We are tasked with solving the inequality:

\[
\frac{x - 2}{x^2 - 3x + 10} < 0
\]

### Steps to solve:

1. **Factor the denominator:**  
   First, we check if \( x^2 - 3x + 10 \) can be factored. Since the discriminant \( \Delta = (-3)^2 - 4(1)(10) = 9 - 40 = -31 \) is negative, the quadratic expression has no real roots. Therefore, \( x^2 - 3x + 10 \) does not factor into real numbers, and it is always positive for all real \(x\), since the parabola opens upwards.

2. **Analyze the numerator:**  
   The numerator is \( x - 2 \), which is zero at \( x = 2 \). It changes its sign at \( x = 2 \). To the left of \( x = 2 \), \( x - 2 \) is negative, and to the right of \( x = 2 \), it is positive.

3. **Sign of the rational function:**  
   Since the denominator \( x^2 - 3x + 10 \) is always positive, the sign of the rational expression \(\frac{x - 2}{x^2 - 3x + 10}\) is determined by the sign of the numerator.

   - For \( x < 2 \), \( x - 2 < 0 \), so the expression is negative.
   - For \( x > 2 \), \( x - 2 > 0 \), so the expression is positive.

4. **Conclusion:**  
   The inequality asks for when the rational expression is **less than zero**, which occurs when \( x < 2 \).

Thus, the solution is:

\[
x < 2
\]

Would you like any further details or explanations?

Here are 5 related questions for further exploration:
1. How would the solution change if the inequality was \( \frac{x - 2}{x^2 - 3x + 10} > 0 \)?
2. What would happen if the discriminant of the denominator was positive?
3. How does factoring a quadratic expression affect solving inequalities?
4. What is the effect of complex roots on the sign of a quadratic?
5. How do rational inequalities compare to polynomial

Learn how to solve the rational inequality x - 2 / x² - 3x + 10 < 0. This solution explores sign analysis, quadratic discriminants, and rational functions. Suitable for students in Grades 10-12 looking to enhance their algebra skills.

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