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Let's analyze the given problem step by step to determine which statements about \( x \), given \( 2 < x < 6 \), are necessarily true.

The given range is: 
\[
2 < x < 6
\]

### a. \( 0 < x < 4 \)
- The lower bound \( 0 < x \) is true because \( x > 2 \).
- However, the upper bound \( x < 4 \) is **not** necessarily true, since \( x \) can go up to 6.

**Conclusion**: This is **not necessarily true**.

### b. \( 0 < x - 2 < 4 \)
- Subtracting 2 from the inequality \( 2 < x < 6 \) gives:
\[
0 < x - 2 < 4
\]
This is the same as the given inequality.

**Conclusion**: This is **necessarily true**.

### c. \( 1 < \frac{x}{2} < 3 \)
- Divide the inequality \( 2 < x < 6 \) by 2:
\[
1 < \frac{x}{2} < 3
\]
This is the same as the given inequality.

**Conclusion**: This is **necessarily true**.

### d. \( \frac{1}{6} < \frac{1}{x} < \frac{1}{2} \)
- Taking the reciprocal of the inequality \( 2 < x < 6 \), we reverse the inequality signs:
\[
\frac{1}{6} < \frac{1}{x} < \frac{1}{2}
\]
This is the same as the given inequality.

**Conclusion**: This is **necessarily true**.

### e. \( 1 < \frac{6}{x} < 3 \)
- Taking the reciprocal of \( 2 < x < 6 \) and multiplying by 6:
\[
\frac{6}{6} > \frac{6}{x} > \frac{6}{2} \quad \Rightarrow \quad 1 < \frac{6}{x} < 3
\]
This is the same as the given inequality.

**Conclusion**: This is **necessarily true**.

### f. \( |x - 4| < 2 \)
- The absolute value \( |x - 4| < 2 \) means \( x \) is within 2 units of 4, which translates to:
\[
2 < x < 6
\]
This is exactly the same as the original inequality.

**Conclusion**: This is **necessarily true**.

### g. \( -6 < -x < 2 \)
- Multiply \( 2 < x < 6 \) by \(-1\), reversing the inequality:
\[
-6 < -x < -2
\]
This is **not** the same as the given inequality \( -6 < -x < 2 \), because the upper bound would be \(-2\), not 2.

**Conclusion**: This is **not necessarily true**.

### h. \( -6 < -x < -2 \)
- This is exactly the same as the result from multiplying the original inequality by \(-1\):
\[
-6 < -x < -2
\]

**Conclusion**: This is **necessarily true**.

### Summary:
- **Necessarily true**: b, c, d, e, f, h
- **Not necessarily true**: a, g

Let me know if you'd like further explanations or more details! Here are 5 related questions to deepen your understanding:

1. How do you reverse inequalities when taking reciprocals?
2. What are the steps to solve inequalities involving absolute values?
3. How do you handle inequalities when multiplying or dividing by negative numbers?
4. Why does subtracting or adding constants not affect the inequality signs?
5. How would you solve for \( x \) if it involved more complex transformations, like squaring both sides?

**Tip**: Always be careful when multiplying or dividing inequalities by negative numbers, as it reverses the inequality sign!

If 2 < x < 6, which of the following statements about x are necessarily true, and which are not necessarily true?

Analyze the inequalities given the range 2 < x < 6, and determine which statements are necessarily true. This problem covers key algebraic concepts like absolute value, reciprocals, and inequality rules. Suitable for Grades 9-12.

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