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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?

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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.

Solve the inequality involving absolute value: |(x - 3)/6| < 4.

Learn how to solve the absolute value inequality |(x - 3)/6| < 4 with step-by-step guidance. This algebra problem covers key concepts like manipulating absolute values and properties of inequalities. Suitable for high school students in Grades 9-10.

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