1. **State the problem:** Solve the equation $\sqrt{6x+1} = 9$ for $x$. 2. **Recall the formula and rule:** To solve an equation involving a square root, we square both sides to eliminate the square root. Remember, squaring both sides can introduce extraneous solutions, so we must check our answers. 3. **Square both sides:** $$\left(\sqrt{6x+1}\right)^2 = 9^2$$ $$6x + 1 = 81$$ 4. **Isolate $x$:** $$6x = 81 - 1$$ $$6x = 80$$ 5. **Divide both sides by 6:** $$x = \frac{80}{6}$$ $$x = \frac{\cancel{80}}{\cancel{6}} \Rightarrow x = \frac{40}{3}$$ 6. **Check the solution:** Substitute $x = \frac{40}{3}$ back into the original equation: $$\sqrt{6 \times \frac{40}{3} + 1} = \sqrt{80 + 1} = \sqrt{81} = 9$$ The solution satisfies the original equation. **Final answer:** $$x = \frac{40}{3}$$