To solve the given equation, we have the expression:

\(\sqrt{1 + \frac{x}{121}} = \frac{12}{11}\)

This implies:

First, square both sides to eliminate the square root:

• \(\left(\sqrt{1 + \frac{x}{121}}\right)^2 = \left(\frac{12}{11}\right)^2\)
• This simplifies to: \(1 + \frac{x}{121} = \frac{144}{121}\)

Next, isolate \(\frac{x}{121}\) on one side:

• Subtract 1 from both sides of the equation: \(\frac{x}{121} = \frac{144}{121} - 1\)
• Convert 1 to the fraction \(\frac{121}{121}\) for subtraction: \(\frac{x}{121} = \frac{144}{121} - \frac{121}{121}\)
• This simplifies to: \(\frac{x}{121} = \frac{144 - 121}{121} = \frac{23}{121}\)

Finally, solve for \(x\) by multiplying both sides by 121:

• \(x = 23\)

Thus, the value of \(x\) is 23.

This matches option: 23.