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GMAT Practice Question (One of Hundreds)

x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity 7^x-1 - 5^x?

z < 25 and w = 7^x
x = 4

A)	Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B)	Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C)	BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D)	EACH statement ALONE is sufficient.
E)	Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Correct Answer: A

The question could be written: x is a positive integer. Is w - z > 5(7(^x-1)) - 5^x+1

Simplify the question: since raising a number to an odd power does not change the sign, x is a positive integer.
The question, is w - z > 5(7^x-1 - 5^x)?, can be simplified to: is w - z > 5*7^x-1 - 5^x+1?
Evaluate Statement (1) alone.
1. Statement (1) allows the following substitution:
  Is 7^x - (a number less than 25) > 5(7^x-1) - 5^x+1?
  Equivalently: Is 7^x - (a number less than 5²) > 5(7^x-1) - 5^x+1?
2. If this question can be answered definitively for all legal values of x (i.e., positive integers), Statement (1) is sufficient. Although this statement is difficult to evaluate algebraically, a little logic makes Statement (1) plainly sufficient. It is helpful to step back and see the logic about to be employed.
  
  a - b will always be greater than c - d if these numbers are positive and a > c and b < d. In this situation, a smaller number (b is smaller than d) is being subtracted from a larger number (a is greater than c). Consequently, if the left side of the equation starts from a larger number and subtracts a smaller number than the right side of the equation, it is quite clear that the difference on the left side will be larger than the difference on the right side of the inequality.
  
  For example: 10 - 2 > 5 - 4
  You are starting with a larger number on the left (i.e., 10 > 5) and subtracting a smaller number on the left (2 < 4). Consequently, it only makes sense that the number on the left is going to be larger.
3. This same logic holds true in the inequality derived in Statement (1). Since x is a positive integer (it is essential to know this), 7^x will be bigger than 5(7^x-1). You know this is true because there will be x sevens on the left side of the inequality and (x-1) sevens on the right side of the inequality. The extra 7 on the left will out-weight the extra 5 on the right, making the left side start with a larger number.
4. Continuing with this logic, (a number less than 5²) will be less than 5^x+1 since x is a positive integer and the smallest possible value for x (i.e., 1) makes 5^x+1 = 5¹⁺¹ = 5² = 25. Since 5^x+1 will always be at least 25, it will always be greater than (a number less than 25). Statement (1) is SUFFICIENT.
  Note: If z were a negative number, which it could be, the inequality would still hold true. It would make the left side of the inequality even larger as we would effectively be adding a number to 7^x.
Evaluate Statement (2) alone.
1. Statement (2) says that x = 4. The inequality can now be re-written:
  is w - z > 5(7^4-1 - 5⁴⁺¹)?
  In other words, is w - z > 1,715 - 3,125?
  Or, to simplify it as much as possible:
  is w - z > -1,410?
  If w = 7⁴ = 2,401 and z = 1, the answer is YES. However, if w = -100,000 (nothing in Statement (2) precludes this possibility—do not import information over from Statement (1)) and z = 1, the answer is NO. Since Statement (2) does not provide enough information to definitively answer the original question, it is NOT SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer A is correct.

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