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INECUACIONES CON EL METDO DE PUNTOS CRTICOS

To solve the inequality \(\frac{x^2 + 2x - 3}{x - 2} < \frac{x^2 - 4}{x - 2}\), we can follow these steps:

### Step 1: Simplify both sides of the inequality.
1. **Factor the numerators**:

   - \(x^2 + 2x - 3\) factors into \((x + 3)(x - 1)\).
   - \(x^2 - 4\) factors into \((x - 2)(x + 2)\).

   So, the inequality becomes:
   \[
   \frac{(x + 3)(x - 1)}{x - 2} < \frac{(x - 2)(x + 2)}{x - 2}
   \]

2. **Cancel out the common factors**:

   Both sides have the common factor \(x - 2\) in the denominator. Since the inequality involves division, we must be cautious of where \(x = 2\) since it would make the expression undefined.

   After canceling, we get:
   \[
   \frac{x + 3}{1} < x + 2
   \]
   Simplifying further:
   \[
   x + 3 < x + 2
   \]

3. **Subtract \(x\) from both sides**:
   \[
   3 < 2
   \]

This statement is false, which means the inequality \(\frac{x^2 + 2x - 3}{x - 2} < \frac{x^2 - 4}{x - 2}\) has no solution under the simplifications made. 

### Conclusion:
Given that \(3 < 2\) is false, none of the given answer options would be correct. Therefore, the correct choice is **"NINGUNA DE LAS ANTERIORES"**.

Would you like more details or have any further questions? Here are some related questions to consider:

1. How do you determine the critical points of a rational inequality?
2. What happens if you simplify an inequality incorrectly?
3. How do you handle cases where the denominator is zero in inequalities?
4. What are the steps for solving inequalities involving rational functions?
5. Can you always cancel out factors in inequalities?
6. What are the potential pitfalls when working with rational inequalities?
7. How does the critical point method work for solving inequalities?
8. What does it mean when an inequality has no solution?

**Tip:** Always check for values that make the denominator zero when simplifying inequalities involving rational functions. These values can affect the domain and the solution set.

Learn how to solve the inequality (x^2 + 2x - 3)/(x - 2) < (x^2 - 4)/(x - 2) using the critical points method. Detailed step-by-step solution with considerations for rational functions and critical points. Suitable for Grades 10-12.

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