Given:
- A number $n$ lies between $m$ and $-m$ :

$$
\(-m<n<m\)
$$


Question: Which statement is definitely true?
Options:
(A) $\(m>0\)$
(B) $\(|n| \geq 0\)$
(C) $\(-n<0\)$

Step-by-Step Analysis
Option (A): $\(m>0\)$
- Initial Thought: For $-m<n<m$ to hold, $m$ must be positive.
- If $\(m=0:-0<n<0 \Rightarrow 0<n<0\)$, which is impossible.
- If $\(m<0\)$ : Say $\(m=-5\)$, then $\(5<n<-5\)$, which is also impossible.
- Conclusion: $\(m>0\)$ seems necessary.

Why This Might Not Be the Answer:
The problem states that $n$ lies between $m$ and $-m$, but it doesn't explicitly say $\(-m<n<m\)$. If interpreted as $n$ is anywhere between $m$ and $-m$ (including $n=m$ or $n=-m$ ), then:
- For $m=0, n=0$ satisfies $n$ being "between" 0 and -0 .
- Thus, $m>0$ is not strictly necessary (though it's the typical case).

This interpretation makes (A) not always true.

Option (B): $\(|n| \geq 0\)$
- Absolute Value Property: For any real number $n,|n| \geq 0$ is always true, regardless of $m$.
- Even if $m=0$ and $\(n=0,|0| \geq 0\)$ holds.
- Conclusion: This is a universal truth and definitely true in all cases.

Option (C): $\(-n<0\)$
- Dependent on $n$ :
- If $\(n>0:-n<0\)$ is true.
- If $n=0$ : $-0<0$ is false.
- If $\(n<0:-n>0\)$, so $\(-n<0\)$ is false.
- Conclusion: Not always true.

Why the Official Answer is (B)
1. Universality of (B):
- $\(|n| \geq 0\)$ is always true, no matter the values of $m$ or $n$.
- It doesn't rely on the condition $\(-m<n<m\)$ to hold.
2. Ambiguity in (A):
- If we strictly interpret "between" as $\(-m<n<m\)$, then $\(m>0\)$ is required.
- However, if "between" allows $n=m$ or $n=-m$, then $m=0$ with $n=0$ is possible, making $m>0$ not definitely true.
3. GRE's Preference:
- The GRE prioritizes absolute truths (like (B)) over context-dependent ones (like (A)).