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We are given a line segment \( I \) to \( K \), with point \( J \) in between. The problem asks for the length of \( JK \). The segment \( IJ \) is labeled as \( x + 6 \), and \( JK \) is labeled as \( 9 + x \). We are also told that the entire length of \( IK \) is 9.

To solve for \( JK \), we use the fact that:

\[
IJ + JK = IK
\]

Substitute the given expressions:

\[
(x + 6) + (9 + x) = 9
\]

Simplifying the equation:

\[
2x + 15 = 9
\]

Now, solve for \( x \):

\[
2x = 9 - 15
\]
\[
2x = -6
\]
\[
x = -3
\]

Now, substitute \( x = -3 \) into the expression for \( JK \):

\[
JK = 9 + x = 9 + (-3) = 6
\]

Thus, the length of \( JK \) is \( \boxed{6} \).

Do you need further details or explanations? Here are 5 related questions you might find helpful:

1. How would you find the length of \( IJ \) given this value of \( x \)?
2. What would change if the total length of \( IK \) were different?
3. How do you interpret negative values for \( x \) in this context?
4. Could we approach this problem geometrically, and how?
5. How could you generalize this process for any values of \( x \)?

**Tip**: Always check the total length to confirm if the individual segment lengths make sense in the context of the problem.

Find the length of JK given that IJ = x + 6, JK = 9 + x, and the total length of IK is 9.

This problem involves finding the length of segment JK using algebraic expressions for IJ and JK and solving for x. We apply the segment addition postulate and solve the linear equation step-by-step.

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