Given the problem, we have a line segment $AC$ with a point $B$ on it. The lengths of the segments are provided as:

• $AC = 2x + 7$
• $BC = x$
• $AB = 5x - 9$

Since $B$ is on the line segment $AC$, the total length $AC$ should be the sum of the lengths of $AB$ and $BC$. Therefore, we can write the equation:

$AB + BC = AC$

Substituting the given expressions:

$(5x - 9) + x = 2x + 7$

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

1. Combine like terms on the left side:

$5x + x - 9 = 2x + 7$

$6x - 9 = 2x + 7$

2. Move all terms involving $x$ to one side by subtracting $2x$ from both sides:

$6x - 2x - 9 = 7$

$4x - 9 = 7$

3. Add 9 to both sides to isolate the term with $x$:

$4x = 16$

4. Divide both sides by 4:

$x = 4$

Now that we have $x = 4$, we can find the numerical length of $AB$ by substituting $x$ back into the expression for $AB$:

$AB = 5x - 9 = 5(4) - 9 = 20 - 9 = 11$

Thus, the numerical length of $AB$ is $\boxed{11}$.

Would you like further details or have any questions?

Here are 5 questions related to this problem:

1. What would the length of $BC$ be if $x = 4$?
2. How does the relationship $AB + BC = AC$ help solve for $x$?
3. What happens if $B$ were not on the line segment $AC$?
4. How would the solution change if $AB = 5x + 9$ instead?
5. Can you verify the solution by calculating the length of $AC$ using $x = 4$?

Tip: Always check your final answer by substituting the value back into the original equation to ensure consistency.