To solve for $x$ and find the length of line segment $JK$, we can use the given information:

• $HK = 20$
• $HJ = 2x - 2$
• $JK = 7x + 4$

Since point $J$ is between $H$ and $K$, the lengths of $HJ$ and $JK$ should add up to the total length of $HK$. Therefore, we can write the equation:

$HJ + JK = HK$

Substituting the given expressions:

$(2x - 2) + (7x + 4) = 20$

Now, let's solve this equation for $x$.

Simplify the equation:

$2x - 2 + 7x + 4 = 20$

Combine like terms:

$9x + 2 = 20$

Subtract 2 from both sides:

$9x = 18$

Divide by 9:

$x = 2$

Now that we have $x = 2$, we can find the length of $JK$:

$JK = 7x + 4 = 7(2) + 4 = 14 + 4 = 18$

Therefore, the length of line segment $JK$ is 18 units.

Would you like more details or have any questions? Here are some related questions to explore further:

1. What is the length of line segment $HJ$ with the solved value of $x$?
2. If $J$ were to be exactly halfway between $H$ and $K$, how would $x$ change?
3. Can you find the coordinates of point $J$ if $H$ is at the origin?
4. How would the equation change if $HK$ was not a straight line but a curve?
5. What would the equation be if $J$ was outside the segment $HK$?

Tip: Always check your substituted values to verify the solution, ensuring there are no calculation errors.