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# If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values

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If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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Updated on: 07 Aug 2018, 05:05
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Question Stats:

48% (02:20) correct 52% (02:32) wrong based on 850 sessions

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If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values can y take?

(A) 0
(B) 1
(C) 2
(D) 4
(E) Cannot be determined

This is Question 4 for

Provide your solution below. Kudos for participation. Happy Solving!

Best Regards
The e-GMAT Team

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Originally posted by EgmatQuantExpert on 20 May 2015, 09:23.
Last edited by EgmatQuantExpert on 07 Aug 2018, 05:05, edited 3 times in total.
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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20 May 2015, 10:12
6
4
|2x – 15| < 2
So 6.5 < x < 8.5

In y = |2 + x| - |2 – x| for above values of X , |2+x| will be equal to 2+X and |2 – x| will be equal to x-2

Substituting
y=2+x-x+2=4

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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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20 May 2015, 13:30
3
EgmatQuantExpert wrote:
If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values can y take?

(A) 0
(B) 1
(C) 2
(D) 4
(E) Cannot be determined

This is Question 4 for

Provide your solution below. Kudos for participation. The Official Answer and Explanation will be posted on 22nd May.

Till then, Happy Solving!

Best Regards
The e-GMAT Team

Here's how to solve it:

First find the range of values for x:

Inequalities involving absolute values - here you have a number case again. First solve like this without the brackets:

2x – 15 < 2
2x < 17
x < 8.5

Second case (FLIP the sign and put a negative sign around the other side of the inequality):

2x – 15 > -2
x>6.5

Hence: 6.5<x<8.5 >>> since x is an integer, only 7 is possible as a value.

Now put in 7 in If y = |2 + x| - |2 – x| to see that there is only 1 solution. ANSWER CHOICE A
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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20 May 2015, 21:12
1
1
Solving |2x – 15| < 2 first to get range of x
(2x – 15) < 2
x < 8.5

-(2x – 15) < 2
x > 6.5

Range of x 6.5<x<8.5

y = |2 + x| - |2 – x| to see that there is only 1 solution.

$$|2 + x| = 2 + x$$ ...... is always +ve , x > 2 as (6.5<x<8.5)

$$|2 – x| = -(2 - x)$$ ...... range of x > 2 , so 2-x will be negative

$$y = |2 + x| - |2 – x|$$
$$y = 2 + x -(-(2 - x))$$
$$y = 4 ;$$
constant value

Ans : A
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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20 May 2015, 21:20
2
reto wrote:

Hence: 6.5<x<8.5 >>> since x is an integer, only 7 is possible as a value.

Now put in 7 in If y = |2 + x| - |2 – x| to see that there is only 1 solution. ANSWER CHOICE A

reto, want to correct one thing in your solution, there are more possible values for x that will result in y as integer

for Example : integer value 8 , and you can take x = 15/2 , 6.9, 7.2 any fraction between (6.5 - 8.5) all result in integer value for Y
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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21 May 2015, 03:21
UJs wrote:
reto wrote:

Hence: 6.5<x<8.5 >>> since x is an integer, only 7 is possible as a value.

Now put in 7 in If y = |2 + x| - |2 – x| to see that there is only 1 solution. ANSWER CHOICE A

reto, want to correct one thing in your solution, there are more possible values for x that will result in y as integer

for Example : integer value 8 , and you can take x = 15/2 , 6.9, 7.2 any fraction between (6.5 - 8.5) all result in integer value for Y

Dear reto

Please note that the question doesn't mention that x is an integer. So, it would be wrong to assume that. In this case, you were able to answer the question correctly even with this wrong assumption. But in a different question, this wrong assumption would have led you to an incorrect answer.

For example, try this variation of the above question:

If y = |2 + x| - |2 – x| and |x – 1| < 1, how many integer values can y take?

(A) None
(B) 1
(C) 2
(D) 3
(E) Cannot be determined

First solve it without using the constraint that x is an integer.
Then solve the question using the constraint that x is an integer.

