Question
Which expression is equivalent to \( \frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}} ? \)
Ask by Stanley Barber. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression simplifies to \( \frac{13-5x}{2x-8} \).
Solution
Calculate or simplify the expression \( \frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}} \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}}\)
- step1: Subtract the terms:
\(\frac{\frac{13-5x}{x-2}}{2-\frac{4}{x-2}}\)
- step2: Subtract the terms:
\(\frac{\frac{13-5x}{x-2}}{\frac{2x-8}{x-2}}\)
- step3: Multiply by the reciprocal:
\(\frac{13-5x}{x-2}\times \frac{x-2}{2x-8}\)
- step4: Reduce the fraction:
\(\left(13-5x\right)\times \frac{1}{2x-8}\)
- step5: Multiply the terms:
\(\frac{13-5x}{2x-8}\)
The expression equivalent to \( \frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}} \) is \( \frac{13-5x}{2x-8} \).
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Simplify this solution The Deep Dive
To simplify the expression \( \frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}} \), we can first combine the fractions in both the numerator and the denominator. The numerator becomes: \[ \frac{3}{x-2} - 5 = \frac{3 - 5(x-2)}{x-2} = \frac{3 - 5x + 10}{x-2} = \frac{13 - 5x}{x-2} \] The denominator becomes: \[ 2 - \frac{4}{x-2} = \frac{2(x-2) - 4}{x-2} = \frac{2x - 4 - 4}{x-2} = \frac{2x - 8}{x-2} = \frac{2(x - 4)}{x-2} \] Now we can rewrite the entire expression: \[ \frac{\frac{13 - 5x}{x-2}}{\frac{2(x - 4)}{x-2}} = \frac{13 - 5x}{2(x - 4)} \] Thus, the expression simplifies to: \[ \frac{13 - 5x}{2(x - 4)} \] This is the equivalent expression.
