What is the value of x in `1+1/(1+1/(1+1/x))=11/7`?

Options

A. 1
B. 3
C. `1/2`
D. `7/11`

Answer

To solve the equation \( 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}} = \frac{11}{7} \), we will follow these steps:

### Step 1: Isolate the Fraction
First, we will isolate the fraction on the left side by subtracting 1 from both sides of the equation.

\[ \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}} = \frac{11}{7} - 1 \]

Calculating the right side:

\[ \frac{11}{7} - 1 = \frac{11}{7} - \frac{7}{7} = \frac{11 - 7}{7} = \frac{4}{7} \]

So we have:

\[ \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}} = \frac{4}{7} \]

### Step 2: Invert Both Sides
Next, we will invert both sides to eliminate the fraction.

\[ 1 + \frac{1}{1 + \frac{1}{x}} = \frac{7}{4} \]

### Step 3: Isolate the Next Fraction
Now, we will isolate the next fraction by subtracting 1 from both sides.

\[ \frac{1}{1 + \frac{1}{x}} = \frac{7}{4} - 1 \]

Calculating the right side:

\[ \frac{7}{4} - 1 = \frac{7}{4} - \frac{4}{4} = \frac{7 - 4}{4} = \frac{3}{4} \]

So we have:

\[ \frac{1}{1 + \frac{1}{x}} = \frac{3}{4} \]

### Step 4: Invert Again
Next, we will invert both sides again.

\[ 1 + \frac{1}{x} = \frac{4}{3} \]

### Step 5: Isolate the Last Fraction
Now, we will isolate the last fraction by subtracting 1 from both sides.

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

Calculating the right side:

\[ \frac{4}{3} - 1 = \frac{4}{3} - \frac{3}{3} = \frac{4 - 3}{3} = \frac{1}{3} \]

So we have:

\[ \frac{1}{x} = \frac{1}{3} \]

### Step 6: Solve for x
Finally, we will take the reciprocal to find \( x \).

\[ x = 3 \]

Thus, the value of \( x \) is \( 3 \).

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