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The image shows a line segment \( JL \) with points \( J \), \( K \), and \( L \) marked on it. The total length of segment \( JL \) is represented by the expression \( 15x + 110 \). The segment from \( J \) to \( K \) is given by \( 6x + 51 \), and the segment from \( K \) to \( L \) is represented by \( 2x + 24 \).

We need to find the value of \( x \) by setting up the equation that represents the entire line segment \( JL \) as the sum of its parts \( JK \) and \( KL \):

\[
JK + KL = JL
\]

Substitute the expressions for \( JK \), \( KL \), and \( JL \):

\[
(6x + 51) + (2x + 24) = 15x + 110
\]

Let's solve this equation for \( x \).The solution to the equation is \( x = -5 \).

Now that we have the value of \( x \), we can find the lengths of segments \( JK \) and \( KL \) by substituting \( x = -5 \) into the expressions for each segment:

\[
JK = 6x + 51, \quad KL = 2x + 24
\]

Would you like to proceed with calculating these lengths or have any other questions? 

Here are some questions you might consider next:

1. What are the lengths of the segments \( JK \) and \( KL \) with \( x = -5 \)?
2. How do we verify that the total length \( JL \) equals the sum of \( JK \) and \( KL \)?
3. What would happen if the expressions for segments were different?
4. How can negative values for \( x \) affect the lengths of geometric figures?
5. Can we interpret this problem in a real-world context?

**Tip:** When solving equations involving geometry, always check if the value makes sense in the context of the problem.

Given a line segment JL with total length 15x + 110, segment JK as 6x + 51, and segment KL as 2x + 24, solve for x.

Learn how to solve the equation for a line segment JL, where JK + KL = JL. This algebraic problem involves solving linear equations for x and is suitable for students in grades 9-10.

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