gmat focus DS -2

[abhasjha]

[GMATinsight]

Question : Is (x/3) + (3/x) > 2?

Statement 1) x<3

Case 1: if x = 2, (x/3) + (3/x) = (2/3) + (3/2) = 0.66+1.5 = 2.16 which is Greater than 2 'YES'
Case 2: if x = 1/2, (x/3) + (3/x) = (1/6) + (3x2) = 6.166 which is Greater than 2 'YES'
Case 3: if x = -1, (x/3) + (3/x) = (-1/3) + (-3) = -3.33 which is lesser than 2 'NO'

INSUFFICIENT

Statement 1) x>1
Case 1: if x = 2, (x/3) + (3/x) = (2/3) + (3/2) = 0.66+1.5 = 2.16 which is Greater than 2 'YES'
Case 2: if x = 3, (x/3) + (3/x) = (3/3) + (3/3) = 1+1 = 2 which is NOT Greater than 2 'NO'

INSUFFICIENT

Combining the two statements
x>1 and x<3
Case 1: if x = 2, (x/3) + (3/x) = (2/3) + (3/2) = 0.66+1.5 = 2.16 which is Greater than 2 'YES'
Case 2: if x = 3/2, (x/3) + (3/x) = (1/2) + (3/(3/2)) = 0.5+2 = 2.5 which is Greater than 2 'YES'

SUFFICIENT

Answer: Option C

[GMATGuruNY]

Is x/3 + 3/x > 2?

1) x<3
2) x>1

Determine the CRITICAL POINTS.
A critical point occurs when the inequality is UNDEFINED or when the two sides of the inequality are EQUAL.

If x=0, then 3/x is undefined.
If we test a few small values for x -- x=1, x=2, x=3 -- we'll quickly find that x/3 + 3/x = 2 when x=3.
Thus, the critical points are x=0 and x=3.

To determine where x/3 + 3/x > 2, test one value to the left and one value to the right of each critical point.

x<0:
If we plug x=-3 into x/3 + 3/x > 2, we get:
-3/3 + 3/-3 > 2
-2 > 2.
Doesn't work.
Thus, x<0 is not a valid range.

0<x<3:
If we plug x=1 into If we plug x=1 into x/3 + 3/x > 2, we get:
1/3 + 3/1 > 2
10/3 > 2.
This works
Thus, 0<x<3 is a valid range.

x>3:
If we plug x=6 into x/3 + 3/x > 2, we get:
6/3 + 3/6 > 2
5/2 > 2.
This works.
Thus, x>3 is a valid range.

Result:
x/3 + 3/x > 2 if 0<x<3 or x>3.

Question rephrased:
Is 0<x<3 or x>3?

Statement 1: x<3
If x=2, then 0<x<3.
If x=-1, then x is not in either valid range.
INSUFFICIENT.

Statement 2: x>1
If x=2, then 0<x<3.
If x=3, then x is not in either valid range.
INSUFFICIENT.

Statements combined:
Since 1<x<3, we know that x is contained with the valid range 0<x<3.
SUFFICIENT.

The correct answer is C.

[Brent@GMATPrepNow]

Is x/3 + 3/x > 2?

1) x<3
2) x>1

My solution is similar to Mitch's.
Here's how mine looks pictorially.


Cheers,
Brent

[[email protected]]

Hi abhasjha,

This DS question can be dealt with in a couple of different ways. Here's a way to approach it that uses a combination of Number Properties and TESTing Values:

We're asked if X/3 + 3/X > 2? This is a YES/NO question.

Since we're not told anything about X, we have to consider all the different possibilities, but there are some fairly straight-forward Number Properties that are hinted at in the question.

If X = ANY Negative number, then both fractions will be negative and the sum will be negative, so the answer to the question will be NO.
If X = A BIG Positive number (anything 6 or greater), then the sum will clearly be greater than 2 and the answer to the question will be YES.
If X = A TINY Positive fraction, then the sum will also be greater than 2 (since 3/(+ fraction) will be > 2) and the answer to the question will be YES.
If X = 3, then we'll end up with 1 + 1, which is NOT greater than 2, so the answer will be NO.

There are some other possibilities that we might consider, but the above 3 options are a great series of patterns to keep in mind.

Fact 1: X < 3

Here, X could be...
Any negative ---> NO answer
Positive fraction ---> YES answer
Fact 1 is INSUFFICIENT

Fact 2: X > 1

Here, X could be....
3 ---> NO answer
6 ---> YES answer
Fact 2 is INSUFFICIENT

Combined, we have 1 < X < 3

We'll have to do a bit more "math" here than normal, but it won't be too difficult...
If...
X = 2, then 2/3 + 3/2 = 15/6 = 2.5 and the answer is YES
X = 1.01, then 1.01/3 + 3/1.01 = something greater than 3 and the answer is YES
X = 2.5, then 2.5/3 + 3/2.5 = 5/6 + 6/5 = 61/30 = greater than 2 and the answer is YES
Combined, SUFFICIENT

Final Answer: C

GMAT assassins aren't born, they're made,
Rich