To solve the inequality [tex]\( |x + 6| < 9 \)[/tex], we need to understand how to work with absolute value inequalities. The absolute value expression [tex]\( |x + 6| \)[/tex] represents the distance of the expression [tex]\( x + 6 \)[/tex] from 0 on the number line.
To remove the absolute value, we split the inequality into two cases based on the definition of absolute value. The inequality [tex]\( |x + 6| < 9 \)[/tex] implies:
[tex]\[ -9 < x + 6 < 9 \][/tex]
Now, we will solve this compound inequality step by step.
1. Start with the inequality:
[tex]\[ -9 < x + 6 < 9 \][/tex]
2. Subtract 6 from each part of the inequality to isolate [tex]\( x \)[/tex]:
[tex]\[ -9 - 6 < x + 6 - 6 < 9 - 6 \][/tex]
This simplifies to:
[tex]\[ -15 < x < 3 \][/tex]
So, the solution to the inequality [tex]\( |x + 6| < 9 \)[/tex] is the open interval [tex]\( (-15, 3) \)[/tex].
Therefore, the correct answer is:
[tex]\[ (-15, 3) \][/tex]
