Given a line segment that contains the points A,B, & C in order, and given that B is the midpoint of AC, if AB = 2x - 5, and BC = x + 7, find x.

1. Break Down the Problem

We need to find the value of $x$ given that point $B$ is the midpoint of line segment $AC$. This means that the length of segment $AB$ is equal to the length of segment $BC$. We are provided with the expressions for the lengths of these segments.

2. Relevant Concepts

Since $B$ is the midpoint of $AC$: $AB = BC$ The given lengths are:

• $AB = 2x - 5$
• $BC = x + 7$

3. Analysis and Detail

Using the midpoint property: $2x - 5 = x + 7$

Now, we will solve for $x$:

1. Subtract $x$ from both sides: $2x - x - 5 = 7$ This simplifies to: $x - 5 = 7$

2. Add 5 to both sides: $x = 7 + 5$ Therefore, $x = 12$

4. Verify and Summarize

To verify, substitute $x = 12$ back into the expressions for $AB$ and $BC$:

1. Calculate $AB$: $AB = 2(12) - 5 = 24 - 5 = 19$

2. Calculate $BC$: $BC = 12 + 7 = 19$

Both lengths are equal, confirming our solution is correct.

Final Answer

The value of $x$ is $\boxed{12}$.