Best Regards

Japinder
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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21 May 2015, 10:33
1
Lucky2783 wrote:
|2x – 15| < 2
So 6.5 < x < 8.5

In y = |2 + x| - |2 – x| for above values of X , |2+x| will be equal to 2+X and |2 – x| will be equal to x-2

Substituting
y=2+x-x+2=4

Also using the graphs we can immediately see that y=4 is only solution.
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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22 May 2015, 03:21
X is between 0 and 2.
y=2x
Does this mean the number of integer values of y can't be determined ? Please suggest.

EgmatQuantExpert wrote:
UJs wrote:
reto wrote:

Hence: 6.5<x<8.5 >>> since x is an integer, only 7 is possible as a value.

Now put in 7 in If y = |2 + x| - |2 – x| to see that there is only 1 solution. ANSWER CHOICE A

reto, want to correct one thing in your solution, there are more possible values for x that will result in y as integer

for Example : integer value 8 , and you can take x = 15/2 , 6.9, 7.2 any fraction between (6.5 - 8.5) all result in integer value for Y

Dear reto

Please note that the question doesn't mention that x is an integer. So, it would be wrong to assume that. In this case, you were able to answer the question correctly even with this wrong assumption. But in a different question, this wrong assumption would have led you to an incorrect answer.

For example, try this variation of the above question:

If y = |2 + x| - |2 – x| and |x – 1| < 1, how many integer values can y take?

(A) None
(B) 1
(C) 2
(D) 3
(E) Cannot be determined

First solve it without using the constraint that x is an integer.
Then solve the question using the constraint that x is an integer.

Best Regards

Japinder
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Joined: 04 Jan 2015
Posts: 2323
Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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22 May 2015, 06:40
Official Explanation

|2x – 15| < 2

Can also be written as |x – 7.5| < 1

The expression |x - 7.5| represents the distance between x and 7.5 on the number line.

The inequality |x - 7.5| < 1 means that the distance between x and 7.5 on the number line is less than 1.

This means, x lies between 6.5 and 8.5, exclusive.

Now, y = |2+x| - | 2 – x |

Rewriting this as
y = |x + 2| - | x – 2|

Here |x+2| represents the distance of x from -2 on the number line
And, |x-2| represents the distance of x from 2 on the number line.

Representing these 2 distances on number line:

It’s easy to see that y = 4.

So, only 1 possible value of y.

Note: the question is asking about HOW MANY values y can have, not WHAT values y can have.
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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22 May 2015, 06:49
1
1
csirishac wrote:
X is between 0 and 2.
y=2x
Does this mean the number of integer values of y can't be determined ? Please suggest.

EgmatQuantExpert wrote:

For example, try this variation of the above question:

If y = |2 + x| - |2 – x| and |x – 1| < 1, how many integer values can y take?

(A) None
(B) 1
(C) 2
(D) 3
(E) Cannot be determined

Dear csirishac

You are right that the highlighted part in the quote above gives us: 0 < x < 2

You're also right that for this range of x, y = 2x

Now, the question is: how many integer values can y have?

Note that we are not given that x is an integer. So, x can take any decimal values as well.

When x = 0.5, y = 1
When x = 1, y = 2
When x = 1.5, y = 3

Thus, y can take 3 integer values only. So, for this question, the correct answer will be D.

Best Regards, Japinder
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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02 Dec 2015, 15:24
EgmatQuantExpert wrote:
Official Explanation

|2x – 15| < 2

Can also be written as |x – 7.5| < 1

The expression |x - 7.5| represents the distance between x and 7.5 on the number line.

The inequality |x - 7.5| < 1 means that the distance between x and 7.5 on the number line is less than 1.

This means, x lies between 6.5 and 8.5, exclusive.

Now, y = |2+x| - | 2 – x |

Rewriting this as
y = |x + 2| - | x – 2|

Here |x+2| represents the distance of x from -2 on the number line
And, |x-2| represents the distance of x from 2 on the number line.

Representing these 2 distances on number line:

It’s easy to see that y = 4.

