To solve the inequality [tex]\( |x + 2| < 2 \)[/tex], we need to break it down into two separate inequalities based on the definition of absolute value. The absolute value [tex]\( |x + 2| \)[/tex] represents the distance of [tex]\( x + 2 \)[/tex] from 0, which must be less than 2. This translates into two inequalities:
1. [tex]\( x + 2 < 2 \)[/tex]
2. [tex]\( x + 2 > -2 \)[/tex]
### Solving the First Inequality:
1. [tex]\( x + 2 < 2 \)[/tex]
2. Subtract 2 from both sides:
[tex]\[
x < 0
\][/tex]
### Solving the Second Inequality:
1. [tex]\( x + 2 > -2 \)[/tex]
2. Subtract 2 from both sides:
[tex]\[
x > -4
\][/tex]
### Combining the Inequalities:
By combining the two inequalities, we obtain:
[tex]\[
-4 < x < 0
\][/tex]
This solution tells us that [tex]\( x \)[/tex] must be greater than [tex]\(-4\)[/tex] and less than [tex]\( 0 \)[/tex].
### Result and Graph:
The solution [tex]\( -4 < x < 0 \)[/tex] means that [tex]\( x \)[/tex] lies within the open interval from [tex]\(-4\)[/tex] to [tex]\( 0\)[/tex], but does not include [tex]\(-4\)[/tex] or [tex]\( 0\)[/tex] themselves.
Thus, the correct answer is:
B. Solution: [tex]\( x>-4 \)[/tex] and [tex]\( x<0 \)[/tex]
