To solve this problem, we need to determine the possible values of $x$ based on the given data and the condition that the median of the seven integers is 4.

Step-by-step analysis:

We are given seven integers:
$3, 3, 4, 5, x, x, -4$

First, sort these integers, including $x$, in ascending order. The ordered list is:

$-4, 3, 3, 4, 5, x, x$

Now, the median of seven numbers is the 4th number in the sorted list, as the median of an odd number of elements is the middle element.

The condition states that the median is 4. Therefore, the 4th number in the sorted list must be 4. So, we have:

$\text{4th element } = 4$

This gives us a critical clue: the value of $x$ cannot affect the 4th element directly because it must be equal to or greater than 4. Let's now check how different values of $x$ can affect the arrangement of the numbers.

Case-by-case checking of the options:

1. $x = 3$
   The ordered list would be: $-4, 3, 3, 3, 4, 5, 5$ In this case, the 4th element is 3, which contradicts the condition that the median should be 4. So, $x = 3$ is not a valid option.

2. $x = 4$
   The ordered list would be: $-4, 3, 3, 4, 4, 4, 5$ Here, the 4th element is 4, which satisfies the condition. So, $x = 4$ is a valid option.

3. $x = 5$
   The ordered list would be: $-4, 3, 3, 4, 5, 5, 5$ Here, the 4th element is 4, which satisfies the condition. So, $x = 5$ is a valid option.

4. $x = 6$
   The ordered list would be: $-4, 3, 3, 4, 5, 6, 6$ Here, the 4th element is 4, which satisfies the condition. So, $x = 6$ is a valid option.

Conclusion:

The only value of $x$ that does not satisfy the condition is $\mathbf{3}$. Therefore, the correct answer is:

$\boxed{3}$

Would you like more details on this solution? Also, here are five related questions to consider:

1. How do you calculate the median for an odd set of numbers?
2. What happens if $x$ were negative in this case?
3. How does changing the number of elements affect the median position?
4. Why does the sorting step help in identifying the correct median?
5. What is the impact of multiple $x$ values on the median when their value changes?

Tip: When determining the median, always arrange the numbers in ascending order first!