We are given the inequality:

$|5x - 1| < 7$

To solve this, we must consider the definition of absolute value. The inequality $|A| < B$ (where $A$ is an expression and $B$ is a positive number) is equivalent to the double inequality:

$-B < A < B$

Step 1: Rewrite the inequality without the absolute value

The inequality $|5x - 1| < 7$ becomes:

$-7 < 5x - 1 < 7$

Step 2: Solve for $x$

First, isolate $5x$ by adding 1 to all parts of the inequality:

$-7 + 1 < 5x < 7 + 1$

$-6 < 5x < 8$

Now, divide the entire inequality by 5 to isolate $x$:

$\frac{-6}{5} < x < \frac{8}{5}$

This simplifies to:

$-\frac{6}{5} < x < \frac{8}{5}$

Step 3: Write the solution in interval notation

The solution is the set of $x$-values between $-\frac{6}{5}$ and $\frac{8}{5}$, not including the endpoints. So, the solution in interval notation is:

$\left( -\frac{6}{5}, \frac{8}{5} \right)$

Let me know if you'd like further details or explanations!

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Here are 5 related questions for you to consider:

1. How do you solve inequalities involving $\geq$ or $\leq$?
2. How does the solution change if the inequality is $|5x - 1| \leq 7$?
3. How would you solve $|5x - 1| > 7$?
4. What is the general method for solving inequalities with absolute values?
5. How do you graph the solution of $|5x - 1| < 7$ on a number line?

Tip: Always check your solution by plugging in values from within and outside your interval to verify if the inequality holds true.