To find the solution to the inequality [tex]\(4|x - 6| \leq 16\)[/tex]:
1. First, isolate the absolute term by dividing both sides by 4:
[tex]\[
|x - 6| \leq 4
\][/tex]
2. This inequality means that the expression inside the absolute value, [tex]\(x - 6\)[/tex], can vary between -4 and 4, inclusive. Therefore, we can write the compound inequality:
[tex]\[
-4 \leq x - 6 \leq 4
\][/tex]
3. Next, solve this compound inequality step-by-step:
a. Add 6 to all parts of the inequality to isolate [tex]\(x\)[/tex]:
[tex]\[
-4 + 6 \leq x - 6 + 6 \leq 4 + 6
\][/tex]
b. Simplify the resulting expression:
[tex]\[
2 \leq x \leq 10
\][/tex]
So, the solution to the inequality [tex]\(4|x - 6| \leq 16\)[/tex] is:
[tex]\[
2 \leq x \leq 10
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{2 \leq x \leq 10}
\][/tex]
