The inequality given is $|3x + 12| < 9$.

To solve this, follow these steps:

Step 1: Remove the absolute value

We rewrite the inequality without the absolute value by considering the two cases of the absolute value:

1. $3x + 12 < 9$
2. $-(3x + 12) < 9$ (which is equivalent to $3x + 12 > -9$)

Step 2: Solve each case separately

Case 1: $3x + 12 < 9$

$3x + 12 < 9$ Subtract 12 from both sides: $3x < 9 - 12$ $3x < -3$ Now, divide both sides by 3: $x < -1$

Case 2: $3x + 12 > -9$

$3x + 12 > -9$ Subtract 12 from both sides: $3x > -9 - 12$ $3x > -21$ Now, divide both sides by 3: $x > -7$

Step 3: Combine the results

The solution to the inequality is the intersection of the two cases, which gives: $-7 < x < -1$

Final Answer:

The solution is $-7 < x < -1$.

Would you like more details on any of the steps or explanations?

Here are 5 related questions for further practice:

1. Solve $|2x - 5| \leq 7$.
2. Solve $|4x + 3| \geq 11$.
3. Solve $|x - 2| + 5 > 8$.
4. Solve $|5x + 4| \leq 6$.
5. Solve $|x + 1| < 10$.

Tip: When solving absolute value inequalities, always consider both the positive and negative cases of the expression inside the absolute value.