We are tasked with solving the inequality $|3x - 5| > 1$ and expressing the solution in interval notation.

Step 1: Break the absolute value inequality into two cases

The inequality $|3x - 5| > 1$ means:

$3x - 5 > 1 \quad \text{or} \quad 3x - 5 < -1$

Step 2: Solve each inequality

Case 1: $3x - 5 > 1$

$3x > 6 \quad \Rightarrow \quad x > 2$

Case 2: $3x - 5 < -1$

$3x < 4 \quad \Rightarrow \quad x < \frac{4}{3}$

Step 3: Combine the results

The solution to $|3x - 5| > 1$ is where $x > 2$ or $x < \frac{4}{3}$.

In interval notation, this is: $(-\infty, \frac{4}{3}) \cup (2, \infty)$

Final Solution:

$(-\infty, \frac{4}{3}) \cup (2, \infty)$

Would you like me to go into further detail about any of the steps?

Here are 5 questions to deepen your understanding:

1. What are the rules for solving absolute value inequalities?
2. How does splitting an absolute value inequality into two cases work?
3. Why is the solution expressed as two separate intervals?
4. How would the solution change if the inequality were $|3x - 5| \leq 1$?
5. How can you verify the solution graphically?

Tip: When solving absolute value inequalities, always double-check the critical points by testing values in each interval.