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We are given the inequality:

\[
\left|\frac{2x - 4}{6}\right| \geq 9
\]

To remove the absolute value, we split the inequality into two cases:

1. \(\frac{2x - 4}{6} \geq 9\)
2. \(\frac{2x - 4}{6} \leq -9\)

Let's solve each case separately:

### Case 1: \(\frac{2x - 4}{6} \geq 9\)
Multiply both sides by 6 to get rid of the denominator:
\[
2x - 4 \geq 54
\]
Now, add 4 to both sides:
\[
2x \geq 58
\]
Finally, divide by 2:
\[
x \geq 29
\]

### Case 2: \(\frac{2x - 4}{6} \leq -9\)
Again, multiply both sides by 6:
\[
2x - 4 \leq -54
\]
Add 4 to both sides:
\[
2x \leq -50
\]
Finally, divide by 2:
\[
x \leq -25
\]

### Conclusion:
The solution is:
\[
x \leq -25 \quad \text{or} \quad x \geq 29
\]

Thus, the correct answer is **option C**, with the blanks filled in as:
\[
x \leq -25 \quad \text{or} \quad x \geq 29
\]

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Do you need more details or have any questions? Here are some related questions for practice:

1. How do you solve absolute value inequalities that involve "less than"?
2. What is the difference between "and" and "or" in compound inequalities?
3. How would the solution change if the inequality was \(\leq\) instead of \(\geq\)?
4. Can you explain the process of removing absolute value symbols in inequalities?
5. What happens if you forget to flip the inequality sign when multiplying by a negative number?

**Tip:** When solving absolute value inequalities, always remember to break the inequality into two cases, one for the positive and one for the negative scenario.

Solve the inequality for x: |(2x - 4)/6| ≥ 9.

Learn how to solve the absolute value inequality |(2x - 4)/6| ≥ 9 by splitting it into two cases. This algebra problem covers solving linear inequalities and absolute value properties, suitable for students in Grades 8-10.

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