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

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

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.

How does your answer change?

Best Regards

Japinder

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

X is between 0 and 2.
y=2x
Does this mean the number of integer values of y can't be determined ? Please suggest.

Official Explanation

Correct Answer: B

|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.

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

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.

Let me try to answer your question.

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|??

Please advise. Your contitube is pure gold!

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.

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:)

integer values possible are 7 and 8.

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 ...

Question 1: If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values can y take?
|2x – 15| < 2 --> 6.5<x<8.5

Take x = 6.5
y = |2+6.5| - |2-6.5| = 8.5 - 4.5 = 4
MIN y = 4

Take x = 8.5
y = |2+8.5| - |2-8.5| = 10.5 - 6.5 = 4
MAX y = 4
So y can only have 1 value, or for any value of x in the range, y = 4



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

Take x = 0
y = |2+0| - |2-0| = 2 - 2 = 0
MIN y = 0

Take x = 2
y = |2+2| - |2-2| = 4 - 0 = 4
MAX y = 4
So y can have 3 int values, 1,2 & 3 for the range 0<x<2



EgmatQuantExpert is it OK to consider the problem as finding the MIN/MAX of y?

Posted from my mobile device





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

Why is 8 not possible as a value? Integers less than 8.5 and greater 6.5 are 7 and 8, is it not?

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,

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