To solve the absolute value inequality [tex]\(2|3x + 9| < 36\)[/tex], follow these steps:
1. Isolate the absolute value expression:
[tex]\[
2|3x + 9| < 36
\][/tex]
2. Divide both sides by 2 to simplify:
[tex]\[
|3x + 9| < 18
\][/tex]
3. Understand the meaning of the absolute value inequality:
[tex]\[
|3x + 9| < 18
\][/tex]
This implies that the expression inside the absolute value, [tex]\(3x + 9\)[/tex], must lie between -18 and 18. Hence, we can write:
[tex]\[
-18 < 3x + 9 < 18
\][/tex]
4. Solve the compound inequality:
- First, subtract 9 from all parts of the inequality:
[tex]\[
-18 - 9 < 3x < 18 - 9
\][/tex]
Simplifies to:
[tex]\[
-27 < 3x < 9
\][/tex]
- Then, divide all parts by 3:
[tex]\[
\frac{-27}{3} < x < \frac{9}{3}
\][/tex]
Simplifies to:
[tex]\[
-9 < x < 3
\][/tex]
Therefore, the solution to the inequality [tex]\(2|3x + 9| < 36\)[/tex] is:
[tex]\[
-9 < x < 3
\][/tex]
The correct answer is:
d. [tex]\(-9 < x < 3\)[/tex]
