In the image, we are given a line segment $AC$, which is divided into two smaller segments $AB$ and $BC$. The length of segment $AB$ is represented as $2x - 6$, and the length of segment $BC$ is given as 18. Additionally, the total length of segment $AC$ is given by the expression $x + 45$.

To find the value of $x$, we can set up the following equation based on the fact that the sum of the lengths of segments $AB$ and $BC$ should equal the total length $AC$:

$AB + BC = AC$

Substitute the given expressions:

$(2x - 6) + 18 = x + 45$

Now, let's solve for $x$:

1. Simplify the left-hand side:

$2x - 6 + 18 = x + 45$

$2x + 12 = x + 45$

2. Subtract $x$ from both sides:

$2x - x + 12 = 45$

$x + 12 = 45$

3. Subtract 12 from both sides:

$x = 45 - 12$

$x = 33$

So, the value of $x$ is $33$.

Would you like more details on this solution or further clarification?

Here are 5 related questions to deepen your understanding:

1. How would the equation change if the length of $BC$ was a different value?
2. Can you create a similar problem where the total length of the segment is given, and you need to solve for $x$?
3. How would you check your solution for correctness after solving for $x$?
4. What other algebraic methods could you use to solve this equation?
5. How does understanding this problem help in solving geometric problems involving line segments?

Tip: When solving algebraic equations, always check your solution by substituting it back into the original equation to verify the accuracy.