1. **Problem Statement:** Describe and graph the locus represented by the inequality $$1 < |z + i| \leq 2$$ where $$z = x + iy$$ is a complex number. 2. **Rewrite the expression:** We have $$|z + i| = |x + iy + i| = |x + i(y + 1)|$$. 3. **Distance interpretation:** The modulus $$|x + i(y + 1)|$$ represents the distance from the point $$(x,y)$$ to the point $$(0,-1)$$ in the complex plane. 4. **Inequality in terms of distance:** The inequality $$1 < |z + i| \leq 2$$ means the distance from $$(x,y)$$ to $$(0,-1)$$ is greater than 1 and less than or equal to 2. 5. **Geometric locus:** This describes an annulus (ring-shaped region) centered at $$(0,-1)$$ with inner radius 1 and outer radius 2. 6. **Mathematical form:** The locus satisfies $$ 1 < \sqrt{(x - 0)^2 + (y + 1)^2} \leq 2 $$ which can be squared to $$ 1^2 < (x - 0)^2 + (y + 1)^2 \leq 2^2 $$ $$ 1 < x^2 + (y + 1)^2 \leq 4 $$ 7. **Summary:** The locus is the set of points $$ (x,y) $$ such that the distance from $$(0,-1)$$ is strictly greater than 1 and at most 2, forming a ring between two concentric circles. **Final answer:** The locus is an annulus centered at $$(0,-1)$$ with inner radius 1 and outer radius 2, described by $$ 1 < x^2 + (y + 1)^2 \leq 4 $$