To solve for the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( f(x) = 18 \)[/tex] given the function [tex]\( f(x) = 3|x-2| + 6 \)[/tex], we should follow these steps:
1. Set up the equation: Start by setting the function equal to 18.
[tex]\[
3|x-2| + 6 = 18
\][/tex]
2. Isolate the absolute value term: Subtract 6 from both sides to isolate the term with the absolute value.
[tex]\[
3|x-2| = 18 - 6
\][/tex]
[tex]\[
3|x-2| = 12
\][/tex]
3. Solve for the absolute value: Divide both sides by 3.
[tex]\[
|x-2| = \frac{12}{3}
\][/tex]
[tex]\[
|x-2| = 4
\][/tex]
4. Solve the absolute value equation: Recall that [tex]\( |a| = b \)[/tex] has two solutions [tex]\( a = b \)[/tex] and [tex]\( a = -b \)[/tex]. Apply this to the equation [tex]\( |x-2| = 4 \)[/tex].
[tex]\[
x - 2 = 4 \quad \text{or} \quad x - 2 = -4
\][/tex]
5. Solve each equation for [tex]\( x \)[/tex]:
[tex]\[
x - 2 = 4 \quad \Rightarrow \quad x = 4 + 2 = 6
\][/tex]
[tex]\[
x - 2 = -4 \quad \Rightarrow \quad x = -4 + 2 = -2
\][/tex]
6. Conclusion: The values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = 18 \)[/tex] are [tex]\( x = 6 \)[/tex] and [tex]\( x = -2 \)[/tex].
Thus, the correct values of [tex]\( x \)[/tex] are:
[tex]\[\boxed{-2 \text{ and } 6}\][/tex]
So, the answer is:
[tex]\[ x = -2, x = 6 \][/tex]
