The problem asks to determine the solution set for the inequality:

$|4x + 9| < 3$

Steps to solve the inequality:

1. Break down the absolute value inequality: $|4x + 9| < 3 \implies -3 < 4x + 9 < 3$

2. Isolate $4x$: Subtract 9 from all sides: $-3 - 9 < 4x < 3 - 9$ $-12 < 4x < -6$

3. Solve for $x$: Divide all terms by 4: $-3 < x < -\frac{3}{2}$

Thus, the solution to the inequality is: $-3 < x < -\frac{3}{2}$

Answer:

From the options, the correct one is:

• Option a: $-3 < x < -\frac{3}{2}$.

Would you like more details on this process or further clarification?

Here are 5 related questions you can explore:

1. How do you solve absolute value inequalities involving a "greater than" condition?
2. What happens if the absolute value inequality has no solution?
3. How do you graph the solution of absolute value inequalities on a number line?
4. How would the solution change if the inequality were $|4x + 9| \leq 3$?
5. What if the expression inside the absolute value were more complex, like $|2x^2 + 3x + 1| < 5$?

Tip: Always remember to break down absolute value inequalities into two inequalities and solve both!