To solve for the length of $JK$, we need to use the fact that $JK + KL = JL$, since $K$ is a point on the line segment $JL$.

Given: $JL = 4x + 2, \quad KL = 5x - 6, \quad JK = 3x$

We can set up the equation: $JK + KL = JL$ Substituting the given expressions: $3x + (5x - 6) = 4x + 2$

Now, simplify the equation: $3x + 5x - 6 = 4x + 2$ $8x - 6 = 4x + 2$

Next, move all terms involving $x$ to one side and the constant terms to the other side: $8x - 4x = 2 + 6$ $4x = 8$

Solve for $x$: $x = 2$

Now that we have $x = 2$, we can find the numerical length of $JK$ by substituting $x$ back into the expression for $JK$: $JK = 3x = 3(2) = 6$

Thus, the numerical length of $JK$ is $\boxed{6}$.

Would you like to know more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the result change if the expressions for $JL$ and $KL$ were different?
2. What if point $K$ were not on the segment $JL$? How would that affect the calculations?
3. How can you verify that your solution is correct?
4. What if $JK$ and $KL$ had overlapping or shared lengths?
5. How can this problem be extended to three dimensions?

Tip: Always double-check the arrangement of your algebraic equation and your solution step-by-step to avoid mistakes.