If x and y are positive integers and |x - 2|

Question

If  x  and  y  are positive integers and  |x - 2| \lt 2 - y , what is the value of  xy ?

A. 1
B. 2
C. 3
D. 6
E. Cannot be determined from the given information

Bunuel · Accepted Answer

Official Solution:

If  x  and  y  are positive integers and  |x - 2| \lt 2 - y , what is the value of  xy ?

A. 1
B. 2
C. 3
D. 6
E. Cannot be determined from the given information

We know that the absolute value is always non-negative, hence, the smallest possible value for  |x-2|  is 0. Therefore, from  |x-2| \lt 2-y , we get  0 \lt 2 - y , which simplifies to  y \lt 2 . Combining this with the fact that  y  is a positive integer, we get  y=1 .

Substituting  y=1  into  |x-2| \lt 2-y , we get  |x-2| \lt 1 , which simplifies to  -1 \lt x-2 \lt 1 . This gives us  1 \lt x \lt 3 , which means  x  can only be 2. Therefore,  xy=2*1=2 .

Answer: B

TOBEATHATPERSON · Answer

Only positive integer

So 2-y can be 1, 2
Can't be 2 so y is 1 x is 2

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gmatophobia · Answer

|x - 2| \lt 2 - y

2 - y \geq 0

y \leq 2

Hence, possible values of y = 1 or 2.

y cannot be equal to 2, as the inequality will then be  |x - 2| \lt 0 . This is not possible

Hence, y = 1

|x - 2| \lt 2 - 1

|x - 2| \lt 1

-1 < x - 2 < 1

As x is in an integer, the only possible value ⇒ x - 2 = 0

x = 2

xy = 2 * 1 = 2

Option B

BrushMyQuant · Answer

Given that x and y are positive numbers and \(|x - 2| \lt 2 - y\) and we need to find the value of xy

As we have \|x-2\| in the equation so we will have two cases
-Case 1: x-2 ≥ 0 or x ≥ 2 => \|x - 2\| = x-2 => x - 2 < 2 - y => x + y < 2 + 2 => x + y < 4 Now x and y are positive integers so y has to be at least 1 and x ≥ 2 => Only possible case is x = 2 and y = 1 => x y = 2*1 = 2	-Case 2: x-2 ≤ 0 or x ≤ 2 => \|x - 2\| = -(x - 2) = 2 - x => 2 - x < 2 - y => x > y Now x and y are positive integers so y has to be at least 1 and x ≤ 2 => Only possible case is x = 2 and y = 1 => x y = 2*1 = 2

So, Answer will be B
Hope it helps!

Watch the following video to learn the Basics of Absolute Values

Joaquinramirez7 · Answer

Why do you need the samllest possible value of left side? I mean, i got that you are trying to make left side =0 to establish upper limit of y (y<2), but you dont get this situation getting smallest value from the left side, yo get this making left side = 0, I'm not sure that alway that i need upper limit of y in equation like this I going to solve it with the smallest value in the other side, but probably i will find it making the other side = 0, do you understand my question?

Kinshook · Answer

If x and y are positive integers and |x - 2| \lt 2 - y , what is the value of xy ? |x-2| < 2-y Case 1: x>=2 x-2 < 2-y x+y < 4 (x,y) = (2,1) Case 2: x<2 2-x < 2-y x >y Not feasible xy = 2*1 = 2 IMO B

Bunuel · Answer

I’m not entirely sure I follow your question, but here’s how the solution works: The left-hand side of the inequality is an absolute value, which is always greater than or equal to 0. Therefore, the right-hand side, which is stated to be greater than the left-hand side, must be positive. This gives us y < 2. Since it’s given that y is an integer, we conclude that y = 1. Using this logic, we were able to determine the value of y. Here are some additional examples of absolute value problems that can be solved using the method described above: https://gmatclub.com/forum/how-many-dif ... 29658.html https://gmatclub.com/forum/the-equation ... 21149.html https://gmatclub.com/forum/if-x-3x-2-th ... l#p2955120 Hope this helps.

bumpbot · Answer

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If x and y are positive integers and |x - 2| < 2 - y, what is the

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