https://file.aiquickdraw.com/mathbot/file/17303893085720c0ebjhy.jpg

To solve this problem, let's analyze the given information.

Since \( K \) is the midpoint of \( JL \), the segments \( JK \) and \( KL \) must be equal in length. The problem states that:
- \( JK = 4x \)
- \( KL = 9x - 6 \)

Since \( JK = KL \) (because \( K \) is the midpoint), we can set up the following equation:
\[
4x = 9x - 6
\]

### Solving for \( x \)
1. Subtract \( 4x \) from both sides:
   \[
   0 = 5x - 6
   \]
2. Add \( 6 \) to both sides:
   \[
   6 = 5x
   \]
3. Divide by \( 5 \) to isolate \( x \):
   \[
   x = \frac{6}{5}
   \]

### Finding \( JK \)
Now that we have \( x = \frac{6}{5} \), substitute this value back into the expression for \( JK \):
\[
JK = 4x = 4 \times \frac{6}{5} = \frac{24}{5}
\]

Thus, \( JK = \frac{24}{5} \), which is already in its simplest form as a proper fraction.

### Final Answer
\[
JK = \frac{24}{5}
\]

Would you like more details on any part of this solution, or do you have any questions?

### Related Questions
1. What if \( K \) wasn’t the midpoint? How would that affect the answer?
2. Can you convert \( \frac{24}{5} \) into a mixed number?
3. How would you solve this if \( JK \) and \( KL \) had different units?
4. What happens if we use decimals instead of fractions in this problem?
5. How could we verify that \( K \) is indeed the midpoint with this value?

### Tip
When you know a point is a midpoint, setting the two segments equal simplifies the setup for finding unknown values.

K is the midpoint of JL. If JK = 4x and KL = 9x - 6, what is JK?

Learn how to solve for the length of segment JK given that K is the midpoint of JL, with JK = 4x and KL = 9x - 6. This algebraic problem involves setting up an equation based on the midpoint theorem. Suitable for students in Grades 7-8.

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