1. **State the problem:** Graph the inequality $x \geq -3 \cdot 2$ for three different sets: integers ($\mathbb{Z}$), natural numbers ($\mathbb{N}$), and real numbers ($\mathbb{R}$). 2. **Simplify the inequality:** Calculate $-3 \cdot 2$: $$-3 \cdot 2 = -6$$ So the inequality becomes: $$x \geq -6$$ 3. **Explain the sets:** - $\mathbb{Z}$: all integers (..., -2, -1, 0, 1, 2, ...) - $\mathbb{N}$: natural numbers (1, 2, 3, ...) - $\mathbb{R}$: all real numbers 4. **Graphing each case:** (i) For $x \in \mathbb{Z}$ and $x \geq -6$: - Mark a closed point at $-6$ on the number line. - Shade all integer points to the right of $-6$ (i.e., $-6, -5, -4, ...$). (ii) For $x \in \mathbb{N}$ and $x \geq -6$: - Since natural numbers start at 1, the smallest natural number satisfying $x \geq -6$ is 1. - Mark points at $1, 2, 3, 4, ...$ on the number line. (iii) For $x \in \mathbb{R}$ and $x \geq -6$: - Mark a closed point at $-6$. - Shade the entire number line to the right of $-6$ continuously. 5. **Summary:** - The inequality simplifies to $x \geq -6$. - The graphs differ by the domain of $x$. Final answer: $$x \geq -6$$