We are given the following from the image:

1. $AB = x + 6$
2. $AC = 3x - 31$
3. $BC = 85$

Since $A$, $B$, and $C$ are collinear, we can use the fact that the total length $AC$ is the sum of $AB$ and $BC$. Therefore, the equation is:

$AB + BC = AC$

Substituting the given expressions for $AB$, $BC$, and $AC$:

$(x + 6) + 85 = 3x - 31$

Now, let's solve this equation for $x$.It seems there was a small issue with formatting the equation. Let me correct that and solve the equation again.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can walk you through the process of solving the equation manually.

We had the equation:

$(x + 6) + 85 = 3x - 31$

Simplify it:

$x + 6 + 85 = 3x - 31$

$x + 91 = 3x - 31$

Now, subtract $x$ from both sides:

$91 = 2x - 31$

Add 31 to both sides:

$122 = 2x$

Now, divide both sides by 2:

$x = \frac{122}{2} = 61$

Thus, $x = 61$.

Would you like any more details or have any further questions?

Here are some related questions:

1. What is the length of $AB$ using $x = 61$?
2. How do we verify that the total distance $AB + BC$ equals $AC$?
3. What other methods could be used to solve this type of problem?
4. How can we generalize this problem to other similar line segments?
5. What are the properties of collinear points used in this problem?

Tip: Always check your solution by substituting it back into the original equation to ensure correctness.