1. **State the problem:** We are given the function $$y = 18 \cdot 3^{2(x-1)} - 54$$ and the inequality $$y \geq 0$$. 2. **Goal:** Find the values of $$x$$ for which $$y \geq 0$$. 3. **Rewrite the inequality:** $$18 \cdot 3^{2(x-1)} - 54 \geq 0$$ 4. **Isolate the exponential term:** Add 54 to both sides: $$18 \cdot 3^{2(x-1)} \geq 54$$ Divide both sides by 18: $$3^{2(x-1)} \geq 3$$ 5. **Rewrite the right side as a power of 3:** Since $$3 = 3^1$$, the inequality becomes: $$3^{2(x-1)} \geq 3^1$$ 6. **Use properties of exponential functions:** Because the base 3 is greater than 1, the function $$3^t$$ is increasing, so the inequality holds if and only if the exponents satisfy: $$2(x-1) \geq 1$$ 7. **Solve the inequality for $$x$$:** $$2x - 2 \geq 1$$ $$2x \geq 3$$ $$x \geq \frac{3}{2}$$ **Final answer:** $$\boxed{x \geq \frac{3}{2}}$$