If n is a negative integer, which statements is always true?

Options

A. `6n^(-2)lt4n^(-1)`
B. `(n)/(4)gt-6n^(-1)`
C. `6n^(-1)lt4n^(-1)`
D. `4n^(-1)gt(6n)^(-1)`

Answer

To determine which statement is always true if \( n \) is a negative integer, we will analyze each statement step by step.

### Step 1: Analyze the first statement
**Statement 1:** \( 6n^{-2} < \frac{4}{n} \)

- Rewrite \( n^{-2} \) as \( \frac{1}{n^2} \):
\[ 6 \cdot \frac{1}{n^2} < \frac{4}{n} \]
- Multiply both sides by \( n^2 \) (note that \( n^2 \) is positive since \( n \) is negative):
\[ 6 < 4n \]
- Divide both sides by 4:
\[ \frac{6}{4} < n \quad \Rightarrow \quad \frac{3}{2} < n \]
- Since \( n \) is a negative integer, this statement is false.

### Step 2: Analyze the second statement
**Statement 2:** \( \frac{n}{4} > -6 \)

- Substitute \( n = -1 \):
\[ \frac{-1}{4} > -6 \]
- This simplifies to:
\[ -0.25 > -6 \]
- This is true, but we need to check if it holds for all negative integers. For \( n = -2 \):
\[ \frac{-2}{4} > -6 \quad \Rightarrow \quad -0.5 > -6 \quad \text{(True)} \]
- However, as we increase \( n \) (e.g., \( n = -10 \)):
\[ \frac{-10}{4} > -6 \quad \Rightarrow \quad -2.5 > -6 \quad \text{(True)} \]
- This statement seems true for negative integers, but let's check further.

### Step 3: Analyze the third statement
**Statement 3:** \( \frac{6}{n} < \frac{4}{n} \)

- Substitute \( n = -1 \):
\[ \frac{6}{-1} < \frac{4}{-1} \quad \Rightarrow \quad -6 < -4 \]
- This is true. Let's check for \( n = -2 \):
\[ \frac{6}{-2} < \frac{4}{-2} \quad \Rightarrow \quad -3 < -2 \quad \text{(True)} \]
- This holds for all negative integers since dividing by a negative number reverses the inequality.

### Step 4: Analyze the fourth statement
**Statement 4:** \( \frac{4}{n} > \frac{1}{6n} \)

- Substitute \( n = -1 \):
\[ \frac{4}{-1} > \frac{1}{6 \cdot -1} \quad \Rightarrow \quad -4 > -\frac{1}{6} \]
- This is false since \(-4 < -\frac{1}{6}\).

### Conclusion
The only statement that is always true for any negative integer \( n \) is:

**Answer:** \( \frac{6}{n} < \frac{4}{n} \)