To solve the equation [tex]\(\frac{1}{3}|x - 3| + 4 = 10\)[/tex], let's work through it step-by-step:
1. Isolate the absolute value term:
[tex]\[
\frac{1}{3}|x - 3| + 4 = 10
\][/tex]
Subtract 4 from both sides:
[tex]\[
\frac{1}{3}|x - 3| = 6
\][/tex]
2. Eliminate the fraction by multiplying both sides by 3:
[tex]\[
|x - 3| = 18
\][/tex]
3. Solve for the expression inside the absolute value:
Recall that if [tex]\(|A| = B\)[/tex], then [tex]\(A = B\)[/tex] or [tex]\(A = -B\)[/tex]. Here, [tex]\(A\)[/tex] is [tex]\(x - 3\)[/tex] and [tex]\(B\)[/tex] is 18. Therefore:
[tex]\[
x - 3 = 18 \quad \text{or} \quad x - 3 = -18
\][/tex]
4. Solve both equations separately:
- For [tex]\(x - 3 = 18\)[/tex]:
[tex]\[
x = 18 + 3 = 21
\][/tex]
- For [tex]\(x - 3 = -18\)[/tex]:
[tex]\[
x = -18 + 3 = -15
\][/tex]
5. Write the solution:
The solutions are [tex]\(x = 21\)[/tex] and [tex]\(x = -15\)[/tex].
So, the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\(\frac{1}{3}|x - 3| + 4 = 10\)[/tex] are [tex]\(\boxed{-15 \text{ and } 21}\)[/tex].
