To solve the compound inequality \( 14 + 17 < x - 9 < 25 + 10 \), we can break it down into two parts.

1. Solve the left part of the inequality:

\[
14 + 17 < x - 9
\]

First, calculate \( 14 + 17 \):

\[
31 < x - 9
\]

Next, add \( 9 \) to both sides:

\[
31 + 9 < x
\]

\[
40 < x
\]

This can be rewritten as:

\[
x > 40
\]

2. Solve the right part of the inequality:

\[
x - 9 < 25 + 10
\]

First, calculate \( 25 + 10 \):

\[
x - 9 < 35
\]

Next, add \( 9 \) to both sides:

\[
x < 35 + 9
\]

\[
x < 44
\]

Now, we can combine both parts of the inequality:

\[
40 < x < 44
\]

Thus, the solution to the inequality is:

\[
x \in (40, 44)
\]