Question 2 (Multiple Choice, Worth 5 points)

What is the solution to [tex]4|x-6| \leq 16[/tex]?

A. [tex]x \leq 2[/tex] or [tex]x \geq 10[/tex]

B. [tex]2 \leq x \leq 10[/tex]

C. [tex]-10 \leq x \leq 2[/tex]

D. [tex]x \leq -10[/tex] or [tex]x \geq 2[/tex]



Answer :

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]