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 $a$.