1. **Stating the problem:** We are asked to find the length $XZ$ on a number line to the nearest sixteenth of an inch. 2. **Understanding the problem:** Points $X$, $Y$, and $Z$ are on a number line with $X$ and $Y$ closer together and $Z$ farther right. We want to find the distance between $X$ and $Z$. 3. **Formula used:** The distance between two points on a number line is the absolute difference of their coordinates: $$XZ = |Z - X|$$ 4. **Measurement and approximation:** Assuming the ruler is divided into eighths of an inch, each eighth can be further divided into two sixteenths. 5. **Greatest possible measurement error:** Since the smallest division is $\frac{1}{8}$ inch, the greatest possible error when measuring is half of that: $$\text{Error} = \frac{1}{2} \times \frac{1}{8} = \frac{1}{16}$$ 6. **Final answer:** The greatest possible measurement error with Jack's ruler is $\frac{1}{16}$ inch. **Note:** Without exact coordinates of points $X$ and $Z$, we cannot calculate the exact length $XZ$, but the error in measurement is $\frac{1}{16}$ inch.