To solve the absolute value inequality [tex]\(2|3x + 9| < 36\)[/tex], let's go through the process step by step.
1. Isolate the absolute value expression:
Divide both sides of the inequality by 2 to isolate the absolute value term.
[tex]\[
\frac{2|3x + 9|}{2} < \frac{36}{2}
\][/tex]
Simplifying this gives:
[tex]\[
|3x + 9| < 18
\][/tex]
2. Rewrite the absolute value inequality as a compound inequality:
Recall that [tex]\(|A| < B\)[/tex] means [tex]\(-B < A < B\)[/tex]. Apply this principle to our problem:
[tex]\[
-18 < 3x + 9 < 18
\][/tex]
3. Solve the compound inequality:
First, we solve the left part of the inequality:
[tex]\[
-18 < 3x + 9
\][/tex]
Subtract 9 from both sides:
[tex]\[
-18 - 9 < 3x
\][/tex]
Simplify:
[tex]\[
-27 < 3x
\][/tex]
Divide by 3:
[tex]\[
-9 < x
\][/tex]
Next, we solve the right part of the inequality:
[tex]\[
3x + 9 < 18
\][/tex]
Subtract 9 from both sides:
[tex]\[
3x < 18 - 9
\][/tex]
Simplify:
[tex]\[
3x < 9
\][/tex]
Divide by 3:
[tex]\[
x < 3
\][/tex]
4. Combine the results:
We have two inequalities:
[tex]\[
-9 < x
\][/tex]
and
[tex]\[
x < 3
\][/tex]
Combining these, we get the final solution as:
[tex]\[
-9 < x < 3
\][/tex]
Therefore, the correct answer is:
b. [tex]\(-9 < x < 3\)[/tex]
