If y = |2 + x| - |2 x| and |2x 15|

Question

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    the e-GMAT Question Series on Absolute Value.

Provide your solution below. Kudos for participation. Happy Solving! :-D

Best Regards
The e-GMAT Team

https://e-gmat.com/blogs/wp-content/uploads/2018/08/Free-Session-Algebra-e1533292089856.jpg

Lucky2783 · Accepted Answer

|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

Answer 1

UJs · Answer

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 2 as (6.5 2 , so 2-x will be negative y = |2 + x| - |2 – x| y = 2 + x -(-(2 - x)) y = 4 ; constant value Ans : A

reto · Answer

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>> 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

UJs · Answer

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

EgmatQuantExpert · Answer

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.

How does your answer change?

Best Regards

Japinder

Lucky2783 · Answer

Also using the graphs we can immediately see that y=4 is only solution.

csirishac · Answer

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 · Answer

Official Explanation

Correct Answer: B

|2x – 15| < 2

Can also be written as |x – 7.5| < 1

https://s17.postimg.org/jatn85l9b/Q5_2.png

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:

https://s12.postimg.org/k1jh0mvpp/image.png 
 
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.

EgmatQuantExpert · Answer

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.

Does this clarify your doubt?

Best Regards, Japinder

energetics · Answer

Question 1: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values can y take? |2x – 15| < 2 --> 6.5 0

SchruteDwight · Answer

Without considering the condition that  13<2x<17 , We are dealing with three ranges (if this is unclear check out this link  https://gmatclub.com/forum/for-how-many-integer-values-of-x-is-x-8-5-x-x-231930.html#p1987331 )

These ranges are

x<-2

-2<x<2

2<x

Now, we get the limitation that  13<2x<17 . Hence, the only range we have to consider is  2<x

In that range,  |2+x|>0  and  |2-x|<0

Therefore,

2+x-(x-2)=y \implies y=4 \implies \ 1 \ value

Jyoti3380 · Answer

I got B as my answer.

How I did it is
1) Solved both the given equations
 Y=2x,-2x,4,-4
 X= Range is between 6.5 & 8.5. Hence, X=7,8 as well.

2) Using these X's value in 1st equation. Y=4,-4,-2x : these will be ruled out as they are out of my range. Hence only way is Y=2x, which is 2*4=8....a possible outcome as per found out range.

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

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