If `(2x)/(1+(1)/(1+(x)/(1-x)))`=1 then find the value of x

Options

A. `(2)/(3)`
B. `(1)/(3)`
C. `(5)/(6)`
D. `(1)/(4)`

Answer

To solve the equation \(\frac{2x}{1 + \frac{1}{1 + \frac{x}{1 - x}}} = 1\), we will follow these steps:

### Step 1: Simplify the Inner Fraction
First, we need to simplify the inner fraction \(\frac{1}{1 + \frac{x}{1 - x}}\).

- The denominator can be rewritten as:
\[ 1 + \frac{x}{1 - x} = \frac{(1 - x) + x}{1 - x} = \frac{1}{1 - x} \]

Thus, we have:
\[ \frac{1}{1 + \frac{x}{1 - x}} = 1 - x \]

### Step 2: Substitute Back into the Equation
Now, substitute this back into the original equation:
\[ \frac{2x}{1 + (1 - x)} = 1 \]

### Step 3: Simplify the Denominator
Now simplify the denominator:
\[ 1 + (1 - x) = 2 - x \]

So, the equation becomes:
\[ \frac{2x}{2 - x} = 1 \]

### Step 4: Cross-Multiply
Cross-multiply to eliminate the fraction:
\[ 2x = 2 - x \]

### Step 5: Solve for \(x\)
Now, add \(x\) to both sides:
\[ 2x + x = 2 \]
\[ 3x = 2 \]

Now, divide both sides by 3:
\[ x = \frac{2}{3} \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{\frac{2}{3}} \]

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