To solve the inequality [tex]\( -6 \leq -2k + 8 \leq 2 \)[/tex], we need to break it into two separate inequalities and solve each one individually.
### Step 1: Solve the first inequality
[tex]\[ -6 \leq -2k + 8 \][/tex]
1. Subtract 8 from both sides:
[tex]\[
-6 - 8 \leq -2k
\][/tex]
Simplify:
[tex]\[
-14 \leq -2k
\][/tex]
2. Divide both sides by -2 (and remember to reverse the inequality sign):
[tex]\[
\frac{-14}{-2} \geq \frac{-2k}{-2}
\][/tex]
Simplify:
[tex]\[
7 \geq k
\][/tex]
This is equivalent to:
[tex]\[
k \leq 7
\][/tex]
### Step 2: Solve the second inequality
[tex]\[ -2k + 8 \leq 2 \][/tex]
1. Subtract 8 from both sides:
[tex]\[
-2k + 8 - 8 \leq 2 - 8
\][/tex]
Simplify:
[tex]\[
-2k \leq -6
\][/tex]
2. Divide both sides by -2 (and remember to reverse the inequality sign):
[tex]\[
\frac{-2k}{-2} \geq \frac{-6}{-2}
\][/tex]
Simplify:
[tex]\[
k \geq 3
\][/tex]
### Step 3: Combine the results
We have:
[tex]\[ k \leq 7 \][/tex]
[tex]\[ k \geq 3 \][/tex]
Combining these two results, we get:
[tex]\[ 3 \leq k \leq 7 \][/tex]
### Conclusion
The solution to the inequality [tex]\( -6 \leq -2k + 8 \leq 2 \)[/tex] is:
[tex]\[ 3 \leq k \leq 7 \][/tex]
Thus, the final answer is:
[tex]\[ 7 \geq k \geq 3 \][/tex]
