To solve the inequality \( |x - 6| < 7 \), we can break it down into two cases because the absolute value inequality implies that the expression inside must be less than 7 and greater than -7.

### Step-by-step solution:
1. Start by rewriting the absolute value inequality:

\[
-7 < x - 6 < 7
\]

2. Solve the inequality by adding 6 to all parts:

\[
-7 + 6 < x < 7 + 6
\]

\[
-1 < x < 13
\]

### Final solution:
The solution to the inequality is:

\[
x \in (-1, 13)
\]

This means \( x \) lies between -1 and 13, excluding the endpoints.

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### Related Questions:
1. How do you solve \( |x - 3| > 4 \)?
2. What is the graphical representation of \( |x - 6| < 7 \)?
3. How do you solve compound inequalities like \( -2 \leq 3x + 1 < 5 \)?
4. What happens if the inequality is \( |x - 6| \geq 7 \)?
5. How do absolute value inequalities relate to distance on a number line?

**Tip:** Absolute value inequalities represent distances. For \( |x - a| < b \), the variable \( x \) is within \( b \) units from \( a \).

Learn how to solve the absolute value inequality |x - 6| < 7 with a detailed step-by-step explanation. This problem involves algebraic concepts like absolute value inequalities and covers important formulas for solving inequalities. Suitable for students in Grades 7-9.

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