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Solve the inequality for [tex]\( x \)[/tex] and identify the graph of its solution.

[tex]\[ 3|x+2|\ \textless \ 6 \][/tex]

Choose the answer that gives both the correct solution and the correct graph.

A. Solution: [tex]\( x\ \textgreater \ -4 \)[/tex] and [tex]\( x\ \textless \ 0 \)[/tex]; graph:
B. Solution: [tex]\( x\ \textless \ 0 \)[/tex] or [tex]\( x\ \textgreater \ 4 \)[/tex]; graph:
C. Solution: [tex]\( x\ \textless \ -4 \)[/tex] or [tex]\( x\ \textgreater \ 0 \)[/tex]; graph:
D. Solution: [tex]\( x\ \textgreater \ -4 \)[/tex] and [tex]\( x\ \textless \ 0 \)[/tex]; graph:



Answer :

GinnyAnswer
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].

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