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.