Which expression is equivalent to the following complex fraction?
[tex]$\frac{\frac{3}{x-1}-4}{2-\frac{2}{x-1}}$[/tex]
Answer (2)
Multiply the numerator and denominator by ( x − 1 ) to eliminate the inner fractions.
Simplify the numerator: 3 − 4 ( x − 1 ) = − 4 x + 7 .
Simplify the denominator: 2 ( x − 1 ) − 2 = 2 ( x − 2 ) .
The equivalent expression is 2 ( x − 2 ) − 4 x + 7 .
Explanation
Initial Analysis and Strategy We are given a complex fraction and asked to find an equivalent expression. The given complex fraction is 2 − x − 1 2 x − 1 3 − 4 To simplify this complex fraction, we can multiply both the numerator and the denominator by ( x − 1 ) to eliminate the inner fractions.
Eliminating Inner Fractions Multiplying the numerator and denominator by ( x − 1 ) , we get ( 2 − x − 1 2 ) ( x − 1 ) ( x − 1 3 − 4 ) ( x − 1 ) = 2 ( x − 1 ) − 2 3 − 4 ( x − 1 ) Now, we simplify the numerator and the denominator separately.
Simplifying Numerator and Denominator The numerator simplifies to: 3 − 4 ( x − 1 ) = 3 − 4 x + 4 = 7 − 4 x = − 4 x + 7 The denominator simplifies to: 2 ( x − 1 ) − 2 = 2 x − 2 − 2 = 2 x − 4 = 2 ( x − 2 ) So, the simplified fraction is 2 ( x − 2 ) − 4 x + 7
Finding the Equivalent Expression Comparing our simplified expression with the given options, we find that it matches the second option: 2 ( x − 2 ) − 4 x + 7 Therefore, the equivalent expression is 2 ( x − 2 ) − 4 x + 7 .
Final Answer Thus, the expression equivalent to the given complex fraction is 2 ( x − 2 ) − 4 x + 7 .
Examples
Complex fractions might seem abstract, but they appear in various real-world scenarios. For instance, when calculating electrical resistance in parallel circuits or analyzing rates of change in chemical reactions, complex fractions can arise. Simplifying these fractions makes the calculations more manageable and helps in understanding the underlying relationships between different variables. Imagine you're designing a circuit and need to determine the total resistance; being able to simplify complex fractions allows you to efficiently find the solution and ensure your circuit functions correctly. The ability to manipulate and simplify such expressions is a valuable skill in many STEM fields.
The equivalent expression to the given complex fraction 2 − x − 1 2 x − 1 3 − 4 is 2 ( x − 2 ) − 4 x + 7 . This was determined by multiplying both the numerator and the denominator by ( x − 1 ) to eliminate the inner fractions and simplifying accordingly.
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