To solve the inequality \( \frac{1}{2 - x} < x \), we'll follow these steps:

1. **Express the inequality**:
   \[
   \frac{1}{2 - x} < x
   \]

2. **Multiply both sides by \(2 - x\)**:  
   Since \(2 - x\) could be positive or negative, we need to consider two cases:  
   - Case 1: \(2 - x > 0\)  (i.e., \(x < 2\))  
   - Case 2: \(2 - x < 0\)  (i.e., \(x > 2\))

### Case 1: \(x < 2\)
- When \(2 - x > 0\), we can multiply both sides by \(2 - x\) without changing the inequality's direction:
  \[
  1 < x(2 - x)
  \]
  Expanding and rearranging the inequality:
  \[
  1 < 2x - x^2
  \]
  \[
  x^2 - 2x + 1 < 0
  \]
  Factor the quadratic:
  \[
  (x - 1)^2 < 0
  \]
  The square of a real number cannot be less than zero, so there is no solution in this case.

### Case 2: \(x > 2\)
- When \(2 - x < 0\), we must reverse the inequality after multiplying:
  \[
  1 > x(2 - x)
  \]
  Expanding and rearranging the inequality:
  \[
  1 > 2x - x^2
  \]
  \[
  x^2 - 2x + 1 > 0
  \]
  Factor the quadratic:
  \[
  (x - 1)^2 > 0
  \]
  The square of a real number is greater than zero for all \(x \neq 1\). Since we're considering \(x > 2\), this inequality is true for all \(x > 2\).

### Conclusion:
The solution to the inequality is \(x > 2\).

Would you like further details or have any questions?

Here are some related questions you might find helpful:
1. How do you solve quadratic inequalities?
2. What is the significance of the direction change when multiplying inequalities by negative numbers?
3. How do you factor quadratic expressions?
4. What are the rules for handling inequalities with fractions?
5. How do you determine the solution set of inequalities?

**Tip:** Always consider the sign of the expression when multiplying both sides of an inequality to avoid incorrect conclusions.

Learn how to solve the inequality 1/(2-x) < x with a detailed step-by-step solution. This problem involves inequalities and quadratic equations, suitable for students in Grades 9-12.

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