If x is an integer, is 9^x + 9^(-x) = b ? (1) 3^x + 3^(-x) = (b + 2) GMAT Data Sufficiency

Question: If x is an integer, is 9^x + 9^(-x) = b?

(1) 3^x + 3^(-x) = √(b+2)
(2) x > 0

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, Is 9^X + 9^(-X) = B ? (1) 3^X + 3^(-X) = (B + 2)” - is a topic that belongs to the GMAT Quantitative reasoning section of GMAT. The question is borrowed from the book “GMAT Quantitative Review”. The GMAT Quant section mainly comprises a set of 31 questions that need to be answered logically. GMAT Data Sufficiency questions include a problem statement that is heeded by two factual arguments. GMAT data sufficiency consists of 15 questions which are two-fifths of the entire sum of 31 GMAT quant questions.

Solution and Explanation

Approach Solution 1:

1. The first factual statement states that: 3^x + 3^(-x) = √(b+2)

By squaring both sides, we can get,
9^x + 2 * 3^x * ⅓^x +9^-x = b+2
Therefore, 9^x + 9^-x = b
Thus the equation given in the question can be proved by the given argument in statement one. Hence the statement alone is sufficient.

2. The second factual statement implies that x > 0.

From this statement, we cannot prove the equation of the given question. Hence the statement alone is not sufficient. Further information is required to prove the equation cited in the question.
Therefore, option A is the correct answer.

Correct Answer: A

Approach Solution 2:

The question suggests that:
if x is an integer, is 9^x + 9^-x = b?
That can be derived as 3^2x + 3^-2x = b

1. The 1st argument cites as 3^x + 3^-x = √(b+2)

3^x + 3^-x = √(b+2)
By squaring both sides, we can get,
(3^x + 3^-x)^2 = b+2
3^2x + 2* 3^x * 3^-x + 3^-2x = b+2
3^2x + 2 + 3^-2x = b+2
3^2x + 3^-2x = b

Therefore, this equation corresponds to the above-simplified equation of the given question.
Hence statement one alone is sufficient to prove the given equation of the question.

2. The second statement demonstrates x > 0. This statement is not justified because it does not possess fundamental facts to decode the equation given in the question.

Therefore, we can conclude that this statement alone is not sufficient to prove the equation of the given question.
Hence, option A is the correct answer.

Correct Answer: A

Approach Solution 3:

1. Since we know that 9^x is the square of 3^x, therefore, we need to square the statement one in order to prove the equation.

The first statement implies the equation: 3^x + 3^-x = √(b+2)
Therefore, by squaring both sides, we get
3^2x + 3^-2x + 2* 3^x* 3^-x =√(b+2)^2
9^x + 9^-x = b + 2 - 2 = b
Hence the statement alone is sufficient to prove the equation mentioned in the question.

2. The second factual statement states x > 0, which is insufficient to prove the equation given in the question.

Hence option A is the correct answer.

Correct Answer: A

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