1. **State the problem:** Solve the equation $$5 - \sqrt[3]{x^2 - 9} = 7$$ for $x$. 2. **Isolate the cube root term:** Subtract 5 from both sides: $$5 - \sqrt[3]{x^2 - 9} = 7 \implies - \sqrt[3]{x^2 - 9} = 7 - 5$$ $$- \sqrt[3]{x^2 - 9} = 2$$ 3. **Multiply both sides by -1 to simplify:** $$\cancel{-} \sqrt[3]{x^2 - 9} = \cancel{-} 2$$ $$\sqrt[3]{x^2 - 9} = -2$$ 4. **Cube both sides to eliminate the cube root:** $$\left(\sqrt[3]{x^2 - 9}\right)^3 = (-2)^3$$ $$x^2 - 9 = -8$$ 5. **Solve for $x^2$:** $$x^2 = -8 + 9$$ $$x^2 = 1$$ 6. **Take the square root of both sides:** $$x = \pm \sqrt{1}$$ $$x = \pm 1$$ 7. **Check solutions in the original equation:** - For $x=1$: $$5 - \sqrt[3]{1^2 - 9} = 5 - \sqrt[3]{1 - 9} = 5 - \sqrt[3]{-8} = 5 - (-2) = 5 + 2 = 7$$ (True) - For $x=-1$: $$5 - \sqrt[3]{(-1)^2 - 9} = 5 - \sqrt[3]{1 - 9} = 5 - \sqrt[3]{-8} = 5 - (-2) = 7$$ (True) **Final answer:** $$x = \pm 1$$