Question: If x is an integer and, what is x?

1. \(4^{x+1}-4^{x-1}>100\)
2. \(4^{x+1} +4^{x-1}>100\)

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“If x is an integer and, what is x?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Answer:

Approach Solution (1):

Given: x is an integer and \(4^x<100\)

So x could be 3/2/1/0/-1/-2/-3… For each of these values, 4x < 100

Question: What is x?

Statement 1: \(4^{x+1}-4^{x-1}>100\)

From the above statement, factor out \(4^{x-1}\). Try and factor out the smallest index to avoid dealing with fractions.

\(4^{x-1}*4^{2-1}>100\)

\(4^{x-1}*15>100\)

For this product to be greater than 100, \(4^{x-1}\) should be atleast \(4^2\) or x must be 3 or greater than 3

Since x is one of 3/2/1/0/-1/…, x must be 3.

Hence sufficient

Statement 2: \(4^{x+1} +4^{x-1}>100\)

From above, factor out \(4^x\)

\(4^x*(4+1)>100\)

\(4^x*5>100\)

For this product to be greater than 100, \(4^x\) should be atleast \(4^3\) or x must be 3 or greater

Since x is one of 3/2/1/0/-1/…, x must be 3.

Hence sufficient

Correct option: D

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