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Which expression is equivalent to the following complex fraction?

[tex]\[
\frac{1-\frac{1}{x}}{2}
\][/tex]

A. [tex]\(\frac{x-1}{2x}\)[/tex]

B. [tex]\(\frac{-1}{2}\)[/tex]

C. [tex]\(\frac{2x-2}{x}\)[/tex]

D. [tex]\(\frac{2x}{x-1}\)[/tex]


Sagot :

To determine which expression is equivalent to the given complex fraction [tex]\(\frac{1 - \frac{1}{x}}{2}\)[/tex], we will simplify the expression step-by-step.

First, let's rewrite the numerator of the complex fraction:

[tex]\[ 1 - \frac{1}{x} \][/tex]

To combine the terms in the numerator, we need a common denominator:

[tex]\[ 1 = \frac{x}{x} \][/tex]

This allows us to write the numerator as:

[tex]\[ 1 - \frac{1}{x} = \frac{x}{x} - \frac{1}{x} = \frac{x - 1}{x} \][/tex]

Now the original complex fraction can be rewritten as:

[tex]\[ \frac{\frac{x - 1}{x}}{2} \][/tex]

Next, we simplify the division of fractions by multiplying by the reciprocal of the denominator:

[tex]\[ \frac{\frac{x - 1}{x}}{2} = \frac{x - 1}{x} \times \frac{1}{2} = \frac{(x - 1) \cdot 1}{2 \cdot x} = \frac{x - 1}{2x} \][/tex]

Thus, the expression [tex]\(\frac{1 - \frac{1}{x}}{2}\)[/tex] is equivalent to:

[tex]\[ \frac{x - 1}{2x} \][/tex]

Therefore, the correct option is:

[tex]\[ \boxed{\frac{x-1}{2 x}} \][/tex]
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