Question: User: $3^{1 - x^2} - \frac{1}{9} < 0$
1. **State the problem:** Solve the inequality $$3^{1 - x^2} - \frac{1}{9} < 0$$.
2. **Rewrite the inequality:** Move $$\frac{1}{9}$$ to the other side:
$$3^{1 - x^2} < \frac{1}{9}$$.
3. **Express $$\frac{1}{9}$$ as a power of 3:**
Since $$9 = 3^2$$, then $$\frac{1}{9} = 3^{-2}$$.
4. **Rewrite the inequality using the same base:**
$$3^{1 - x^2} < 3^{-2}$$.
5. **Use the property of exponential functions:**
Since the base 3 is greater than 1, the function $$3^t$$ is increasing. Therefore, the inequality
$$3^{a} < 3^{b}$$ implies $$a < b$$.
6. **Apply this to our inequality:**
$$1 - x^2 < -2$$.
7. **Solve the inequality for $$x$$:**
$$1 - x^2 < -2$$
$$-x^2 < -3$$
Multiply both sides by $$-1$$ (remember to reverse the inequality sign):
$$x^2 > 3$$.
8. **Find the solution set:**
$$x^2 > 3$$ means
$$x < -\sqrt{3}$$ or $$x > \sqrt{3}$$.
**Final answer:**
$$\boxed{x < -\sqrt{3} \text{ or } x > \sqrt{3}}$$