We are solving the inequality:

$3|x + 5| - 6 > 21$

Step-by-Step Solution:

1. Isolate the absolute value term:

$3|x + 5| - 6 > 21$

Add 6 to both sides:

$3|x + 5| > 27$

2. Divide both sides by 3:

$|x + 5| > 9$

3. Solve the absolute value inequality:

For $|x + 5| > 9$, the expression splits into two cases:

• Case 1: $x + 5 > 9$
• Case 2: $x + 5 < -9$

Case 1:

$x + 5 > 9$ Subtract 5 from both sides: $x > 4$

Case 2:

$x + 5 < -9$ Subtract 5 from both sides: $x < -14$

Final Solution:

The solution is: $x > 4 \quad \text{or} \quad x < -14$

Solution in interval notation:

$x \in (-\infty, -14) \cup (4, \infty)$

Would you like further details on any step?

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Here are 5 related questions to consider:

1. How do you solve inequalities involving both absolute values and quadratic terms?
2. Can you explain how absolute value inequalities work in general?
3. What happens if the inequality had been $\leq$ or $\geq$ instead of $>$?
4. How would the solution change if there were a different coefficient multiplying the absolute value?
5. How can you graph the solution to the inequality on a number line?

Tip: When dealing with absolute value inequalities, always remember to split the inequality into two cases—positive and negative—before solving.