https://file.aiquickdraw.com/mathbot/file/1728384455891se1j96i9.jpg

The given expression in the image is:

\[
\frac{\frac{1}{r - 1}}{\frac{1}{r} - 2(1 - r)}
\]

### Step 1: Simplify the denominator
The denominator of the main fraction is:

\[
\frac{1}{r} - 2(1 - r)
\]

Let's expand and simplify this part first:

\[
2(1 - r) = 2 - 2r
\]

Thus, the denominator becomes:

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

Now, we can combine the terms over a common denominator. For \(\frac{1}{r}\), the common denominator is \(r\), so rewrite everything as:

\[
\frac{1}{r} - 2 + 2r = \frac{1 - 2r + 2r^2}{r}
\]

So, the full denominator is now:

\[
\frac{1 - 2r + 2r^2}{r}
\]

### Step 2: Simplify the entire expression
Now we have the expression:

\[
\frac{\frac{1}{r - 1}}{\frac{1 - 2r + 2r^2}{r}}
\]

To simplify this, multiply by the reciprocal of the denominator:

\[
\frac{1}{r - 1} \times \frac{r}{1 - 2r + 2r^2} = \frac{r}{(r - 1)(1 - 2r + 2r^2)}
\]

Thus, the simplified expression is:

\[
\boxed{\frac{r}{(r - 1)(1 - 2r + 2r^2)}}
\]

Would you like further explanation on this? Let me know if you have any questions!

### Related questions:
1. How do you find the common denominator when combining fractions?
2. What is the process of multiplying fractions with numerators and denominators?
3. How do you handle negative terms in simplification of fractions?
4. Can this expression be factored further?
5. What is the significance of rational expressions in algebra?

**Tip**: Always simplify complex fractions by working on the numerator and denominator separately before combining!

Simplify the following expression: \( \frac{\frac{1}{r - 1}}{\frac{1}{r} - 2(1 - r)} \)

Learn how to simplify the complex rational expression \( \frac{\frac{1}{r - 1}}{\frac{1}{r} - 2(1 - r)} \) using fraction division and simplification techniques. This example focuses on key algebraic skills for students in Grades 10-12.

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