1. **State the problem:** Solve for $x$ in the equation $$\sqrt{x} - \sqrt{6x} - 9 + \sqrt{x} + \sqrt{6x} - 9 = 6.$$\n\n2. **Simplify the equation:** Combine like terms on the left side. Notice that $\sqrt{6x}$ terms cancel out: $$\sqrt{x} - \sqrt{6x} - 9 + \sqrt{x} + \sqrt{6x} - 9 = 2\sqrt{x} - 18.$$\n\n3. **Rewrite the equation:** $$2\sqrt{x} - 18 = 6.$$\n\n4. **Isolate the square root term:** Add 18 to both sides: $$2\sqrt{x} = 24.$$\n\n5. **Divide both sides by 2:** $$\cancel{2}\sqrt{x} = \frac{24}{\cancel{2}} \implies \sqrt{x} = 12.$$\n\n6. **Square both sides to solve for $x$:** $$\left(\sqrt{x}\right)^2 = 12^2 \implies x = 144.$$\n\n7. **Check the solution:** Substitute $x=144$ back into the original equation to verify it satisfies the equation.\n\n**Final answer:** $$\boxed{144}.$$