To solve the absolute value inequality [tex]\( |x + 8| \leq 2 \)[/tex], we need to consider the definition of absolute value inequalities. Specifically, [tex]\( |A| \leq B \)[/tex] implies [tex]\( -B \leq A \leq B \)[/tex].
In this case, [tex]\( A = x + 8 \)[/tex] and [tex]\( B = 2 \)[/tex].
Therefore, we can write the inequality as:
[tex]\[
-2 \leq x + 8 \leq 2
\][/tex]
This compound inequality can be separated into two simpler inequalities:
1. [tex]\(-2 \leq x + 8\)[/tex]
2. [tex]\(x + 8 \leq 2\)[/tex]
Now we will solve each inequality for [tex]\( x \)[/tex].
### Solving the first inequality [tex]\( -2 \leq x + 8 \)[/tex]:
Subtract 8 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
-2 - 8 \leq x
\][/tex]
[tex]\[
-10 \leq x
\][/tex]
or
[tex]\[
x \geq -10
\][/tex]
### Solving the second inequality [tex]\( x + 8 \leq 2 \)[/tex]:
Subtract 8 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x + 8 \leq 2
\][/tex]
[tex]\[
x \leq 2 - 8
\][/tex]
[tex]\[
x \leq -6
\][/tex]
### Combining the Results
To satisfy both inequalities, [tex]\( x \)[/tex] must be between [tex]\(-10\)[/tex] and [tex]\(-6\)[/tex]. Therefore, the solution to the inequality [tex]\( |x + 8| \leq 2 \)[/tex] is:
[tex]\[
-10 \leq x \leq -6
\][/tex]
In summary:
- [tex]\( x \leq -6 \)[/tex]
- [tex]\( x \geq -10 \)[/tex]
Thus, the solution is [tex]\( x \leq -6 \)[/tex] and [tex]\( x \geq -10 \)[/tex], which can be written more compactly as:
[tex]\[
-10 \leq x \leq -6
\][/tex]
