Is n<0? GMAT Data Sufficiency

Question: Is n<0?

1)n−1<0
2)|3−n|>|n+5|

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.

“Is n<0?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:
Approach Solution 1:
From statement 1 we have n<1. insufficient

From statement 2 we have (3-n)^2>(n+5)^2

9+n^2-6n>n^2+25+10n
-16>16n
ie n<-1
Sufficient

Correct Answer: B

Approach Solution 2:
Condition 1) : n−1<0⇔n<1

Since the range of the question, n < 0 does not include that of the condition 1), n<1, the condition 1) is not sufficient.

Condition 2) :
|3−n|>|n+5|
⇔|3−n|^2 >|n+5|^2
⇔(3−n)^2 >(n+5)^2
⇔n^2 −6n+9>n2+10n+25
⇔−16>16n
⇔n<−1

Correct Answer: B

Approach Solution 3:
This is a different approach to statement 2:
Statement 2: |3−n| just means 'the distance between 3 and n on a number line'.
For example, if n = 2, then |3-n| = |3-2| = |1| = 1. The distance between 3 and 2 on a number line is 1.
If n = 10, then |3-n| = |3-10| = |-7| = 7. And likewise, the distance between 3 and 10 on a number line is 7.
This is just a fact about absolute values of differences.

Similarly, |n + 5| can be read as |n - (-5)|. That makes it the absolute value of a difference. So, we can say this one as 'the distance between n and -5 on a number line'.

That means we can fully translate statement 2 like this:

The distance between n and 3 on a number line is greater than the distance between n and -5 on a number line. Meaning n is closer to -5 than it is to 3.

The numbers to the right of 3 will all be closer to 3, so those don't work:

Now look at the numbers in between -5 and 3:

All of the ones that are closer to -5, are negative numbers.
We can conclude that if a number is closer to -5 than it is to 3, it's definitely got to be a negative number.

That makes statement 2 sufficient.

Correct Answer: B

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