To find the values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = 3|x - 2| + 6 \)[/tex] equals 18, we need to solve the equation [tex]\( 3|x - 2| + 6 = 18 \)[/tex].
1. Start by isolating the absolute value expression:
[tex]\[
3|x - 2| + 6 = 18
\][/tex]
Subtract 6 from both sides:
[tex]\[
3|x - 2| = 12
\][/tex]
2. Next, divide both sides by 3 to further isolate [tex]\( |x - 2| \)[/tex]:
[tex]\[
|x - 2| = 4
\][/tex]
3. The absolute value equation [tex]\( |x - 2| = 4 \)[/tex] can be split into two separate linear equations:
- Case 1: [tex]\( x - 2 = 4 \)[/tex]
- Case 2: [tex]\( x - 2 = -4 \)[/tex]
4. Solve each of the linear equations:
- For [tex]\( x - 2 = 4 \)[/tex]:
[tex]\[
x = 4 + 2
\][/tex]
[tex]\[
x = 6
\][/tex]
- For [tex]\( x - 2 = -4 \)[/tex]:
[tex]\[
x = -4 + 2
\][/tex]
[tex]\[
x = -2
\][/tex]
Thus, the values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = 18 \)[/tex] are [tex]\( x = 6 \)[/tex] and [tex]\( x = -2 \)[/tex].
Therefore, the correct answer is:
[tex]\[ x = -2, x = 6 \][/tex]
So, the correct choice is:
[tex]\[ x = -2, x = 6 \][/tex]
