- Simplify the numerator by finding a common denominator to get \(\frac{12x-52}{(x-3)(x-4)}\).
- Simplify the denominator by finding a common denominator to get \(\frac{4x-11}{x-3}\).
- Divide the simplified numerator by the simplified denominator and cancel the common term \((x-3)\).
- Multiply the remaining terms to find the result, which is \(\boxed{\frac{12x-52}{4x^2-27x+44}}\).
Explanation
- Simplifying the Numerator
To simplify the complex fraction \[\frac{\frac{16}{x-3}-\frac{4}{x-4}}{5-\frac{x-4}{x-3}}\] we start by simplifying the numerator and the denominator separately.
First, let's simplify the numerator:
\[\frac{16}{x-3} - \frac{4}{x-4} = \frac{16(x-4) - 4(x-3)}{(x-3)(x-4)}\]
Expanding the terms in the numerator:
\[16(x-4) - 4(x-3) = 16x - 64 - 4x + 12 = 12x - 52\]
So, the numerator becomes \(\frac{12x - 52}{(x-3)(x-4)}\).
- Simplifying the Denominator
Next, we simplify the denominator of the complex fraction:
\[5 - \frac{x-4}{x-3} = \frac{5(x-3) - (x-4)}{x-3}\]
Expanding the terms in the numerator of this fraction:
\[5(x-3) - (x-4) = 5x - 15 - x + 4 = 4x - 11\]
So, the denominator becomes \(\frac{4x - 11}{x-3}\).
- Final Combination and Simplification
Now, we combine the simplified numerator and denominator to evaluate the full complex fraction:
\[\frac{\frac{12x - 52}{(x-3)(x-4)}}{\frac{4x - 11}{x-3}} = \frac{12x - 52}{(x-3)(x-4)} \cdot \frac{x-3}{4x - 11}\]
We can cancel the common factor \((x-3)\) in the numerator and the denominator:
\[\frac{12x - 52}{(x-4)(4x - 11)}\]
Expanding the denominator:
\[(x-4)(4x - 11) = 4x^2 - 11x - 16x + 44 = 4x^2 - 27x + 44\]
Therefore, the expression simplifies to \(\frac{12x - 52}{4x^2 - 27x + 44}\), which corresponds to option E.
Examples
Simplifying complex fractions is a fundamental skill in algebra with applications in fields like engineering and economics. For example, when analyzing the total resistance in an electrical circuit containing parallel components, you often end up with complex fractional expressions that must be simplified to understand the circuit's behavior. Mastering these techniques allows you to reduce complex models into simpler, more manageable forms for calculation and design.