To solve the inequality [tex]\( |x+2| < 2 \)[/tex], we need to understand the properties of absolute values. The absolute value inequality [tex]\( |A| < B \)[/tex] can be rewritten as two combined inequalities: [tex]\( -B < A < B \)[/tex].
### Step-by-Step Solution:
1. Rewrite the Inequality:
[tex]\[
|x + 2| < 2
\][/tex]
This can be rewritten as:
[tex]\[
-2 < x + 2 < 2
\][/tex]
2. Isolate [tex]\(x\)[/tex] in the Inequality:
To isolate [tex]\(x\)[/tex], subtract 2 from all parts of the inequality:
[tex]\[
-2 - 2 < x + 2 - 2 < 2 - 2
\][/tex]
Simplifying this gives:
[tex]\[
-4 < x < 0
\][/tex]
### Solution:
The solution to the inequality is:
[tex]\[
-4 < x < 0
\][/tex]
### Graphing the Solution:
On a number line, this solution translates to:
- An open interval from [tex]\( -4 \)[/tex] to [tex]\( 0 \)[/tex], meaning [tex]\( x \)[/tex] lies between [tex]\( -4 \)[/tex] and [tex]\( 0 \)[/tex], but does not include [tex]\( -4 \)[/tex] and [tex]\( 0 \)[/tex] themselves.
### Match with Given Choices:
- A states: [tex]\( x > -4 \)[/tex] and [tex]\( x < 0 \)[/tex].
While the description [tex]\(x > -4\)[/tex] and [tex]\(x < 0\)[/tex] correctly describes the solution, this repetition suggests there might be a typo hence the same option repeated is considered again.
- B states: [tex]\( x > -4 \)[/tex] and [tex]\( x < 0 \)[/tex].
Similar point again as in A.
- C incorrectly states: [tex]\( x < -4 \)[/tex] or [tex]\( x > 0 \)[/tex]. This describes a different set of solutions outside of our desired interval.
Therefore, the correct choice is:
A. Solution: [tex]\( x > -4 \)[/tex] and [tex]\( x < 0 \)[/tex]
