Is |x-2|+|x+2|

Question

There's something wrong with this question; it is mathematically impossible for the answer to be 'yes'. The minimum possible value of |x-2| + |x + 2| is 4, and that will happen whenever -2  <  x  <  2. If x is not in that range, then the value of the expression will be greater than 4. It can never be less than 4.

IanStewart · Answer

The answer to the question is 'no'. You don't need any Statements at all, so it doesn't matter what the Statements say. Even if Statement 2 said 'apples are red', we'd still know the answer to the question is 'no'. I suppose that makes the answer D, but it never happens on real GMAT questions that you can answer the question without needing the Statements, which is why I said that the setup of the question doesn't make any sense.

Now, if instead the question included a 'less than or equal' sign - that is, if the question asked "Is |x-2|+|x+2|  <  4?" then the answer is B, since that question is simply asking whether x is between -2 and 2 (inclusive). You can see why that's so either by using the standard cases approach to absolute value, or by interpreting the absolute values as distances; |x-2| is the distance between x and 2, and |x + 2| = |x - (-2)| is the distance between x and -2. So provided x is somewhere in the middle of 2 and -2, the sum of those distances will be 4, as you can see on a number line:

---------(-2)-----x------------2-------------

shinbhu · Answer

Solving the modulus gives us two solutions x<2 and x>-2, with two other solutions that are invalid.

1. x<0. Unsuff since x>2 according to the stem.
2. x is between -2 and 2. Clearly, the stem shows exactly that.

B.

Bunuel · Answer

I completely agree with Ian here. Neither statement is needed to answer the question, stem is enough to do so. So technically the answer to this question is D. Though you'll never see such a question on real a test. Having said that I've seen one DS question from GMAT Prep where also only stem was enough to answer the question, though I think that it was just a flawed question (if-zy-xy-0-is-x-z-x-z-101210.html#p783615). To elaborate more on the given problem: Is |x-2|+|x+2|<4? We have two check points -2 for |x+2| and 2 for |x-2| (check points - the value of x for which the expression in || equals to zero). Thus we have 3 ranges to check: --------{-2}--------{2}-------- A. x<-2 --> as for this range x-2=negative and x+2=negative then we'll have: -(x-2)-(x+2)=-2x . Now, as we are considering the range where x<-2 then -2x>4 and the answer to the question "is |x-2|+|x+2|<4 " is NO; B. -2\leq{x}\leq{2} --> as for this range x-2=negative and x+2=positive then we'll have: -(x-2)+(x+2)=4 . Again, the answer to the question "is |x-2|+|x+2|<4 ?" is NO; C. x>2 --> as for this range x-2=positive and x+2=positive then we'll have: (x-2)+(x+2)=2x . The same here, as we are considering the range where x>2 then 2x>4 and the answer to the question "is |x-2|+|x+2|<4 " is NO. So we can see that in ALL 3 ranges the answer to the question is NO: |x-2|+|x+2|\geq{4} (the minimum possible value of the given expression is 4 when -2\leq{x}\leq{2} ). Therefore even if statement (1) were "Roses are red" and statement (2) were "Violets are blue" we could still answer definite NO to the question based on the stem alone and thus would have D as an answer. Again as Ian noted if the question were: "is |x-2|+|x+2|\leq{4} " then yes, the answer to the question would be B, as the question in this case can be rephrased as "is -2\leq{x}\leq{2} ?". Hope it's clear.

mydreammba · Answer

intially marked B but Now with D

Is |x-2|+|x+2|<4?

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