Question

(1/x+2/(x-1))/(4/x-1)

Answer 1

To simplify the expression \((\frac{1}{x} + \frac{2}{x-1}) / (\frac{4}{x} - 1)\), we will work step by step.

First, simplify the numerator:

\[
\frac{1}{x} + \frac{2}{x-1}
\]

To combine these fractions, we need a common denominator. The common denominator is \(x(x-1)\).

Rewriting each term under this common denominator gives:

\[
\frac{1}{x} = \frac{(x-1)}{x(x-1)}
\]
\[
\frac{2}{x-1} = \frac{2x}{x(x-1)}
\]

Now combine:

\[
\frac{(x-1) + 2x}{x(x-1)} = \frac{3x - 1}{x(x-1)}
\]

Now we simplify the denominator:

\[
\frac{4}{x} - 1
\]

Again, let's rewrite \(1\) under the same denominator \(x\):

\[
1 = \frac{x}{x}
\]

Thus, we write:

\[
\frac{4}{x} - 1 = \frac{4 - x}{x}
\]

Now we substitute both parts back into the original expression:

\[
\frac{\frac{3x - 1}{x(x-1)}}{\frac{4 - x}{x}}
\]

Dividing by a fraction is equivalent to multiplying by its reciprocal:

\[
\frac{3x - 1}{x(x-1)} \cdot \frac{x}{4 - x}
\]

The \(x\) in the numerator of the second fraction cancels with \(x\) in the denominator of the first fraction, assuming \(x \neq 0\):

\[
\frac{3x - 1}{(x-1)(4 - x)}
\]

So the simplified expression is:

\[
\frac{3x - 1}{(x-1)(4 - x)}
\]

And that is the final answer:

\[
\frac{3x - 1}{(x-1)(4 - x)}
\]