To solve the equation [tex]\(\frac{1}{3}|x - 3| + 4 = 10\)[/tex], we need to isolate the absolute value expression and then consider the different cases it presents. Let's go through the solution step-by-step.
1. Subtract 4 from both sides to isolate the absolute value term:
[tex]\[
\frac{1}{3}|x - 3| + 4 - 4 = 10 - 4
\][/tex]
[tex]\[
\frac{1}{3}|x - 3| = 6
\][/tex]
2. Multiply both sides by 3 to eliminate the fraction:
[tex]\[
3 \cdot \frac{1}{3}|x - 3| = 6 \cdot 3
\][/tex]
[tex]\[
|x - 3| = 18
\][/tex]
3. Consider the two cases for the absolute value expression:
[tex]\[
x - 3 = 18 \quad \text{or} \quad x - 3 = -18
\][/tex]
4. Solve each case separately:
- For [tex]\(x - 3 = 18\)[/tex]:
[tex]\[
x = 18 + 3
\][/tex]
[tex]\[
x = 21
\][/tex]
- For [tex]\(x - 3 = -18\)[/tex]:
[tex]\[
x = -18 + 3
\][/tex]
[tex]\[
x = -15
\][/tex]
Therefore, the solutions to the equation are [tex]\(x = -15\)[/tex] and [tex]\(x = 21\)[/tex].
Thus, the correct answer from the choices provided is:
- [tex]\( x = -15 \)[/tex] and [tex]\( x = 21 \)[/tex]
So, the correct choice is:
[tex]\[ \boxed{x = -15 \text{ and } x = 21} \][/tex]
