachment:
WolframAlpha--yx5-x-5_--2014-07-08_0728.png
As you can see y is a continuous function from -10 to 10, inclusive.

Try NEW absolute value DS question.

I think the answer is 11. Options C.
The highlighted section should have 9 values instead of 19.

Have you tried to write down possible values? I guess you are missing 0 and 9 negative values.
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3158
Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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5
8
Here's a visual way of thinking through this question.

First of all, we see that the given expression for y contains two modulus expressions: |x + 5| and |x - 5|

What do these expressions mean?

|x-5| represents the distance of point x from the point 5 on the number line.

|x+5| can be written as |x - (-5)|. So, |x+5| represents the distance of point x from the point -5 on the number line.

So, let's plot the points 5 and -5 on the number line, and consider the different possible values of x.

Case 1: x < -5

Depicting this case on the number line: It's clear that in this case, |x-(-5)| - |x-5| = -(distance between points - 5 and 5 on the number line) = -10. The minus sign here comes because the bigger distance (|x-5|) is subtracted from the smaller distance (|x+5|)

Case 2: -5 < = x <= 5

Depicting this case on the number line: Let's say x = -5. In that case, it's easy to see that |x-(-5)| - |x-5| = -10

If x = -4.5, then |x-(-5)| - |x-5| = 0.5 - 9.5 = -9
.
.
.

If x = 5, then |x-(-5)| - |x-5| = 10

Thus, we see that in Case 2, all integral values between -10 and 10, inclusive, are possible for y.

Case 3: x > 5

Depicting this case on the number line: It's easy to see that in this case, |x-(-5)| - |x-5| = +10

Combining all cases

We see that the following integral values of y are possible: -10, -9, -8 . . . 8, 9, 10

That is, 21 values in all.

Hope this visual way of thinking through this and other similar absolute value questions was helpful for you! Japinder
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Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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I have been taught that if we have a |x+b| = c. The midpoint is always -b.

I tried applying it here: so |x+5| gives us midpoint -5. and |x-5| gives us midpoint 5.

However - how do I know the actual range? I'm not actually given something like "|x+5| < 2" or whatever.
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Joined: 04 Jan 2015
Posts: 3158
Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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1
erikvm wrote:
I have been taught that if we have a |x+b| = c. The midpoint is always -b.

I tried applying it here: so |x+5| gives us midpoint -5. and |x-5| gives us midpoint 5.

However - how do I know the actual range? I'm not actually given something like "|x+5| < 2" or whatever.

Dear erikvm

The way to read the equation '|x+b| = c is: "the distance of x from the point -b on the number line is c'. So 2 values of x are possible: one that is c units to the left of -b and the second that is c units to the right of -b. As you can see, '-b' is indeed the mid-point between the two possible values of x. But note that I didn't start my analysis of |x+b| = c from this result. I started from the fact that |x+b| represents the distance of x from the point -b on the number line.

Let's now come to the question at hand:

Start your analysis by considering that |x+5| represents the distance of x from -5 and |x-5| represents the distance of x from 5.

Try solving the question from here on. Go through the solution I've posted above, and if something is still not clear, I'll be happy to help! Best Regards

Japinder
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If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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EgmatQuantExpert wrote:
erikvm wrote:
I have been taught that if we have a |x+b| = c. The midpoint is always -b.

I tried applying it here: so |x+5| gives us midpoint -5. and |x-5| gives us midpoint 5.

However - how do I know the actual range? I'm not actually given something like "|x+5| < 2" or whatever.

Dear erikvm

The way to read the equation '|x+b| = c is: "the distance of x from the point -b on the number line is c'. So 2 values of x are possible: one that is c units to the left of -b and the second that is c units to the right of -b. As you can see, '-b' is indeed the mid-point between the two possible values of x. But note that I didn't start my analysis of |x+b| = c from this result. I started from the fact that |x+b| represents the distance of x from the point -b on the number line.

