To simplify the expression \( 2 - \frac{2}{2 - \frac{1}{x}} \), follow these steps:

### Step 1: Simplify the inner fraction

The inner fraction is \( \frac{1}{x} \). Substitute this into the expression:

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

### Step 2: Combine the expression in the denominator

The denominator in the larger fraction is \( 2 - \frac{1}{x} \). To simplify this, find a common denominator:

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

Now, the expression becomes:

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

### Step 3: Simplify the fraction

Now simplify the fraction \( \frac{2}{\frac{2x - 1}{x}} \). This is equivalent to multiplying 2 by the reciprocal of \( \frac{2x - 1}{x} \):

\[
\frac{2}{\frac{2x - 1}{x}} = 2 \cdot \frac{x}{2x - 1} = \frac{2x}{2x - 1}
\]

So the expression becomes:

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

### Step 4: Combine the terms

To combine these terms, get a common denominator:

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

Now subtract:

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

### Final Answer:

The simplified expression is:

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

Would you like any further explanation or have questions about any step? Here are some related topics you could explore:

1. How to find and simplify complex fractions.
2. Techniques for finding common denominators in algebraic expressions.
3. Steps to simplify expressions involving nested fractions.
4. How to distribute and combine like terms in algebra.
5. Methods for checking the accuracy of algebraic simplifications.

**Tip:** Always be careful with signs when dealing with multiple fractions, as small mistakes in sign changes can lead to incorrect simplifications.

Learn how to simplify the algebraic expression 2 - (2 / (2 - (1 / x))). This problem involves complex fraction simplification and algebraic manipulation techniques. Suitable for Grades 9-12 students.

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