If x is not equal to 1

[vipulgoyal]

If x is not equal to 1, is x^2/(x-1) greater than x?
(1) x is not an integer.
(2) x is positive.

I think OE needs another view, OE as follows..

Is \frac{x^2}{x-1}>x --> is \frac{x^2}{x-1}-x>0? --> is \frac{x^2-x^2+x}{x-1}>0 --> is \frac{x}{x-1}>0 --> is x<0 or x>1?

(1) x is not an integer --> clearly insufficient: for example if x=1.5 the the answer will be YES but if x=0.5 then the answer will be NO.

(2) x is positive. Also not sufficient.

(1)+(2) The values of x from (1) also satisfy (2) (x can be positive fraction from the range (0,1) or some non integer more than 1) thus even taken together statements are not sufficient.

OA Answer: E.

As B says x is +ve , which means range, from 0 to positive "excluding 0"
from a and b , range becomes 0<x<1 "not including 0 and 1"
in that case ans should be C

[VivianKerr]

I agree with you on zero, but you need to consider different fractions.

Here's how I'd break this one down:

- This is a Yes/No question, sufficiency is attained if a statement allows us to answer 100% YES or 100% NO. It doesn't matter HOW we answer it, but a statement should REMOVE the other answer as a possibility.

- Givens: x is NOT equal to 1. What can x be? Negative integers, Fractions (both + and -), Positive integers, and Zero.

- The QUESTION is whether x^2 / (x-1) > x?

Let's consider some real world numbers.

If x = 2, then x^2 / (x-1) = 4, since 4 > 2, the answer is YES.
If x = 4, then x^2 / (x-1) = 16/3, since 16/3 > 4, the answer is YES.

Obviously, if x is a positive integer, the answer will always be YES.

Let's look at some different types of numbers:

If x = 1/2, then x^2 / (x-1) = 1/4 / (-1/2) = 1/4 x -2 = -2/4 = -1/2; since -1/2 < 1/2, the answer is NO.

If x = 3/2, then x^2 / (x-1) = 9/4 / 1/2 = 9/4 x 2 = 18/4; since 18/4 > 3/2, the answer is YES.

Now that we've seen some patterns with real-world numbers, we can address the statements.

1) x is not an integer. As we saw, if x = 1/2, the answer is NO. If x = 3/2, the answer is YES. Not sufficient.

2) x is positive. Our same two examples make this one insufficient as well.

If we combine, it still will not be sufficient since BOTH 1/2 and 3/2 are NOT integers and ARE positive, and yet yield two different answers: one YES, and one NO.

The answer is (E).

[GMATGuruNY]

If x is not equal to 1, is x²/(x-1) greater than x?
(1) x is not an integer.
(2) x is positive.

CRITICAL POINT approach:

CRITICAL POINTS occur when an inequality is UNDEFINED or when the left side is EQUAL to the right side.
x²/(x-1) is undefined when x=1.
x²/(x-1) = x when x=0.
Thus, the critical points are x=0 and x=1.

To determine the ranges where x²/(x-1) > x, test one value to the left and right of each critical point.

x<0:
If we plug x=-1 into x²/(x-1) > x, we get:
(-1)²/(-1-1) > -1
-1/2 > -1.
This works.
Thus, x<0 is a valid range.

0<x<1:
If we plug x=1/2 into x²/(x-1) > x, we get:
(1/2)²/(1/2 - 1 ) > 1/2
-1/2 > 1/2.
Doesn't work.
Thus, 0<x<1 is NOT a valid range.

x>1:
If we plug x=2 into x²/(x-1) > x, we get:
(2)²/(2-1) > 2
4 > 2.
This works.
Thus, x>0 is a valid range.

Result:
x²/(x-1) > x if x<0 or x>1.

Question stem, rephrased:
Is x<0 or x>1?

Statements combined:
It's possible that x=1/2, in which case the answer to the question stem is NO.
It's possible that x=1.5, in which case the answer to the question stem is YES.
Thus, the two statements combined are INSUFFICIENT.

The correct answer is E.