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To find any holes in the function \( f(x) = \frac{x^2 - 11x + 24}{2x - 1} \), we need to determine if there are any values of \( x \) that make both the numerator and the denominator equal to zero simultaneously. 

### Step-by-Step Solution

1. **Factor the Numerator**:
   The numerator is \( x^2 - 11x + 24 \). We need to factor it:

   \[
   x^2 - 11x + 24 = (x - 3)(x - 8)
   \]

2. **Identify the Denominator**:
   The denominator is \( 2x - 1 \). Set the denominator equal to zero to find the values of \( x \) that cause division by zero:

   \[
   2x - 1 = 0 \implies x = \frac{1}{2}
   \]

3. **Check for Common Factors**:
   We check if \( x = \frac{1}{2} \) is also a root of the numerator. Since the numerator factors are \( (x - 3)(x - 8) \), neither factor equals zero when \( x = \frac{1}{2} \). Therefore, there are no common factors between the numerator and the denominator.

4. **Conclusion**:
   Since there are no common factors, there is no cancellation that would indicate a hole. The only discontinuity at \( x = \frac{1}{2} \) is a vertical asymptote, not a hole.

### Final Answer

There are no holes in the function \( f(x) = \frac{x^2 - 11x + 24}{2x - 1} \). The function has a vertical asymptote at \( x = \frac{1}{2} \).

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### Related Questions

1. How do you find vertical asymptotes of a rational function?
2. What is the difference between a hole and a vertical asymptote?
3. Can a function have both a hole and a vertical asymptote?
4. How do you find horizontal asymptotes of a rational function?
5. How can you determine the domain of a rational function?

### Tip

Always check both the numerator and the denominator for common factors when looking for holes in a rational function.

Find all holes of the following function. If a hole exists, write your answer as a coordinate point in simplest form. f(x) = (x^2 - 11x + 24) / (2x - 1)

Learn how to identify holes in the rational function f(x) = (x^2 - 11x + 24) / (2x - 1). This problem involves key algebraic concepts, including factorization and determining vertical asymptotes. Suitable for students in Grades 9-11.

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