- Simplify the numerator \(\frac{3}{x-2} - 5\) to \(\frac{13-5x}{x-2}\).
- Simplify the denominator \(2 - \frac{4}{x-2}\) to \(\frac{2x-8}{x-2}\).
- Divide the simplified numerator by the simplified denominator, canceling the term \((x-2)\).
- Determine the equivalent expression is \(\boxed{\frac{13-5x}{2x-8}}\).
Explanation
- Analysis of the Expression
To simplify the given complex rational expression \(\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}}\), we must first note that the expression is defined only when the denominators are non-zero. Specifically, we must have \(x-2
eq 0\) (so \(x
eq 2\)) and \(2-\frac{4}{x-2}
eq 0\). The second condition implies \(\frac{2(x-2)-4}{x-2}
eq 0\), which simplifies to \(\frac{2x-8}{x-2}
eq 0\), so \(x
eq 4\). Our goal is to perform algebraic manipulation to combine the numerator and denominator into a single fraction and simplify the resulting ratio.
- Simplifying the Numerator
Let's begin by simplifying the numerator:
\[\frac{3}{x-2} - 5.\]
To combine these, find a common denominator of \(x-2\):
\[\frac{3}{x-2} - \frac{5(x-2)}{x-2} = \frac{3 - (5x - 10)}{x-2} = \frac{3 - 5x + 10}{x-2} = \frac{13 - 5x}{x-2}.\]
- Simplifying the Denominator
Next, we simplify the denominator:
\[2 - \frac{4}{x-2}.\]
Again, use a common denominator of \(x-2\):
\[\frac{2(x-2)}{x-2} - \frac{4}{x-2} = \frac{2x - 4 - 4}{x-2} = \frac{2x - 8}{x-2}.\]
- Dividing and Simplifying the Fraction
Now, we divide the simplified numerator by the simplified denominator:
\[\frac{\frac{13-5x}{x-2}}{\frac{2x-8}{x-2}}.\]
To divide by a fraction, we multiply by its reciprocal:
\[\frac{13-5x}{x-2} \cdot \frac{x-2}{2x-8}.\]
Since \(x
eq 2\), the factor \(x-2\) cancels out, leaving us with:
\[\frac{13-5x}{2x-8}.\]
- Final Result
Comparing our final expression \(\frac{13-5x}{2x-8}\) to the given options:
- \(\frac{2x-8}{13-5x}\)
- \(\frac{-5x-8}{2x-8}\)
- \(\frac{-5x-4}{x-2}\)
- \(\frac{13-5x}{2x-8}\)
The expression matches the fourth option. Therefore, the equivalent expression is \(\frac{13-5x}{2x-8}\). The final answer is \(\boxed{\frac{13-5x}{2x-8}}\).
Examples
Simplifying complex rational expressions is a fundamental skill in algebra, often used in fields like physics and engineering when dealing with complex circuits or variable-based formulas. For example, if you are modeling the total resistance of a circuit with variable components represented as \(\frac{1}{R_1} + \frac{1}{R_2}\), you would use these algebraic simplification methods to rewrite the formula into a form that is easier to compute or analyze. This allows researchers to isolate specific variables and understand how they affect the behavior of the overall system.