To determine which of the given expressions is equivalent to the complex fraction [tex]\(\frac{1 - \frac{1}{x}}{2}\)[/tex], follow these steps:
1. Simplify the complex fraction [tex]\(\frac{1 - \frac{1}{x}}{2}\)[/tex]:
- Start by simplifying the numerator: [tex]\(1 - \frac{1}{x}\)[/tex].
- Rewrite [tex]\(1\)[/tex] as [tex]\(\frac{x}{x}\)[/tex]: [tex]\[\frac{x}{x} - \frac{1}{x}\][/tex]
- Combine the fractions in the numerator: [tex]\[\frac{x - 1}{x}\][/tex]
So, the original complex fraction becomes:
[tex]\[
\frac{\frac{x-1}{x}}{2}
\][/tex]
2. Simplify the entire expression:
- This can be rewritten as:
[tex]\[
\frac{x-1}{2x}
\][/tex]
3. Compare with the given options:
Let's evaluate each given option step-by-step to find the match:
- Option 1: [tex]\(\frac{x-1}{2x}\)[/tex]:
[tex]\[
\frac{x-1}{2x}
\][/tex]
This matches the simplified fraction exactly.
- Option 2: [tex]\(-\frac{1}{2}\)[/tex]:
[tex]\[
\frac{1 - \frac{1}{x}}{2} \neq -\frac{1}{2}
\][/tex]
Clearly, this does not match the simplified fraction.
- Option 3: [tex]\(\frac{2x-2}{x}\)[/tex]:
[tex]\[
\frac{2(x-1)}{x} = 2\left(\frac{x-1}{x}\right)
\][/tex]
This is not equivalent to the original simplified fraction.
- Option 4: [tex]\(\frac{2x}{x-1}\)[/tex]:
[tex]\[
\frac{2x}{x-1}
\][/tex]
Again, this does not match the simplified fraction.
Therefore, the correct equivalent expression is:
[tex]\[
\boxed{\frac{x-1}{2x}}
\][/tex]
Hence, the correct option that matches the simplified fraction is the first one:
[tex]\[
\boxed{1}
\][/tex]
