To solve the inequality [tex]\( |x + 2| < 1 \)[/tex], we need to understand the properties of absolute value functions.
The absolute value inequality [tex]\( |x + 2| < 1 \)[/tex] means the expression inside the absolute value, [tex]\( x + 2 \)[/tex], lies within a distance of 1 unit from 0 on the number line. This can be expressed without the absolute value as:
[tex]\[
-1 < x + 2 < 1
\][/tex]
Next, we solve this compound inequality by isolating [tex]\( x \)[/tex]. To do this, we subtract 2 from all parts of the inequality:
[tex]\[
-1 - 2 < x + 2 - 2 < 1 - 2
\][/tex]
Simplifying each part, we get:
[tex]\[
-3 < x < -1
\][/tex]
Thus, the solution to the inequality [tex]\( |x + 2| < 1 \)[/tex] is the interval:
[tex]\[
-3 < x < -1
\][/tex]
Looking at the answer choices provided:
A. [tex]\( 1 < x < 3 \)[/tex] - Incorrect, since this interval does not satisfy the inequality.
B. [tex]\( -3 < x < -1 \)[/tex] - Correct, as it matches our solution.
C. [tex]\( x > 3 \)[/tex] or [tex]\( x < 1 \)[/tex] - Incorrect, since this describes regions outside our interval.
D. [tex]\( x > -1 \)[/tex] or [tex]\( x < -3 \)[/tex] - Incorrect, since this also describes regions outside our interval.
Therefore, the correct answer is:
B. [tex]\( -3 < x < -1 \)[/tex]
