To solve the absolute value inequality [tex]\(\frac{|x+6|}{8} \leq 1\)[/tex], we follow these steps:
1. Start with the given inequality:
[tex]\[
\frac{|x+6|}{8} \leq 1
\][/tex]
2. Isolate the absolute value expression:
To remove the fraction, multiply both sides of the inequality by 8:
[tex]\[
|x+6| \leq 8
\][/tex]
3. Understand the absolute value inequality:
The inequality [tex]\(|x+6| \leq 8\)[/tex] can be interpreted as:
[tex]\[
-8 \leq x+6 \leq 8
\][/tex]
4. Break down the compound inequality into two separate inequalities:
[tex]\[
-8 \leq x+6 \quad \text{and} \quad x+6 \leq 8
\][/tex]
5. Solve each inequality separately:
- For [tex]\(-8 \leq x+6\)[/tex]:
Subtract 6 from both sides:
[tex]\[
-8 - 6 \leq x
\][/tex]
Simplifying, we get:
[tex]\[
-14 \leq x
\][/tex]
- For [tex]\(x+6 \leq 8\)[/tex]:
Subtract 6 from both sides:
[tex]\[
x \leq 8 - 6
\][/tex]
Simplifying, we get:
[tex]\[
x \leq 2
\][/tex]
6. Combine the two inequalities to find the solution set:
[tex]\[
-14 \leq x \leq 2
\][/tex]
Therefore, the value for [tex]\(x\)[/tex] such that [tex]\(\frac{|x+6|}{8} \leq 1\)[/tex] falls within the interval [tex]\([-14, 2]\)[/tex].
Given that [tex]\(x \leq 2\)[/tex] and [tex]\(x \geq [?]\)[/tex], we find that:
[tex]\[
x \geq -14
\][/tex]
So, the inequality:
[tex]\[
\frac{|x+6|}{8} \leq 1
\][/tex]
is equivalent to:
[tex]\[
-14 \leq x \leq 2
\][/tex]
Therefore, [tex]\(x \geq -14\)[/tex].
