To solve the inequality [tex]\( |2x - 8| \leq 6 \)[/tex], follow these steps:
1. Understand the definition of the absolute value inequality. The expression [tex]\( |2x - 8| \leq 6 \)[/tex] means that the distance of [tex]\( 2x - 8 \)[/tex] from 0 is at most 6. Therefore, we can rewrite it as a compound inequality:
[tex]\[
-6 \leq 2x - 8 \leq 6
\][/tex]
2. Solve the compound inequality by breaking it down into two inequalities:
[tex]\[
-6 \leq 2x - 8
\][/tex]
and
[tex]\[
2x - 8 \leq 6
\][/tex]
3. First, solve the inequality [tex]\( -6 \leq 2x - 8 \)[/tex]:
- Add 8 to both sides:
[tex]\[
-6 + 8 \leq 2x \implies 2 \leq 2x
\][/tex]
- Divide both sides by 2:
[tex]\[
1 \leq x
\][/tex]
4. Next, solve the inequality [tex]\( 2x - 8 \leq 6 \)[/tex]:
- Add 8 to both sides:
[tex]\[
2x - 8 + 8 \leq 6 + 8 \implies 2x \leq 14
\][/tex]
- Divide both sides by 2:
[tex]\[
x \leq 7
\][/tex]
5. Combine the results from both inequalities:
[tex]\[
1 \leq x \leq 7
\][/tex]
Therefore, the correct solution is [tex]\( 1 \leq x \leq 7 \)[/tex].
So, the correct answer is:
B. [tex]\( 1 \leq x \leq 7 \)[/tex]
