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Inequalities

by Uva@90 » Fri Oct 25, 2013 8:10 pm
If (x/y)>2, is 3x+2y<18?
(1) x-y is less than 2
(2) y-x is less than 2

OA A

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by GMATGuruNY » Fri Oct 25, 2013 8:24 pm
If x/y >2, is 3x+2y<18?

1. x-y is less than 2
2. y-x is less than 2
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.
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by Uva@90 » Fri Oct 25, 2013 8:32 pm
GMATGuruNY wrote:
If x/y >2, is 3x+2y<18?

1. x-y is less than 2
2. y-x is less than 2
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.
Mitch,
Thanks a ton :)

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Uva.
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by faraz_jeddah » Sat Oct 26, 2013 11:08 am
GMATGuruNY wrote:
If x/y >2, is 3x+2y<18?

1. x-y is less than 2
2. y-x is less than 2
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.
Great algebraic approach.
How can we solve by plugging in values?
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by [email protected] » Sat Oct 26, 2013 1:46 pm
Hi Faraz et al.,

This DS question has some useful Number Properties in it and can be solved with TESTing Values. Here's how:

We're told X/Y > 2 which means that X and Y are either BOTH positive or BOTH negative.

The question asks if 3X + 2Y < 18? This is a YES/NO question.

If both X and Y are NEGATIVE, then the answer to the question is NO (and there'd be no reason to even do the math).
If both X and Y are POSITIVE, then the math IS required because the answer to the question COULD be YES or NO.

Fact 1: X - Y is less than 2

This tells us that X and Y must be relatively "close" to one another, BUT we also know that X/Y > 2, which means that X is MORE THAN TWICE Y. These 2 Facts severely LIMIT the possibilities...

X = 2, Y = 1/2....3(2) + 2(1/2) IS < 18 The answer is YES
X = 3, Y = 1.1....3(3) + 2(1.1) IS < 18 The answer is YES
X CAN'T = 4 (or larger) because there's no value for Y that "fits" both Facts
If X and Y are negative, then we get another YES
Fact1 is SUFFICIENT

Fact 2: Y - X is less than 2

Here, we can use any of our TESTs from Fact 1
X = 2, Y = 1/2 ....The answer is YES

But we also need to consider any other possibilities...
X = 100, Y = 1....3(100 + 2(1) is NOT < 18 and the answer is NO
Fact 2 is INSUFFICIENT

Final Answer: A

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