1. **State the problem:** Solve the equation $$-6\sqrt[3]{10x} + 11 = -19$$ and check the solution(s). 2. **Isolate the cube root term:** Subtract 11 from both sides: $$-6\sqrt[3]{10x} + 11 - 11 = -19 - 11$$ $$-6\sqrt[3]{10x} = -30$$ 3. **Divide both sides by -6:** $$\cancel{-6}\sqrt[3]{10x} = \frac{-30}{\cancel{-6}}$$ $$\sqrt[3]{10x} = 5$$ 4. **Cube both sides to eliminate the cube root:** $$\left(\sqrt[3]{10x}\right)^3 = 5^3$$ $$10x = 125$$ 5. **Solve for x:** $$x = \frac{125}{10}$$ $$x = 12.5$$ 6. **Check the solution:** Substitute $x=12.5$ back into the original equation: $$-6\sqrt[3]{10(12.5)} + 11 = -6\sqrt[3]{125} + 11 = -6(5) + 11 = -30 + 11 = -19$$ The left side equals the right side, so $x=12.5$ is a valid solution. **Final answer:** $$x = 12.5$$