To solve the inequality $\frac{|x - 3|}{6} < 4$, we can break down the process as follows:

Step 1: Remove the Fraction

First, eliminate the fraction by multiplying both sides by 6: $|x - 3| < 24$

Step 2: Solve the Absolute Value Inequality

The inequality $|x - 3| < 24$ means that the distance between $x$ and 3 is less than 24. This can be rewritten as a compound inequality: $-24 < x - 3 < 24$

Step 3: Solve for $x$

Add 3 to each part of the inequality to isolate $x$: $-24 + 3 < x < 24 + 3$ $-21 < x < 27$

Step 4: Write the Solution in Interval Notation

The solution in interval notation is: $(-21, 27)$

Thus, the solution to the inequality is: $(-21, 27)$

Would you like a more detailed explanation, or do you have any questions about this process?

Here are some related questions to expand on this topic:

1. How do you solve absolute value inequalities of the form $|ax + b| > c$?
2. How would the solution change if the inequality was $\frac{|x - 3|}{6} \leq 4$?
3. What happens if there is a negative coefficient inside the absolute value, like $|-x + 3| < 4$?
4. How do you graph the solution of an absolute value inequality on a number line?
5. What is the difference between strict ($<$ or $>$) and non-strict ($\leq$ or $\geq$) inequalities in absolute value problems?

Tip: When solving absolute value inequalities, always consider the two possible cases (positive and negative) that result from removing the absolute value bars.