To simplify the expression [tex]\(\frac{1}{2}(x-1)-\frac{1}{3}\left(\frac{1}{2} x-1\right)\)[/tex], let's break it down step by step:
1. Distribute the constants inside the parentheses:
[tex]\[
\frac{1}{2}(x - 1) = \frac{1}{2}x - \frac{1}{2}
\][/tex]
[tex]\[
\frac{1}{3}\left(\frac{1}{2} x - 1\right) = \frac{1}{3} \cdot \frac{1}{2} x - \frac{1}{3} \cdot 1 = \frac{1}{6} x - \frac{1}{3}
\][/tex]
2. Rewrite the expression using these results:
[tex]\[
\frac{1}{2}x - \frac{1}{2} - \left(\frac{1}{6}x - \frac{1}{3}\right)
\][/tex]
3. Distribute the negative sign to the terms inside the parenthesis:
[tex]\[
\frac{1}{2}x - \frac{1}{2} - \frac{1}{6}x + \frac{1}{3}
\][/tex]
4. Combine like terms:
- Combine [tex]\( \frac{1}{2}x \)[/tex] and [tex]\(- \frac{1}{6}x \)[/tex]:
[tex]\[
\frac{1}{2}x - \frac{1}{6}x = \frac{3}{6}x - \frac{1}{6}x = \frac{2}{6}x = \frac{1}{3}x
\][/tex]
- Combine [tex]\(-\frac{1}{2}\)[/tex] and [tex]\(+\frac{1}{3}\)[/tex]:
[tex]\[
-\frac{1}{2} + \frac{1}{3} = -\frac{3}{6} + \frac{2}{6} = -\frac{1}{6}
\][/tex]
5. Put the simplified terms together:
[tex]\[
\frac{1}{3}x - \frac{1}{6}
\][/tex]
So, the simplified form of the expression [tex]\(\frac{1}{2}(x-1)-\frac{1}{3}\left(\frac{1}{2} x-1\right)\)[/tex] is:
[tex]\[
\boxed{\frac{1}{3}x - \frac{1}{6}}
\][/tex]
By analyzing the given answer choices, option C matches our simplified result:
C. [tex]\(\frac{1}{6}(2x - 1)\)[/tex]
This expression simplified can be written as:
[tex]\[
\frac{2x - 1}{6} = \frac{1}{3}x - \frac{1}{6}
\][/tex]
Thus, the correct answer is:
C. [tex]\(\frac{1}{6}(2 x - 1)\)[/tex]
