The problem asks to solve the absolute value inequality:

$|x - 4| < 9$

To solve this, we use the property of absolute value inequalities. For any inequality of the form $|A| < B$, the equivalent inequality is:

$-B < A < B$

Step 1: Apply the property

For the inequality $|x - 4| < 9$, we rewrite it as:

$-9 < x - 4 < 9$

Step 2: Solve the compound inequality

Now, solve for $x$ by adding 4 to all parts of the inequality:

$-9 + 4 < x < 9 + 4$ $-5 < x < 13$

Final Solution:

The solution to the inequality is:

$x \in (-5, 13)$

This means that $x$ lies between $-5$ and $13$, but does not include these endpoints.

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Would you like more details, or do you have any further questions?

Here are five related questions you might find helpful:

1. How do you solve an absolute value inequality when it's greater than a number?
2. What are the differences between strict and non-strict inequalities in absolute values?
3. How can absolute value inequalities be used in real-life applications?
4. What happens if there's a negative number on the right side of an absolute value inequality?
5. Can absolute value inequalities have no solution? If so, when?

Tip: Always check if you can simplify the inequality or equation before solving it. This makes solving easier and reduces errors.