To find the smallest possible difference between two 4-digit numbers formed by splitting the digits $2, 3, 4, 5, 6, 7, 8,$ and $9$ into two groups of equal size, while ensuring that both numbers are larger than 6000, follow these steps:

Step 1: Identify the possible digit groups

We need to split the digits into two groups, each containing 4 digits. The sum of all the digits is:

$2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 44$

So, each group must have a sum of:

$\frac{44}{2} = 22$

Step 2: Consider possible groups where both numbers are larger than 6000

For both numbers to be larger than 6000, the digits 6, 7, 8, or 9 must be in the thousands place. Let's consider different configurations where the difference between the two resulting numbers can be minimized:

Possible Groupings:

1. Group 1: {6, 8, 4, 4}, Group 2: {7, 9, 3, 3}.
2. Group 1: {6, 9, 3, 4}, Group 2: {7, 8, 2, 5}.

However, the correct configuration requires the numbers to be close together, which means their sums should be as close as possible while maintaining the condition that both are larger than 6000.

Group 1: {6, 8, 4, 4}, Group 2: {7, 9, 3, 3}.

For better distribution of the smallest possible difference, we use one high and one low number:

• Number 1: Use the digits 6, 8, 4, 4.
• Number 2: Use the digits 7, 9, 3, 3.

Let's calculate:

Step 3: Arrange numbers to get closest results

Now, we arrange the digits in both groups to form two 4-digit numbers:

• Group 1: {6, 8, 4, 2} → Smallest possible = 6428
• Group 2: {7, 9, 5, 3} → Largest possible = 7539

This configuration would result in:

$7539 - 6428 = 1111$

Thus, the closest combination would result in a difference of 1111 between the two groups, which cannot be any smaller.

Final Answer:

The smallest possible value of the difference between these two 4-digit numbers, ensuring both are larger than 6000, is 1111.

Would you like further clarification or more details on any part of the solution? Here are some related questions:

1. What is the largest possible difference between two 4-digit numbers formed from the digits 1 to 9?
2. If only one number must be larger than 6000, what is the smallest possible difference?
3. How does the sum of the digits affect the possible differences between the two 4-digit numbers?
4. How can the digits be arranged to maximize the difference while both numbers remain larger than 6000?
5. What is the smallest possible difference if we consider numbers larger than 7000 instead of 6000?

Tip: When trying to minimize differences, focus on creating the two numbers as close as possible, keeping their structure and place value in mind.