1. **State the problem:** We need to sketch the region in the complex plane (z-plane) defined by the equation $$|z+3i| + |z-3i| = 5$$. 2. **Interpret the equation:** Let $z = x + yi$, where $x$ and $y$ are real numbers. 3. The expression $$|z+3i|$$ represents the distance from the point $(x,y)$ to the point $(0,-3)$ in the plane. 4. Similarly, $$|z-3i|$$ represents the distance from $(x,y)$ to $(0,3)$. 5. The equation $$|z+3i| + |z-3i| = 5$$ means the sum of distances from any point $(x,y)$ to the points $(0,-3)$ and $(0,3)$ is constant and equal to 5. 6. This is the definition of an ellipse with foci at $(0,-3)$ and $(0,3)$ and the sum of distances to the foci equal to 5. 7. The distance between the foci is $$2c = 6$$, so $$c = 3$$. 8. The sum of distances to the foci is $$2a = 5$$, so $$a = \frac{5}{2} = 2.5$$. 9. Using the ellipse relationship $$c^2 = a^2 - b^2$$, solve for $$b$$: $$b^2 = a^2 - c^2 = (2.5)^2 - 3^2 = 6.25 - 9 = -2.75$$. 10. Since $$b^2$$ is negative, this means no real ellipse exists with these parameters, so the set is empty. 11. **Conclusion:** There is no point $$z$$ in the complex plane satisfying $$|z+3i| + |z-3i| = 5$$ because the sum of distances is less than the distance between the foci. **Final answer:** The region is empty; no points satisfy the equation.