The image shows a line segment with points labeled $A$, $B$, and $C$, with distances represented by expressions involving $x$. Specifically:

• The distance between points $A$ and $B$ is given by $3x + 2$.
• The distance between points $B$ and $C$ is given by $4x + 5$.
• The total distance from $A$ to $C$ is given as $17$.

To find $x$, you can set up the equation by adding the distances between $A$ and $B$ and between $B$ and $C$ to equal the total distance from $A$ to $C$:

$3x + 2 + 4x + 5 = 17$

Now, combine like terms:

$7x + 7 = 17$

Next, subtract 7 from both sides:

$7x = 10$

Finally, divide by 7:

$x = \frac{10}{7}$

So, $x = \frac{10}{7}$ or approximately $1.43$.

Would you like any further details or have any questions?

Here are some follow-up questions you might find helpful:

1. What is the total length of segment $AB$ when $x = \frac{10}{7}$?
2. What is the total length of segment $BC$ when $x = \frac{10}{7}$?
3. How do we check that the sum of segments $AB$ and $BC$ equals 17?
4. What would happen if the total distance $AC$ were different?
5. Can you solve a similar problem where $AC$ is given, and the expressions for $AB$ and $BC$ are different?

Tip: When solving linear equations, always combine like terms before isolating the variable to simplify the process.