1. **Problem statement:** We want to understand the region defined by the polar coordinate condition $|r| > 0.5$ for the angle range $\theta \in \left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$. 2. **Recall polar coordinates:** A point in polar coordinates is given by $(r, \theta)$ where $r$ is the radius (distance from origin) and $\theta$ is the angle from the positive x-axis. 3. **Condition $|r| > 0.5$ means:** The radius $r$ can be either greater than $0.5$ or less than $-0.5$. Since radius is usually non-negative, negative $r$ means the point is in the opposite direction of angle $\theta$. 4. **Angle range:** $\theta$ varies from $-\frac{\pi}{3}$ to $\frac{\pi}{4}$, which is from $-60^\circ$ to $45^\circ$. 5. **Interpretation:** The region includes all points whose distance from the origin is more than $0.5$ units, in the directions between $-60^\circ$ and $45^\circ$, and also points with radius less than $-0.5$ which correspond to points in the opposite directions of those angles. 6. **Graphically:** This forms two sectors (one for positive $r$ and one for negative $r$) excluding the circle of radius $0.5$. 7. **Final:** The curve is the boundary where $|r|=0.5$ for $\theta$ in the given range, which are two circular arcs of radius $0.5$ at angles $\theta$ and $\theta + \pi$. $$r = 0.5, \quad \theta \in \left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$$ and $$r = -0.5, \quad \theta \in \left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$$ These correspond to the same circle but opposite directions.