To solve the inequality $|6x - 9| < 15$, follow these steps:

Step 1: Rewrite the inequality without the absolute value bars

The definition of absolute value inequality $|A| < B$ translates to: $-B < A < B$ Here, $A = 6x - 9$ and $B = 15$. So the inequality becomes: $-15 < 6x - 9 < 15$

Step 2: Solve the compound inequality

Break it into two parts and solve:

1. Solve $-15 < 6x - 9$: $-15 + 9 < 6x$ $-6 < 6x$ $x > -1$

2. Solve $6x - 9 < 15$: $6x < 15 + 9$ $6x < 24$ $x < 4$

Step 3: Combine the solutions

The solution is: $-1 < x < 4$

Final Answer:

The solution in interval notation is: $(-1, 4)$

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Related Questions:

1. How can absolute value inequalities differ when the inequality is $\geq$ instead of $<$?
2. What steps would change if the inequality were $|6x - 9| > 15$?
3. Can you describe how to graph the solution $-1 < x < 4$?
4. What are the implications of the inequality's bounds in a real-world context?
5. How would the solution change if the absolute value inequality included an equals sign?

Tip:

When solving absolute value inequalities, always remember to split them into two cases and handle them as a compound inequality.