We are given the inequality:

$| 2x + 1 | < 9$

To solve this absolute value inequality, we'll break it into two separate inequalities because $|A| < B$ means that $-B < A < B$.

Step-by-Step Solution:

1. Set up the compound inequality: $-9 < 2x + 1 < 9$

2. Solve the left inequality: $-9 < 2x + 1$ Subtract 1 from both sides: $-10 < 2x$ Now divide by 2: $-5 < x$

3. Solve the right inequality: $2x + 1 < 9$ Subtract 1 from both sides: $2x < 8$ Now divide by 2: $x < 4$

4. Combine the two inequalities: $-5 < x < 4$

Final Answer:

$x \in (-5, 4)$

Would you like more details or have any questions about the solution?

Here are 5 related questions to practice:

1. Solve $|3x - 2| < 7$.
2. Solve $|x + 4| > 3$.
3. Solve $|5x + 6| \geq 12$.
4. Solve $2|x - 1| + 3 < 9$.
5. Solve $|x - 2| \leq 5$.

Tip: Always split absolute value inequalities into two cases, ensuring you flip the inequality sign for the negative case.