We are given the line segment $\overline{HJ}$, with point $I$ lying between $H$ and $J$. The following relationships are provided:

$IJ = 3x + 3, \quad HI = 3x - 1, \quad HJ = 3x + 8$

Since $HJ = HI + IJ$ (the total length of the segment is the sum of its parts), we can set up the equation:

$HJ = HI + IJ$

Substitute the given expressions for $HI$ and $IJ$:

$3x + 8 = (3x - 1) + (3x + 3)$

Now, let's solve this equation step-by-step:

$3x + 8 = 3x - 1 + 3x + 3$ $3x + 8 = 6x + 2$ $8 - 2 = 6x - 3x$ $6 = 3x$ $x = 2$

Now that we have $x = 2$, we substitute this value back into the equation for $HJ$:

$HJ = 3x + 8 = 3(2) + 8 = 6 + 8 = 14$

Thus, the numerical length of $HJ$ is $\boxed{14}$.

Would you like further details or have any questions? Here are some related questions to explore:

1. How would the solution change if the equation for $HJ$ had been different?
2. What is the significance of the point $I$ in this problem?
3. Can this method of solving be applied to any line segment with intermediate points?
4. How would you solve the system if the equations were quadratic instead of linear?
5. What happens to the solution if the variables are changed into constants?

Tip: Always check the equation setup to ensure that all parts of a segment are accounted for when solving problems involving line segments.