We are given the inequality:

$|3x - 5| \geq 1$

This is an absolute value inequality. To solve it, we break it into two cases based on the definition of absolute value:

1. Case 1: $3x - 5 \geq 1$
2. Case 2: $3x - 5 \leq -1$

Solving Case 1:

$3x - 5 \geq 1$ Add 5 to both sides: $3x \geq 6$ Divide by 3: $x \geq 2$

Solving Case 2:

$3x - 5 \leq -1$ Add 5 to both sides: $3x \leq 4$ Divide by 3: $x \leq \frac{4}{3}$

Final Solution:

Combining both cases, we get the solution:

$x \geq 2 \quad \text{or} \quad x \leq \frac{4}{3}$

In interval notation, this is written as:

$x \in \left(-\infty, \frac{4}{3}\right] \cup [2, \infty)$

Would you like more details on absolute value inequalities or how to graph the solution?

Here are five related questions for further exploration:

1. How do you solve absolute value inequalities that involve quadratic expressions?
2. What is the difference between "greater than" and "greater than or equal to" when solving inequalities?
3. How can you represent this solution on a number line?
4. What happens if the inequality sign is reversed (i.e., $|3x - 5| \leq 1$)?
5. How does solving inequalities change when dealing with negative coefficients?

Tip: Always remember to break absolute value inequalities into two cases—one for the positive scenario and one for the negative scenario!