If x < 3, is (x + 1)/(x - 3)

[Brent@GMATPrepNow]

Here's one I just made up.
Estimated difficulty level: 650

If x < 3, is (x + 1)/(x - 3) > 1/3?

(1) x < 2
(2) x > -1

Answer: B

Cheers,
Brent

[DavidG@VeritasPrep]

Here's one I just made up.
Estimated difficulty level: 650

If x < 3, is (x + 1)/(x - 3) > 1/3?

(1) x < 2
(2) x > -1

Answer: B

Cheers,
Brent

Brent original!

We could always do a little number-picking.

S1: Say x = 0; (x + 1)/(x - 3) = 1/-3 = -(1/3). Not greater than 1/3 so we get a NO.
x = -10; (x + 1)/(x - 3) = -9/-13 = 9/13. Greater than 1/3, so we get a YES. S1 is not sufficient.

S2: We already know that x = 0 gives us a NO. What about 1? (x + 1)/(x - 3) = 2/-2 = -1. Not greater than 1/3. Again, we get a NO. Not a coincidence.

If -1 < x < 3, the numerator will always be positive and the denominator will always be negative, so we're dealing with a negative number, which, of course, is less than 1/3. Because the answer is always NO, this statement alone is sufficient. Answer is B

[Brent@GMATPrepNow]

If x < 3, is (x + 1)/(x - 3) > 1/3?

(1) x < 2
(2) x > -1

Given: x < 3

Target question: Is (x + 1)/(x - 3) > 1/3?
This is a good candidate for rephrasing the target question.
Aside: Here's a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Since x < 3, we know that (x - 3) will always be NEGATIVE.
So, let's take (x + 1)/(x - 3) > 1/3, and multiply both sides by (x - 3)
We get: x + 1 < (1/3)(x - 3) [ASIDE: since we multiplied both sides by a NEGATIVE value, we REVERSED the inequality sign]
Now take x + 1 < (1/3)(x - 3) and multiply both sides by 3 to get: 3(x + 1) < x - 3
Expand to get: 3x + 3 < x - 3
Subtract x from both sides: 2x + 3 < -3
Subtract 3 from both sides: 2x < -6
Divide both sides by 2 to get: x < -3

This equivalent inequality is much easier to work with. So, ....
REPHRASED target question: Is x < -3?

Statement 1: x < 2
There are several values of x that satisfy statement 1. Here are two:
Case a: x = -7, in which case x < -3
Case b: x = 0, in which case x > -3
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x > -1
If x is greater than -1, then we can be 100% certain that x IS NOT less than -3
Since we can answer the REPHRASED target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent