We are given the following information about the line segment $HK$ with point $J$ between $H$ and $K$:

• $HK = 26$
• $HJ = 7x - 2$
• $JK = 3x + 8$

Since $J$ is between $H$ and $K$, we can express $HK$ as the sum of the lengths of $HJ$ and $JK$: $HK = HJ + JK$ Substituting the given expressions: $26 = (7x - 2) + (3x + 8)$ Now, combine like terms: $26 = 7x - 2 + 3x + 8$ $26 = 10x + 6$ Next, solve for $x$: $26 - 6 = 10x$ $20 = 10x$ $x = 2$

Now that we have $x = 2$, we can substitute this value into the expression for $JK$ to find its length: $JK = 3x + 8 = 3(2) + 8 = 6 + 8 = 14$

Thus, the solutions are: $x = 2$ $JK = 14$

Would you like more details on any step or do you have any questions?

Here are 5 related questions to expand on this problem:

1. What if $JK$ was a different function of $x$? How would it affect the solution?
2. How would you solve the problem if $HJ$ were expressed in terms of a different variable?
3. What is the length of $HJ$ when $x = 2$?
4. How would the solution change if $J$ were not exactly between $H$ and $K$?
5. Can you verify that the sum of $HJ$ and $JK$ equals $HK$ with the calculated values?

Tip: Always check the sum of the segments after solving for $x$ to ensure the solution satisfies the original equation!