We are given three equations for the line segment $\overline{HJ}$:

• $IJ = 3x + 3$
• $HI = 3x - 1$
• $HJ = 3x + 8$

Since $HJ$ is the total length of the segment and point $I$ is on the segment, the following relationship holds:

$HJ = HI + IJ$

Now let's substitute the expressions for $HI$ and $IJ$ into this equation:

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

Let's simplify the right-hand side:

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

$3x + 8 = 6x + 2$

Now, solve for $x$:

$3x + 8 = 6x + 2$

Subtract $3x$ from both sides:

$8 = 3x + 2$

Subtract 2 from both sides:

$6 = 3x$

Finally, divide by 3:

$x = 2$

Now that we know $x = 2$, substitute it back into the expression for $HJ$:

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

Thus, the length of $HJ$ is 14.

Let me know if you need more details or have any questions!

Here are some related questions to further your understanding:

1. What would happen if the expressions for $HI$ and $IJ$ were different?
2. How can you check your solution using a different method?
3. Can this problem be solved graphically by placing points on a coordinate plane?
4. How does the position of point $I$ affect the segment lengths?
5. What if $IJ$ were expressed in a more complex function involving $x^2$?

Tip: Always substitute your solution back into the original equation to verify your result.