# Manhattan GMAT Challenge Problem of the Week - 2 Aug 2011

by Manhattan Prep, Aug 2, 2011

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## Question

If #x# represents the smallest even integer greater than or equal to [pmath]x^2[/pmath], is #x# less than 12?

(1) |x 3| 1

(2) |x 1| 2

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. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

First, figure out the value of the strange symbol for different small values of x.

#x# = smallest even integer greater than or equal to [pmath]x^2[/pmath].

If x = 1, then we get [pmath]1^2[/pmath] = 1, and the smallest even integer greater than or equal to that is 2. So #1# = 2, which is less than 12.

If x = 2, then we get [pmath]2^2[/pmath] = 4, and the smallest even integer greater than or equal to that is 4. So #2# = 4, which is less than 12.

If x = 3, then we get [pmath]3^2[/pmath] = 9, and the smallest even integer greater than or equal to that is 10. So #3# = 10, which is less than 12.

If x = 4, then we get [pmath]4^2[/pmath] = 16, and the smallest even integer greater than or equal to that is 16. So #4# = 16, which is NOT less than 12.

Larger values of x will also give us results greater than 12.

The value of x is not restricted to integers (or to positives, for that matter), but we should have a sense of what values work. If x is 1, 2, or 3, we get an answer of Yes. If x is 4 or bigger, we get No.

Statement 1: INSUFFICIENT. The condition |x 3| 1 can be translated as follows: x is no more than 1 unit away from 3 on a number line. Thus, the largest value of x is 4, while the smallest is 2. (Plug back in and verify this.) Since we have possible values of 2, 3, and 4, we have some Yess and some Nos.

Statement 2: SUFFICIENT. The condition |x 1| 2 can be translated to this: x is no more than 2 units away from 1. Thus, the largest value of x is 3, while the smallest is 1. All the values in this range have an [pmath]x^2[/pmath] less than or equal to 9, so the smallest even integer above those values is 10, which is definitely less than 12. In other words, for every value of x allowed by this statement, the answer to the question is Yes. Thats enough to answer the question definitively.