To solve the absolute value inequality [tex]\( |x + 2| \leq 3 \)[/tex], we start by understanding that the absolute value inequality [tex]\( |A| \leq B \)[/tex] can be rewritten as two simultaneous inequalities: [tex]\(-B \leq A \leq B\)[/tex].
For the given inequality [tex]\( |x + 2| \leq 3 \)[/tex]:
1. We split the absolute value inequality into two inequalities:
[tex]\[ -3 \leq x + 2 \leq 3 \][/tex]
2. We now solve each part of this compound inequality separately.
3. First, solve the left side of the compound inequality:
[tex]\[ -3 \leq x + 2 \][/tex]
Subtract 2 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ -3 - 2 \leq x \][/tex]
[tex]\[ -5 \leq x \][/tex]
4. Next, solve the right side of the compound inequality:
[tex]\[ x + 2 \leq 3 \][/tex]
Subtract 2 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ x \leq 3 - 2 \][/tex]
[tex]\[ x \leq 1 \][/tex]
5. Combine both inequalities to get the final solution:
[tex]\[ -5 \leq x \leq 1 \][/tex]
Thus, the solution to the absolute value inequality [tex]\( |x + 2| \leq 3 \)[/tex] is:
[tex]\[ -5 \leq x \leq 1 \][/tex]
From the given options, the correct interval is:
[tex]\[ -5 \leq x \leq 1 \][/tex]
Therefore, the correct answer is:
[tex]\[ -5 \leq x \leq 1 \][/tex]
