The problem provided states:

• $AB = x + 2$,
• $BC = 2x - 3$,
• $AC = 5x - 7$,
• B is between A and C.

Since B is between A and C, we can use the property of segment addition, which states:

$AB + BC = AC$

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

$(x + 2) + (2x - 3) = 5x - 7$

Now, let's solve for $x$.

$x + 2 + 2x - 3 = 5x - 7$ $3x - 1 = 5x - 7$ $-1 + 7 = 5x - 3x$ $6 = 2x$ $x = 3$

Now that we know $x = 3$, substitute it back into the expression for $AB$:

$AB = x + 2 = 3 + 2 = 5$

Thus, the length of $AB$ is 5.

Would you like further details or explanations? Here are some related questions to deepen your understanding:

1. What is the length of $BC$ for this problem?
2. Can you verify that $AB + BC = AC$ using the values of $x$ and the corresponding segment lengths?
3. How does the segment addition postulate apply in geometry?
4. What happens to the length of $AB$ if $x$ were a different value?
5. Can we apply a similar process if $B$ were not between $A$ and $C$?

Tip: When solving for unknowns in geometric problems, always check if all parts of the equation add up correctly using substitution.