Let's now come to the question at hand:

Start your analysis by considering that |x+5| represents the distance of x from -5 and |x-5| represents the distance of x from 5.

Try solving the question from here on. Go through the solution I've posted above, and if something is still not clear, I'll be happy to help! Best Regards

Japinder

Hey, thanks for trying to explain, but I still don't understand this logic :[

I do understand that we get two different midpoints, somewhere on the number line, but I don't understand how we can decide on their range. How do I know that the range for for the two values is 5 steps to the right/left of |x+5| and |x-5|
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Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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1
Bunuel wrote:

If $$y=|x+5|-|x-5|$$, then $$y$$ can take how many integer values?

A. 5
B. 10
C. 11
D. 20
E. 21

Kudos for a correct solution.

This is how you solve such questions:
Attachment: Unbenannt.jpg [ 38.03 KiB | Viewed 16656 times ]

So just tick a box, but don't take it too serious Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9848
Location: Pune, India
Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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erikvm wrote:
I have been taught that if we have a |x+b| = c. The midpoint is always -b.

I tried applying it here: so |x+5| gives us midpoint -5. and |x-5| gives us midpoint 5.

However - how do I know the actual range? I'm not actually given something like "|x+5| < 2" or whatever.

Actually, don't think of it from the mid-point perspective because then it is not useful in many circumstances. The actual logic of absolute value is this:
|x - a| gives you the "distance of x from point a on the number line"

You might want to check out this post to understand this perspective:
http://www.veritasprep.com/blog/2011/01 ... edore-did/

So y = |x+5|−|x−5| gives you the difference between "distance of x from -5" and "distance of x from 5"
So if x = 0, difference between "distance of x from -5" and "distance of x from 5" will be "5" - "5" = 0. This gives you y = 0.

Similarly, you can handle something like this: |x+5| + |x−5|
This is sum of "distance of x from -5" and "distance of x from 5".

Using this method, you can solve this question like this: if-y-x-5-x-5-then-y-can-take-how-many-integer-173626.html#p1379135
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Karishma
Veritas Prep GMAT Instructor

Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9848
Location: Pune, India
Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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1
reto wrote:

This is how you solve such questions:
Attachment:
Unbenannt.jpg

So just tick a box, but don't take it too serious Amusing!
Though on a serious note, try to understand the concept behind it and the HARD question will be bumped up to at least MEDIUM in no time!
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Re: If y = |x + 5|-|x - 5|, then y can take how many integer  [#permalink]

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VeritasPrepKarishma wrote:
erikvm wrote:
I have been taught that if we have a |x+b| = c. The midpoint is always -b.

I tried applying it here: so |x+5| gives us midpoint -5. and |x-5| gives us midpoint 5.

However - how do I know the actual range? I'm not actually given something like "|x+5| < 2" or whatever.

Actually, don't think of it from the mid-point perspective because then it is not useful in many circumstances. The actual logic of absolute value is this:
|x - a| gives you the "distance of x from point a on the number line"

You might want to check out this post to understand this perspective:
http://www.veritasprep.com/blog/2011/01 ... edore-did/

So y = |x+5|−|x−5| gives you the difference between "distance of x from -5" and "distance of x from 5"
So if x = 0, difference between "distance of x from -5" and "distance of x from 5" will be "5" - "5" = 0. This gives you y = 0.

Similarly, you can handle something like this: |x+5| + |x−5|
This is sum of "distance of x from -5" and "distance of x from 5".

Using this method, you can solve this question like this: if-y-x-5-x-5-then-y-can-take-how-many-integer-173626.html#p1379135

VeritasPrepKarishma

I am thinking the number of integer solutions to y= | x+5| + | x-5|

In this case we will have three cases:
1. When x <-5

here y can will have 2x values.

2. when -5<=x <= +5

then we have only one integral solution.

3. when x > 5

y will have 2x values

So essentially it will have infinite integral values.

Please let me know if my understanding is correct, Re: If y = |x + 5|-|x - 5|, then y can take how many integer   [#permalink] 04 Jul 2015, 13:13

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