1. The problem asks to sketch the set $S$ of points in the complex plane satisfying the inequalities for problems 13, 15, 17, 23, 14, 16, 18, and 24. 2. Each problem describes a region in the complex plane based on the real part $\operatorname{Re}(z)$, imaginary part $\operatorname{Im}(z)$, or modulus $|z - w|$ for some complex number $w$. 3. We will analyze each problem separately and describe the set $S$: **Problem 13: $\operatorname{Re}(z) < -1$** - This is the half-plane to the left of the vertical line $x = -1$. - The set is open, unbounded, connected, and a domain. **Problem 15: $\operatorname{Im}(z) > 3$** - This is the half-plane above the horizontal line $y = 3$. - The set is open, unbounded, connected, and a domain. **Problem 17: $2 < \operatorname{Re}(z) < 4$** - This is the vertical strip between $x=2$ and $x=4$, excluding boundaries. - The set is open, unbounded, connected, and a domain. **Problem 23: $1 \leq |z - 1| < 2$** - This is the annulus centered at $1$ on the real axis with inner radius 1 (including boundary) and outer radius 2 (excluding boundary). - The set is neither open nor closed (because of mixed inequalities), bounded, connected, and a domain. **Problem 14: $|\operatorname{Re}(z)| > 2$** - This is the union of two half-planes: $x < -2$ or $x > 2$. - The set is open, unbounded, disconnected (two separate parts), and a domain. **Problem 16: $\operatorname{Re}((2 + i)z + 1) > 0$** - Let $z = x + iy$, then $$\operatorname{Re}((2 + i)(x + iy) + 1) = \operatorname{Re}((2 + i)x + (2 + i)iy + 1)$$ $$= \operatorname{Re}(2x + ix + 2iy - y + 1) = 2x - y + 1$$ - Inequality: $2x - y + 1 > 0$ or $y < 2x + 1$. - This is the half-plane below the line $y = 2x + 1$. - The set is open, unbounded, connected, and a domain. **Problem 18: $1 < \operatorname{Im}(z) < 4$** - This is the horizontal strip between $y=1$ and $y=4$, excluding boundaries. - The set is open, unbounded, connected, and a domain. **Problem 24: $2 \leq |z - 3 + 4i| \leq 5$** - This is the closed annulus centered at $(3,4)$ with inner radius 2 and outer radius 5. - The set is closed, bounded, connected, and a domain. 4. The combined sketch would show these regions on the complex plane with vertical and horizontal lines, strips, half-planes, and annuli as described. 5. The sets are mostly open or closed half-planes, strips, or annuli, with properties as noted. Final answer: Each problem corresponds to a specific region in the complex plane as described above, with their openness, boundedness, connectedness, and domain properties.