Given the function [tex]f(x)=3|x-2|+6[/tex], for what values of [tex]x[/tex] is [tex]f(x)=18[/tex]?

A. [tex]x=-2, x=-8[/tex]
B. [tex]x=-2, x=-6[/tex]
C. [tex]x=-2, x=6[/tex]
D. [tex]x=-2, x=8[/tex]



Answer :

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]