So, only 1 possible value of y.

Note: the question is asking about HOW MANY values y can have, not WHAT values y can have.

Hi Japinder,

Everything is clear to me untill the point you say that clearly y=4. Can you pls show me how we derive that y=4?

Thanks.

Regards.
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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02 Dec 2015, 17:25
1
elisabettaportioli wrote:

Hi Japinder,

Everything is clear to me untill the point you say that clearly y=4. Can you pls show me how we derive that y=4?

Thanks.

Regards.

I am assuming that you were able to understand how you got 6.5<x<8.5 from |2x-15|<2.

Once you get this, you realize that for the interval 6.5 < x < 8.5, |2 + x| = 2+x and 2-x = -(2-x) = -2+x ....as |x-a| = x-a for x $$\geq$$ a and = -(x-a) for x < a

So putting the above values in the equation for y, you get,

y = |2+x|-|2-x| = 2+x - (x-2) = 2+x -x +2 = 4

Hope this helps.
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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04 Dec 2015, 01:23
1
Engr2012 wrote:
elisabettaportioli wrote:

Hi Japinder,

Everything is clear to me untill the point you say that clearly y=4. Can you pls show me how we derive that y=4?

Thanks.

Regards.

I am assuming that you were able to understand how you got 6.5<x<8.5 from |2x-15|<2.

Once you get this, you realize that for the interval 6.5 < x < 8.5, |2 + x| = 2+x and 2-x = -(2-x) = -2+x ....as |x-a| = x-a for x $$\geq$$ a and = -(x-a) for x < a

So putting the above values in the equation for y, you get,

y = |2+x|-|2-x| = 2+x - (x-2) = 2+x -x +2 = 4

Hope this helps.

ok, let me see If I got it right...

If 6.5<x<8.5, then x is positive.

Now, with this understanding we try to calculate the following equation y=|2+x|-|2-x| and to do so we need to know the value of |2+x| and |2-x| but since we know that x>0, we can consider only the positive case for |2+x| that leads to 2+x and the positive case for |2-x| that leads to 2-x.

Now we can substitute the values for the mods in the equation and we get

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

The outcome I get is not 4 as it you should be. I only get 4 if I rearrange the sign inside the second mod, and precisely:
|2-x|=|x-2|

indeed y=2+x-(x-2)=2+x-x+2=4

Now my question is WHY SHOULD WE ASSUME THAT TO GET THE CORRECT ANSWER WE NEED TO ARRANGE THE SIGN IN |2-x|??

Thanks.
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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04 Dec 2015, 04:42
1
elisabettaportioli wrote:
Engr2012 wrote:

I am assuming that you were able to understand how you got 6.5<x<8.5 from |2x-15|<2.

Once you get this, you realize that for the interval 6.5 < x < 8.5, |2 + x| = 2+x and 2-x = -(2-x) = -2+x ....as |x-a| = x-a for x $$\geq$$ a and = -(x-a) for x < a

So putting the above values in the equation for y, you get,

y = |2+x|-|2-x| = 2+x - (x-2) = 2+x -x +2 = 4

Hope this helps.

ok, let me see If I got it right...

If 6.5<x<8.5, then x is positive.

Now, with this understanding we try to calculate the following equation y=|2+x|-|2-x| and to do so we need to know the value of |2+x| and |2-x| but since we know that x>0, we can consider only the positive case for |2+x| that leads to 2+x and the positive case for |2-x| that leads to 2-x.

Now we can substitute the values for the mods in the equation and we get

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

The outcome I get is not 4 as it you should be. I only get 4 if I rearrange the sign inside the second mod, and precisely:
|2-x|=|x-2|

indeed y=2+x-(x-2)=2+x-x+2=4

Now my question is WHY SHOULD WE ASSUME THAT TO GET THE CORRECT ANSWER WE NEED TO ARRANGE THE SIGN IN |2-x|??

Thanks.

