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To find the domain of the function 

\[
f(x) = \frac{\sqrt{7 - 5x}}{x^2 - 64},
\]

we need to consider the constraints imposed by both the square root in the numerator and the division by the denominator.

### Step-by-Step Analysis:

1. **Square Root Constraint**:
   The expression under the square root, \(7 - 5x\), must be non-negative (since we cannot take the square root of a negative number in the real number system):

   \[
   7 - 5x \geq 0 \implies x \leq \frac{7}{5}.
   \]

2. **Denominator Constraint**:
   The denominator \(x^2 - 64\) cannot be zero (since division by zero is undefined):

   \[
   x^2 - 64 \neq 0 \implies x \neq \pm 8.
   \]

3. **Combining Constraints**:
   We combine the constraints from both the numerator and the denominator:
   - \(x \leq \frac{7}{5}\)
   - \(x \neq -8, 8\)

   The first constraint, \(x \leq \frac{7}{5}\), limits the domain to all \(x\) values less than or equal to \(\frac{7}{5}\). The second constraint removes \(x = -8\) and \(x = 8\). Since \(x = 8\) is already greater than \(\frac{7}{5}\), it is not within the domain specified by the first constraint. However, \(x = -8\) is less than \(\frac{7}{5}\), so we must exclude it from the domain.

4. **Interval Notation**:
   The domain of \(f(x)\) in interval notation is:

   \[
   (-\infty, -8) \cup (-8, \frac{7}{5}].
   \]

### Conclusion:

The domain of the function \(f(x) = \frac{\sqrt{7 - 5x}}{x^2 - 64}\) in interval notation is:

\[
\boxed{(-\infty, -8) \cup (-8, \frac{7}{5}]}.
\]

Would you like more details or have any questions?

Here are some related questions you might find useful:

1. How do you solve inequalities involving square roots?
2. What is the domain of a function, and how is it determined?
3. How do you handle rational functions when finding the domain?
4. What happens to the graph of a function when there are restrictions on its domain?
5. How would the domain change if there were different expressions under the square root or in the denominator?

**Tip:** Always check both numerator and denominator constraints when finding the domain of a function that involves both a square root and division!

Find the domain of the function \(f(x) = \frac{\sqrt{7 - 5x}}{x^2 - 64}\) and write your answer in interval notation.

Learn how to find the domain of the function f(x) = sqrt(7 - 5x) / (x^2 - 64). This problem involves understanding inequalities, square roots, and rational functions. Suitable for high school students in Grades 10-12.

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