Is x – 1 < 1 for all integers x? (1) (x – 1)² > 1 (2) x < 0
A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. D. Either statement BY ITSELF is sufficient to answer the question. E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.
(D) If we take the square root of both sides of the inequality in Statement (1), we get: x – 1 > 1.
This tells us that 1 is definitely not greater than x – 1, so Statement (1) is sufficient.
Remember: When you take the square root of a variable expression that has already been squared, you must take the absolute value of the squared expression in the process. A simple example is a² = 9. Let’s solve this: a² = 9 a = 3 a = 3 or a = 3.
The same rule holds when dealing with inequalities, as we saw above.
The easiest way to evaluate the sufficiency of Statement (2) is probably to pick numbers. We are told that x is negative.
If x = 1, then the question becomes: Is 1 > 1 – 1? Is 1 > 2? Is 1 > 2? The answer is NO.
If x = 2, then the question becomes: Is 1 > 2 – 1? Is 1 > 3? Is 1 > 3? Again, the answer is NO.
The answer will always be NO if x is negative. The more negative the value of x, the greater the value of x – 1. So Statement (2) is also sufficient.
Since both statements are sufficient individually, the correct answer is choice (D).
