Question

1/x+2/(x-1))/(4/x-1). The incorrect answer is listed here: (3x-1)/(x-1)(4-x). The numerator is correct, but not the denominater

Answer 1

Let's simplify the expression \(\frac{\frac{1}{x} + \frac{2}{x-1}}{\frac{4}{x} - 1}\) step by step.

### Step 1: Simplify the Numerator
The numerator is \(\frac{1}{x} + \frac{2}{x-1}\). To combine these fractions, we need a common denominator, which will be \(x(x-1)\):

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

Now combine them:

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

### Step 2: Simplify the Denominator
Now consider the denominator \(\frac{4}{x} - 1\). To combine these terms, we rewrite \(1\) as \(\frac{x}{x}\):

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

### Step 3: Put It Together
Now we can rewrite the entire expression:

\[
\frac{\frac{1}{x} + \frac{2}{x-1}}{\frac{4}{x} - 1} = \frac{\frac{3x - 1}{x(x-1)}}{\frac{4 - x}{x}}
\]

When we divide by a fraction, we multiply by its reciprocal:

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

The \(x\) in the numerator and denominator cancels out:

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

Thus, the simplified form of the expression is:

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

### Conclusion
The numerator is indeed correct as \(3x - 1\) but the denominator is correctly simplified as \((x-1)(4-x)\). Hence, the final answer is:

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

Make sure to double-check the denominator as the original incorrect statement had a mistake.