- Determine the domain constraints for the expression, identifying \(x
eq 1\) and \(x
eq 2\).
- Simplify the numerator \(\frac{3}{x-1} - 4\) to \(\frac{7-4x}{x-1}\) and the denominator \(2 - \frac{2}{x-1}\) to \(\frac{2x-4}{x-1}\).
- Perform division of the numerator by the denominator, canceling out common terms to reach \(\frac{-4x+7}{2(x-2)}\).
- Final simplified result is \(\boxed{\frac{-4x+7}{2(x-2)}}\).
Explanation
- Analysis of the Expression
To simplify the given complex fraction, we first analyze its components. The expression is given by:
\[ \frac{\frac{3}{x-1}-4}{2-\frac{2}{x-1}} \]
The expression is defined for all \(x\) except where denominators are zero, meaning \(x
eq 1\) and \(2 - \frac{2}{x-1}
eq 0\). Solving the latter for \(x\), we find \(2(x-1) - 2
eq 0\), which simplifies to \(2x - 4
eq 0\), thus \(x
eq 2\). Therefore, the domain is all real numbers \(x
eq 1, 2\).
- Simplifying Numerator and Denominator
Next, we simplify the numerator and the denominator separately.
For the numerator:
\[ \frac{3}{x-1} - 4 = \frac{3 - 4(x-1)}{x-1} = \frac{3 - 4x + 4}{x-1} = \frac{7 - 4x}{x-1} \]
For the denominator:
\[ 2 - \frac{2}{x-1} = \frac{2(x-1) - 2}{x-1} = \frac{2x - 2 - 2}{x-1} = \frac{2x - 4}{x-1} \]
Now the entire expression becomes:
\[ \frac{\frac{7 - 4x}{x-1}}{\frac{2x - 4}{x-1}} \]
- Simplifying the Fraction and Final Result
Now we simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator:
\[ \frac{7 - 4x}{x-1} \cdot \frac{x-1}{2x - 4} \]
Assuming \(x
eq 1\), the factor \((x-1)\) in the numerator and denominator cancels out, leaving:
\[ \frac{7 - 4x}{2x - 4} \]
Finally, we factor the denominator \(2x - 4\) as \(2(x-2)\) to match the given options:
\[ \frac{7 - 4x}{2(x - 2)} \]
Rewriting the numerator as \(-4x+7\), we get the final simplified form:
\[ \frac{-4x+7}{2(x-2)} \]
This matches the second option provided.
Examples
Simplifying rational expressions is essential in fields like physics and engineering, where complex formulas often describe the relationship between variables such as resistance, velocity, or pressure. For instance, when designing electrical circuits, you might encounter formulas with complex fractions that determine the total resistance or current. Reducing these expressions to a simpler form makes it much easier to calculate final values and understand the underlying behavior of the system, just like we did with our expression here.