Ok. Couple of things. Text in red above is not correct. Second, you need to brush up your knowledge of absolute values. Look here math-absolute-value-modulus-86462.html

Once you get that 6.5<x<8.5, this means that x is not just positive but this range of 'x' will also give you |2-x| = -(2-x) and NOT 2-x

This is because, |a-b| = -(a-b) if a < b. So in our example as all values of 'x' will be >2 , |2-x| = -(2-x).

Text in greegn above "WHY SHOULD WE ASSUME THAT TO GET THE CORRECT ANSWER WE NEED TO ARRANGE THE SIGN IN |2-x|??" is absolutely the wrong way of looking at the correct logic behind this question. We are not assuming anything. We are going by the rules of absolute values.

Hope this helps.
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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16 May 2017, 09:32
EgmatQuantExpert wrote:
If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values can y take?

(A) 0
(B) 1
(C) 2
(D) 4
(E) Cannot be determined

This is Question 4 for

Provide your solution below. Kudos for participation. Happy Solving!

Best Regards
The e-GMAT Team

Hey..i got it in 1:49:)

from mod(2x+15)<2 we get
lx+7.5l<1 i.e. distance of x from -7.5 or 7.5 is less than 1

therefore -8.5<x<-6.5 or 6.5<x<8.5 ....as the question is screaming integer...test first the integer value say 7
wwe get y = 4

now test any decimal value say 8.4 and 6.6...we get non integer value
just to be sure try another non integer value say -8.3 and -6.6 we get no integer vakue

so only one integer value...

But i need a concept clarity
it is said that one should interpret lxl as distance of x from zero i.e. lx-0l..so drawing a parallel we can interpret lx-7l as distance of x from 7
my question, so should we interpret lx+7l as distance of x from (-) 7??...had i been sure about it..i would not have tried above question from both positive and negative values..

Pl help:)
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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13 Jul 2017, 07:25
1
reto wrote:
EgmatQuantExpert wrote:
If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values can y take?

(A) 0
(B) 1
(C) 2
(D) 4
(E) Cannot be determined

This is Question 4 for

Provide your solution below. Kudos for participation. The Official Answer and Explanation will be posted on 22nd May.

Till then, Happy Solving!

Best Regards
The e-GMAT Team

Here's how to solve it:

First find the range of values for x:

Inequalities involving absolute values - here you have a number case again. First solve like this without the brackets:

2x – 15 < 2
2x < 17
x < 8.5

Second case (FLIP the sign and put a negative sign around the other side of the inequality):

2x – 15 > -2
x>6.5

Hence: 6.5<x<8.5 >>> since x is an integer, only 7 is possible as a value.

Now put in 7 in If y = |2 + x| - |2 – x| to see that there is only 1 solution. ANSWER CHOICE A

integer values possible are 7 and 8.
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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26 Jul 2017, 12:59
elisabettaportioli wrote:
EgmatQuantExpert wrote:
Official Explanation

|2x – 15| < 2

Can also be written as |x – 7.5| < 1

The expression |x - 7.5| represents the distance between x and 7.5 on the number line.

The inequality |x - 7.5| < 1 means that the distance between x and 7.5 on the number line is less than 1.

This means, x lies between 6.5 and 8.5, exclusive.

Now, y = |2+x| - | 2 – x |

Rewriting this as
y = |x + 2| - | x – 2|

Here |x+2| represents the distance of x from -2 on the number line
And, |x-2| represents the distance of x from 2 on the number line.

Representing these 2 distances on number line:

It’s easy to see that y = 4.

So, only 1 possible value of y.

Note: the question is asking about HOW MANY values y can have, not WHAT values y can have.

Hi Japinder,

Everything is clear to me untill the point you say that clearly y=4. Can you pls show me how we derive that y=4?

Thanks.

Regards.

hi

as far as the number line is concerned, just subtract -2 from 2, a difference revealing the number of numbers y can take...

hope this helps ...
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values  [#permalink]

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12 Aug 2018, 04:18
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Re: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values &nbs [#permalink] 12 Aug 2018, 04:18
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