1. **State the problem:** We have a data set: 3, 9, 1, 1, 1, 6, 10, and an unknown value $x$. The mean of all these numbers is given as 4.5. 2. **Formula for mean:** The mean of a set of numbers is given by $$\text{mean} = \frac{\text{sum of all values}}{\text{number of values}}$$ 3. **Apply the formula:** There are 8 numbers in total (7 known + 1 unknown $x$). So, $$4.5 = \frac{3 + 9 + 1 + 1 + 1 + 6 + 10 + x}{8}$$ 4. **Calculate the sum of known values:** $$3 + 9 + 1 + 1 + 1 + 6 + 10 = 31$$ 5. **Set up the equation:** $$4.5 = \frac{31 + x}{8}$$ 6. **Multiply both sides by 8:** $$4.5 \times 8 = 31 + x$$ $$36 = 31 + x$$ 7. **Isolate $x$:** $$x = 36 - 31$$ $$x = 5$$ **Final answer:** $$\boxed{5}$$