Data Sufficiency

If x/y >2, is 3x+2y<18?

a. x-y is less than 2
b. y-x is less than 2


oa a

question to experts : i know that we can add inequalities when both have same ">" signs but can we substruct inequalitities?

mariah wrote:If x/y >2, is 3x+2y<18?

a. x-y is less than 2
b. y-x is less than 2


oa a

question to experts : i know that we can add inequalities when both have same ">" signs but can we substruct inequalitities?

Since x/y > 2, x and y have the same sign.
If x and y are both negative, then 3x + 2y < 18, and the answer is a definite YES.
Thus, our concern here is what happens to the value of 3x+2y when x and y are both positive.
If x and y are both positive, then x>2y -- information that we can use when we evaluate each statement.
To add inequalities, the <> must face the same direction in each inequality.

Statement 1: x-y<2.
Thus, y+2 > x.
Adding this inequality to x>2y, we get:
(y+2) + x > x + 2y
2 > y.
Since y<2, and x<y+2, x<4.
This means that the upper limit of 3x+2y = 3(4) + 2(2) = 16.
Thus, when x and y are both positive, 3x+2y < 18.
SUFFICIENT.

Statement 2: y-x<2.
Thus, x+2 > y.
Adding this inequality to x>2y, we get:
(x+2) + x > y+ 2y
3y-2x < 2.
If y=2 and x=3, then 3x+2y = 12.
If y=10 and x=15, then 3x+2y = 60.
Since in the first case 3x+2y<18 and in the second case 3x+2y>18, INSUFFICIENT.

The correct answer is A.

Many DS questions that involve inequalities are best solved using a combination of algebra and plugging in values.
Algebra often is the most efficient way to prove that a statement is SUFFICIENT.
Plugging in values often is the most efficient way to prove that a statement is INSUFFICIENT.

I would advise against looking for ways to subtract one inequality from another.

We want x and y to be positive numbers in order to try to make the second equation be > 18.

1. If x - y < 2 and x/y > 2 then we know x < 3 because 3-1=2 and larger numbers separated by "almost 2" will fail the x/y > 2 check.

So we can test a number pair into the second equation (such as x = 2.9 and y = 1) which just barely satisfies x - y < 2 and x/y > 2, and see how close to 18 we get:

3*(2.9) + 2*(1) = 10.7

Larger number pairs can not satisfy both equations, and 10.7 much less than 18, so I'm comfortable with 1 being sufficient by itself.

2. If y - x < 2 and x/y > 2 then x and y can be very large numbers (x=100, y=20 works) or small numbers. So it's easy to see that conditions exist where the second equation is both greater than and less than 18, by inspection.

Therefore [spoiler]2 is insufficient by itself, and it contradicts 1 so we can't use 2[/spoiler].

Answer: A

Another approach, without adding or subtracting inequalities.

Once you reduce the question to both 'x' and 'y' being +ve.

We have, x>2y - - - - - (1)

Statement 1

x-y < 2

x < y + 2 - - - - - - (2)

Combining we have,

2y < x < y+2

2y < y+2

y < 2

Now if y < 2, we have x < 4 from (2) Sufficient

Statement 2

y - x < 2

x > y - 2 - - - - - - - (3)

From (1) and (3)

x > y - 2 (and) x > 2y

From here we can have infinite possibilities which satisfy these 2 equations. Insufficient.