To solve the inequality $|6 + 3x| < 9$, we need to consider the definition of absolute value. Recall that for any expression $|A| < B$, this implies $-B < A < B$.

Step-by-Step Solution:

1. Set up the compound inequality:

   Given $|6 + 3x| < 9$, we can write: $-9 < 6 + 3x < 9$

2. Solve the inequality in two parts:

   Let's solve the two inequalities separately:

   • For the left inequality: $-9 < 6 + 3x$
   • For the right inequality: $6 + 3x < 9$

   Solve $-9 < 6 + 3x$:

   $-9 - 6 < 3x$

   $-15 < 3x$

   Divide both sides by 3:

   $-5 < x$

   Solve $6 + 3x < 9$:

   $3x < 9 - 6$

   $3x < 3$

   Divide both sides by 3:

   $x < 1$

3. Combine the results:

   From both parts, we have:

   $-5 < x < 1$

   This represents the solution to the inequality.

Final Answer:

$\boxed{-5 < x < 1}$

This means $x$ can take any value between $-5$ and $1$, not including $-5$ or $1$.

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Would you like more details on this solution, or do you have any other questions?

Related Questions:

1. How do you solve absolute value inequalities with greater than signs, such as $|x - 4| > 7$?
2. What are the steps to solve a quadratic inequality?
3. How do you solve compound inequalities with different operations?
4. Can you explain how to graph the solution set of $-5 < x < 1$ on a number line?
5. What changes when the absolute value inequality has a "greater than or equal to" sign?

Tip: Always check your solutions by plugging the boundary values back into the original inequality to ensure they satisfy the condition.