Which expression is equivalent to the following complex fraction?
$\frac{1-\frac{1}{x}}{2}$
Answer (2)
Find a common denominator for the terms in the numerator: 1 − x 1 = x x − 1 .
Substitute the simplified numerator back into the complex fraction: 2 x x − 1 .
Divide the fraction by 2: x x − 1 ⋅ 2 1 = 2 x x − 1 .
The equivalent expression is 2 x x − 1 .
Explanation
Understanding the Problem We are asked to find an expression equivalent to the complex fraction 2 1 − x 1 . To do this, we need to simplify the complex fraction and see which of the given options matches our simplified expression.
Simplifying the Numerator First, let's simplify the numerator of the complex fraction. The numerator is 1 − x 1 . To combine these terms, we need a common denominator, which is x . So we rewrite 1 as x x . Thus, the numerator becomes x x − x 1 .
Combining Fractions Now, we can combine the fractions in the numerator: x x − x 1 = x x − 1 .
Substituting Back Substitute the simplified numerator back into the complex fraction: 2 x x − 1 .
Dividing by 2 To divide by 2, we can multiply by 2 1 : x x − 1 ⋅ 2 1 = 2 x x − 1 .
Finding the Equivalent Expression Now, we compare our simplified expression 2 x x − 1 with the given options. We see that the first option, 2 x x − 1 , matches our simplified expression.
Final Answer Therefore, the expression equivalent to the given complex fraction is 2 x x − 1 .
Examples
Complex fractions can appear when calculating average speeds over a distance when the speed changes. For example, if you travel half a distance at speed x and the other half at speed 1, the average speed is represented by a complex fraction that simplifies to x + 1 2 x . Simplifying complex fractions helps in understanding and calculating such real-world quantities efficiently.
The expression equivalent to the complex fraction 2 1 − x 1 is 2 x x − 1 .
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