1. **State the problem:** Solve the equation $11^{-1}x = 10$ for $x$. 2. **Recall the rule:** $11^{-1}$ means the reciprocal of 11, so $11^{-1} = \frac{1}{11}$. 3. **Rewrite the equation:** Replace $11^{-1}$ with $\frac{1}{11}$: $$\frac{1}{11}x = 10$$ 4. **Isolate $x$:** Multiply both sides by 11 to get rid of the denominator: $$x = 10 \times 11$$ 5. **Calculate the product:** $$x = 110$$ **Final answer:** $x = 110$