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21 Apr 2024, 10:17
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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
A. 1
B. 2
C. 3
D. 6
E. Cannot be determined from the given information
21 Aug 2024, 21:43
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|x - 2| < 2 - y,
- 2 + y < x - 2 < 2 - y
now,
- 2 + y < x - 2
so x - y > 0 --(i)
and,
x - 2 < 2 - y
so x + y < 4 --(ii)
from (i) we can deduce that x > y,
from (ii) we can deduce that (x,y) must be (2,1) (since we can take only positive integers, we don't take pairs with 0s)
thus, xy = 2.
Is this a valid solution Bunuel ?
- 2 + y < x - 2 < 2 - y
now,
- 2 + y < x - 2
so x - y > 0 --(i)
and,
x - 2 < 2 - y
so x + y < 4 --(ii)
from (i) we can deduce that x > y,
from (ii) we can deduce that (x,y) must be (2,1) (since we can take only positive integers, we don't take pairs with 0s)
thus, xy = 2.
Is this a valid solution Bunuel ?
General Discussion
21 Apr 2024, 10:17
Kudos
Bookmarks
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
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
