\((x-7)^6 - 9\)

To analyze the expression \(|(x-7)^6 - 9|\), we first set the inside of the absolute value equal to zero to find critical points.

1. Set the expression equal to zero:
\[
(x-7)^6 - 9 = 0
\]
\[
(x-7)^6 = 9
\]
\[
x-7 = \pm 3 \quad \text{(taking the sixth root)}
\]

2. Solve for \(x\):
- \(x - 7 = 3 \Rightarrow x = 10\)
- \(x - 7 = -3 \Rightarrow x = 4\)

Thus, the critical points are \(x = 4\) and \(x = 10\).

3. Next, we analyze the intervals around these points:
- Interval 1: \((- \infty, 4)\)
- Interval 2: \((4, 10)\)
- Interval 3: \((10, \infty)\)

4. Choose test points in each interval to determine the sign of \((x - 7)^6 - 9\).

- For \(x < 4\) (e.g., \(x = 0\)):
\[
(0-7)^6 - 9 > 0 \quad (\text{since } (7^6) - 9 > 0)
\]

- For \(4 < x < 10\) (e.g., \(x = 5\)):
\[
(5-7)^6 - 9 = (-2)^6 - 9 = 64 - 9 > 0
\]

- For \(x > 10\) (e.g., \(x = 11\)):
\[
(11 - 7)^6 - 9 = (4)^6 - 9 > 0 \quad (\text{since } 4096 - 9 > 0)
\]

5. Now summarize:
- \((x-7)^6 - 9 > 0\) for all \(x\) (the expression is always greater than zero).

Since the expression inside the absolute value is always positive, we have:

\[
|(x-7)^6 - 9| = (x-7)^6 - 9
\]

This expression can be used to evaluate further or solve specific equations or inequalities if needed.