We are given that:

• Point $J$ is on line segment $IK$.
• The length of $IK$ is $5x$, i.e., $IK = 5x$.
• The length of $IJ$ is 4, i.e., $IJ = 4$.
• The length of $JK$ is $4x$, i.e., $JK = 4x$.

We know that $IJ + JK = IK$. Substituting the given lengths into this equation:

$IJ + JK = IK$

$4 + 4x = 5x$

Now, solve for $x$:

$4 = 5x - 4x$

$4 = x$

So, $x = 4$.

Now that we have the value of $x$, we can determine the numerical length of $IK$, which is given by $IK = 5x$:

$IK = 5 \times 4 = 20$

Thus, the length of $IK$ is $20$.

Do you want further details or have any questions?

Here are some related questions:

1. What would be the length of $IK$ if $IJ$ were doubled?
2. How would the equation change if $JK$ were $6x$ instead of $4x$?
3. What is the midpoint of $IK$ if $x = 4$?
4. If $IJ$ were $5$ and $JK$ remained $4x$, what would $x$ be?
5. Can we find the coordinates of $I, J, K$ given $x$?

Tip: When solving for a variable, always isolate it on one side of the equation to simplify the problem.