1. **State the problem:** Solve the equation $$-2\sqrt{6x} + 3 = -18$$ for $x$. 2. **Isolate the square root term:** Subtract 3 from both sides: $$-2\sqrt{6x} + 3 - 3 = -18 - 3$$ $$-2\sqrt{6x} = -21$$ 3. **Divide both sides by -2:** $$\cancel{-2}\sqrt{6x} = \frac{-21}{\cancel{-2}}$$ $$\sqrt{6x} = \frac{21}{2}$$ 4. **Square both sides to eliminate the square root:** $$\left(\sqrt{6x}\right)^2 = \left(\frac{21}{2}\right)^2$$ $$6x = \frac{441}{4}$$ 5. **Solve for $x$ by dividing both sides by 6:** $$x = \frac{441}{4 \times 6} = \frac{441}{24}$$ 6. **Simplify the fraction:** $$x = \frac{441 \div 3}{24 \div 3} = \frac{147}{8}$$ **Final answer:** $$x = \frac{147}{8}$$