To solve the inequality $|2x - 8| < 14$, we need to consider the two cases for absolute value inequalities. This can be split into two separate inequalities:

1. $2x - 8 < 14$
2. $2x - 8 > -14$

Solving the first inequality:

$2x - 8 < 14$ Add 8 to both sides: $2x < 22$ Now, divide both sides by 2: $x < 11$

Solving the second inequality:

$2x - 8 > -14$ Add 8 to both sides: $2x > -6$ Now, divide both sides by 2: $x > -3$

Combine both inequalities:

The solution to the inequality $|2x - 8| < 14$ is: $-3 < x < 11$

So, the solution is $x \in (-3, 11)$.

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Here are five follow-up questions related to this:

1. How do you solve absolute value equations instead of inequalities?
2. What is the difference between strict inequalities and non-strict inequalities?
3. How do you graph the solution to an absolute value inequality?
4. Can this type of problem appear in real-world situations? If so, how?
5. How would you handle absolute value inequalities that have a greater-than condition?

Tip: When solving absolute value inequalities, always split the inequality into two cases: one for the positive case and one for the negative case.