To solve the inequality:

$|x+6| - 4 < 5$

Step 1: Isolate the absolute value expression

First, add $4$ to both sides:

$|x+6| < 9$

Step 2: Set up two cases for the absolute value

The expression $|x+6| < 9$ means that the expression inside the absolute value is between $-9$ and $9$:

$-9 < x + 6 < 9$

Step 3: Solve for $x$

Now subtract $6$ from all three parts of the inequality:

$-9 - 6 < x < 9 - 6$ $-15 < x < 3$

Solution:

The solution to the inequality is:

$x \in (-15, 3)$

This means $x$ lies between $-15$ and $3$, but does not include these endpoints.

Would you like further details, or have any questions?

Here are 5 related questions for you:

1. How do you solve inequalities involving absolute value with greater than instead of less than?
2. What changes if the inequality was $|x+6| - 4 \leq 5$?
3. How can we graph the solution set of this inequality?
4. Can you explain the difference between a strict inequality and a non-strict inequality?
5. What happens if the absolute value inequality results in a negative number?

Tip: Always isolate the absolute value expression first before handling inequalities.