To solve the inequality [tex]\( 3|x+2|<6 \)[/tex], let's go through the steps methodically:
1. Isolate the absolute value:
First, divide both sides of the inequality by 3:
[tex]\[
3|x + 2| < 6 \implies |x + 2| < 2
\][/tex]
2. Remove the absolute value:
To handle an absolute value inequality [tex]\( |x + 2| < 2 \)[/tex], rewrite it as two separate inequalities:
[tex]\[
-2 < x + 2 < 2
\][/tex]
3. Solve the inequalities:
Subtract 2 from all three parts of the inequality:
[tex]\[
-2 - 2 < x + 2 - 2 < 2 - 2
\][/tex]
[tex]\[
-4 < x < 0
\][/tex]
Therefore, the solution to the inequality is:
[tex]\[
x > -4 \quad \text{and} \quad x < 0
\][/tex]
This represents the interval [tex]\((-4, 0)\)[/tex].
Comparing the solutions provided in the options:
- Option A: Solution: [tex]\(x > -4\)[/tex] and [tex]\(x < 0\)[/tex]; graph:
- Option B: Solution: [tex]\(x < 0\)[/tex] or [tex]\(x > 4\)[/tex]; graph:
- Option C: Solution: [tex]\(x < -4\)[/tex] or [tex]\(x > 0\)[/tex]; graph:
- Option D: Solution: [tex]\(x > -4\)[/tex] and [tex]\(x < 0\)[/tex]; graph:
Options A and D provide the correct solution [tex]\(x > -4\)[/tex] and [tex]\(x < 0\)[/tex].
So the correct choice is:
A or D with the solution range [tex]\(x > -4\)[/tex] and [tex]\(x < 0\)[/tex].
