1. **State the problem:** Solve for $x$ in the equation $$2.17 = -66.67\left(e^{-0.0003x} - e^{-0.0002x}\right) + 1.17 e^{-0.0002x}.$$\n\n2. **Rewrite the equation:** Distribute and group terms involving $e^{-0.0002x}$ and $e^{-0.0003x}$:\n$$2.17 = -66.67 e^{-0.0003x} + 66.67 e^{-0.0002x} + 1.17 e^{-0.0002x}.$$\n\n3. **Combine like terms:**\n$$2.17 = -66.67 e^{-0.0003x} + (66.67 + 1.17) e^{-0.0002x} = -66.67 e^{-0.0003x} + 67.84 e^{-0.0002x}.$$\n\n4. **Isolate terms:** Move all terms to one side to set equation to zero:\n$$0 = -66.67 e^{-0.0003x} + 67.84 e^{-0.0002x} - 2.17.$$\n\n5. **Substitute:** Let $a = e^{-0.0002x}$, then $e^{-0.0003x} = e^{-0.0002x} e^{-0.0001x} = a e^{-0.0001x}$. Define $b = e^{-0.0001x}$, so $e^{-0.0003x} = a b$.\n\n6. **Rewrite equation in terms of $a$ and $b$:**\n$$0 = -66.67 a b + 67.84 a - 2.17.$$\n\n7. **Divide both sides by $a$ (since $a > 0$):**\n$$0 = -66.67 b + 67.84 - \frac{2.17}{a}.$$\n\n8. **Rearranged:**\n$$66.67 b = 67.84 - \frac{2.17}{a}.$$\n\n9. **Recall $b = e^{-0.0001x}$ and $a = e^{-0.0002x} = (e^{-0.0001x})^2 = b^2$. Substitute $a = b^2$:**\n$$66.67 b = 67.84 - \frac{2.17}{b^2}.$$\n\n10. **Multiply both sides by $b^2$ to clear denominator:**\n$$66.67 b^3 = 67.84 b^2 - 2.17.$$\n\n11. **Bring all terms to one side:**\n$$66.67 b^3 - 67.84 b^2 + 2.17 = 0.$$\n\n12. **Solve cubic equation for $b$:** Use numerical methods or approximation.\nApproximate root: $b \approx 1.01$ (since $b = e^{-0.0001x} > 0$).\n\n13. **Solve for $x$:**\n$$b = e^{-0.0001x} = 1.01 \implies -0.0001x = \ln(1.01) \implies x = -\frac{\ln(1.01)}{0.0001}.$$\n\n14. **Calculate:**\n$$\ln(1.01) \approx 0.00995,$$\nso\n$$x = -\frac{0.00995}{0.0001} = -99.5.$$\n\n15. **Interpretation:** The solution is approximately $$x \approx -99.5.$$