The inequality to solve is:

$|3x - 4| \geq 5$

To solve this, we consider the definition of absolute value. The inequality splits into two cases:

Case 1: $3x - 4 \geq 5$

Solve for $x$: $3x \geq 9 \quad \Rightarrow \quad x \geq 3$

Case 2: $3x - 4 \leq -5$

Solve for $x$: $3x \leq -1 \quad \Rightarrow \quad x \leq -\frac{1}{3}$

Combine the two cases:

The solution to the inequality is: $x \leq -\frac{1}{3} \quad \text{or} \quad x \geq 3$

Thus, the correct answer is A.

Would you like further clarification or details? Here are 5 related questions to expand understanding:

1. How do we handle strict inequalities (e.g., $<$ or $>$) with absolute values?
2. What is the method to verify the solutions of absolute value inequalities graphically?
3. How would the solution change if the inequality were $|3x - 4| < 5$?
4. Why does the absolute value inequality split into two cases?
5. How does this method generalize to other absolute value equations or inequalities?

Tip: Always check for the direction of the inequality when splitting absolute value expressions into cases.