To understand why \( x \) cannot be between 1 and 2, let's carefully reconsider the problem and the process of solving the inequality.

### The Inequality:
\[
\frac{1}{2 - x} < x
\]

### Step 1: Analyze the Expression
- The expression \( \frac{1}{2 - x} \) is undefined when \( 2 - x = 0 \), so \( x = 2 \) is not in the domain of the inequality.
- Thus, \( x \) cannot be equal to 2. The inequality is only valid for \( x < 2 \) and \( x > 2 \).

### Step 2: Consider the Cases

1. **Case 1: \( x < 2 \) (i.e., \( 2 - x > 0 \))**

   When \( x < 2 \), the inequality becomes:
   \[
   1 < x(2 - x)
   \]
   Expanding the right-hand side:
   \[
   1 < 2x - x^2
   \]
   Rearranging:
   \[
   x^2 - 2x + 1 < 0
   \]
   Factoring:
   \[
   (x - 1)^2 < 0
   \]
   **Analysis:** 
   - The expression \( (x - 1)^2 \) is the square of a real number, which is always non-negative. 
   - The inequality \( (x - 1)^2 < 0 \) implies a negative value, which is impossible for any real \( x \).
   - Hence, **no real value of \( x \) between 1 and 2** can satisfy this inequality.

2. **Case 2: \( x > 2 \) (i.e., \( 2 - x < 0 \))**

   When \( x > 2 \), the inequality \( 1 < x(2 - x) \) is not directly relevant because the multiplication would change the direction of the inequality. However, solving as before results in the valid solution \( x > 2 \).

### Conclusion:
- **No value between 1 and 2** satisfies the inequality \( \frac{1}{2 - x} < x \). The only valid solution exists when \( x > 2 \), where the inequality holds.

Would you like further clarification, or do you have any other questions?

Here are some related questions you might find helpful:
1. What happens when you square both sides of an inequality?
2. How do you determine the domain of an inequality?
3. What is the impact of undefined points in solving inequalities?
4. Why can't the square of a real number be negative?
5. How do absolute values impact inequalities?

**Tip:** Always consider the entire domain of the variable when solving inequalities, especially when dealing with fractions or square roots.

Explore why the variable x cannot be between 1 and 2 in the inequality 1 / (2 - x) < x. This detailed explanation covers the domain of functions and inequalities, providing clarity on why certain values of x are invalid. Suitable for high school students studying inequalities and function domains.

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