Is x a positive number?

[BTGmoderatorDC]

Is x a positive number?

(1) (x - 2)^2 > 2
(2) 2^x > 3^x

[Jay@ManhattanReview]

Is x a positive number?

(1) (x - 2)^2 > 2
(2) 2^x > 3^x

Question: Is x a positive number?

Let's take each statement one by one.

(1) (x - 2)^2 > 2

x can take any value: positive, 0, or negative. Insufficient.

Say, x = 0, then (x - 2)^2 > 2 => (0 - 2)^2 > 2 => 4 > 2.
Say, x = 4, then (x - 2)^2 > 2 => (4 - 2)^2 > 2 => 4 > 2.
Say, x = -1, then (x - 2)^2 > 2 => (-1 - 2)^2 > 2 => 9 > 2.

(2) 2^x > 3^x

We see that x cannot be 0 since at x = 0, 2^x > 3^x => 2^0 > 3^0 => 1 = 1. Not valid.

We note that the base (2) of the number 2^x is less than the base (3) of the number 3^x, while the exponents are the same, x each. For 2^x to be greater than 3^x, x must be negative. Let's take an example.

Say x = -2, then 2^x > 3^x => 2^(-1) > 3^(-1) => 1/4 > 1/9

You may test this with any positive number between 0 and 1.

Say x = 1/2, then 2^x > 3^x => 2^(1/2) > 3^(1/2) => 1.414 < 1.732. Not valid.

The correct answer: B

Hope this helps!

-Jay
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[Brent@GMATPrepNow]

Is x a positive number?

(1) (x - 2)^2 > 2
(2) 2^x > 3^x

Target question: Is x a positive number?

Statement 1: (x - 2)^2 > 2
Let's TEST some values.
There are several values of x that satisfy statement 1. Here are two:
Case a: x = 4 works, since (4 - 2)^2 > 2. In this case, the answer to the target question is YES, x is positive
Case b: x = -1 works, since (-1 - 2)^2 > 2. In this case, the answer to the target question is NO, x is not positive
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 2^x > 3^x
Since 2^x is always positive, we can safely divide both sides of the inequality by 2^x to get: 1 < (3^x)/(2^x)
Simplify to get: 1 < (3/2)^x
Now notice that, when x = 0, (3/2)^x equals 1
When x is a NEGATIVE integer, then (3/2)^x will be less than 1
And, when x is a POSITIVE integer, then (3/2)^x will be greater than 1
So, x must be positive.
In other words, the answer to the target question is YES, x is positive
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent