We are given that point $K$ is on line segment $JL$, and the lengths of $JL$, $KL$, and $JK$ are represented by the following expressions:

• $JL = 4x + 2$
• $KL = 5x - 6$
• $JK = 3x$

Since point $K$ is on line segment $JL$, we know that the total length of $JL$ is the sum of the lengths of $JK$ and $KL$. Therefore, we can set up the equation:

$JK + KL = JL$

Substitute the given expressions for $JK$, $KL$, and $JL$:

$3x + (5x - 6) = 4x + 2$

Now, solve for $x$:

$3x + 5x - 6 = 4x + 2$ $8x - 6 = 4x + 2$ $8x - 4x = 2 + 6$ $4x = 8$ $x = 2$

Now that we know $x = 2$, we can find the length of $JK$. Substitute $x = 2$ into the expression for $JK$:

$JK = 3x = 3(2) = 6$

Thus, the length of $JK$ is 6 units.

Would you like further clarification or more details?

Here are 5 related questions to expand your understanding:

1. How would the problem change if the length of $KL$ were negative?
2. What is the total length of $JL$ when $x = 2$?
3. Can you find the length of $KL$ with $x = 2$?
4. What would happen if the value of $x$ were fractional?
5. How can we use geometric concepts to prove that point $K$ divides $JL$ into two segments?

Tip: Always check the relationship between the parts and the whole in geometry problems involving segments.