- Simplify the numerator: \(\frac{3}{x-1} - 4 = \frac{-4x+7}{x-1}\).
- Simplify the denominator: \(2 - \frac{2}{x-1} = \frac{2(x-2)}{x-1}\).
- Divide the simplified numerator by the simplified denominator and cancel the common factor \((x-1)\).
- The simplified expression is \(\frac{-4x+7}{2(x-2)}\), which is the second option.
- Final Answer: \(\boxed{\frac{-4x+7}{2(x-2)}}\)
Explanation
- Analysis of the Expression and Domain
To simplify the complex fraction \(\frac{\frac{3}{x-1}-4}{2-\frac{2}{x-1}}\), we first identify the domain of the expression. For the expression to be well-defined, the denominators must not be zero. Thus, we must have \(x-1
eq 0\), which implies \(x
eq 1\). Additionally, the denominator of the main fraction must be non-zero, \(2 - \frac{2}{x-1}
eq 0\), which implies \(\frac{2}{x-1}
eq 2\), so \(x-1
eq 1\), and \(x
eq 2\). We assume \(x
eq 1\) and \(x
eq 2\) throughout the simplification.
- Simplifying the Numerator
Next, we simplify the numerator, \(\frac{3}{x-1} - 4\). To combine these, we use a common denominator of \(x-1\):
\[\frac{3}{x-1} - 4 = \frac{3 - 4(x-1)}{x-1} = \frac{3 - 4x + 4}{x-1} = \frac{-4x + 7}{x-1}\]
- Simplifying the Denominator
Now, we simplify the denominator of the main expression, \(2 - \frac{2}{x-1}\). Using the common denominator \(x-1\), we have:
\[2 - \frac{2}{x-1} = \frac{2(x-1) - 2}{x-1} = \frac{2x - 2 - 2}{x-1} = \frac{2x - 4}{x-1} = \frac{2(x-2)}{x-1}\]
- Final Simplification
Finally, we rewrite the original complex fraction using the simplified numerator and denominator:
\[\frac{\frac{-4x+7}{x-1}}{\frac{2(x-2)}{x-1}}\]
By multiplying by the reciprocal of the denominator, we get:
\[\frac{-4x+7}{x-1} \cdot \frac{x-1}{2(x-2)}\]
Canceling the common term \((x-1)\), we obtain:
\[\frac{-4x+7}{2(x-2)}\]
Comparing this to the given options, we find it matches the second option.
Examples
Simplifying complex fractions is a fundamental skill in algebra that appears frequently when solving systems of equations, analyzing functions, or performing unit conversions in physics. For example, if you are calculating the effective resistance of a complex circuit or the average speed over a journey with varying segments, you often encounter ratios of sums or differences. Mastering these steps ensures that you can reduce complex models to simpler, more manageable expressions.