1. **State the problem:** Solve the proportion $1,11 = \frac{1}{x}$ for $x$. 2. **Rewrite the problem:** The comma in $1,11$ likely represents a decimal point, so the problem is $1.11 = \frac{1}{x}$. 3. **Use the formula for proportions:** If $a = \frac{b}{c}$, then $c = \frac{b}{a}$. 4. **Apply the formula:** Here, $a = 1.11$, $b = 1$, and $c = x$, so $$x = \frac{1}{1.11}$$ 5. **Calculate the value:** $$x = \frac{1}{1.11} = \frac{1}{\cancel{1.11}} \times \frac{\cancel{1}}{1.11} = 0.9009...$$ 6. **Interpret the result:** The value of $x$ is approximately $0.9009$. **Final answer:** $$x \approx 0.9009$$