1. **State the problem:** We are given two systems of inequalities: System 1: $$y \geq -x + 1$$ $$y < -3x + 2$$ System 2: $$y > -x + 1$$ $$y \leq -3x + 2$$ We want to understand the solution regions for these inequalities and how they relate to the graph with points F, G, A, B, D, E, H. 2. **Recall the meaning of inequalities:** - For $y \geq -x + 1$, the solution region is the area on or above the line $y = -x + 1$. - For $y < -3x + 2$, the solution region is the area strictly below the line $y = -3x + 2$. - For $y > -x + 1$, the solution region is the area strictly above the line $y = -x + 1$. - For $y \leq -3x + 2$, the solution region is the area on or below the line $y = -3x + 2$. 3. **Find the intersection points of the boundary lines:** Set the lines equal to find their intersection: $$-x + 1 = -3x + 2$$ Add $3x$ to both sides: $$-x + 3x + 1 = 2$$ Simplify: $$2x + 1 = 2$$ Subtract 1: $$2x = 1$$ Divide both sides by 2: $$x = \frac{1}{2}$$ Substitute $x=\frac{1}{2}$ into $y = -x + 1$: $$y = -\frac{1}{2} + 1 = \frac{1}{2}$$ So the lines intersect at point $$\left(\frac{1}{2}, \frac{1}{2}\right)$$. 4. **Analyze the solution regions:** - For System 1, the solution is the region where $y$ is greater than or equal to $-x + 1$ and less than $-3x + 2$. - For System 2, the solution is the region where $y$ is strictly greater than $-x + 1$ and less than or equal to $-3x + 2$. 5. **Interpret the graph:** The lines passing through points F and E correspond to one line, and points G and H correspond to the other. The intersection region around points A and B represents the solution set where both inequalities hold. 6. **Summary:** The solution regions are the areas between the two lines, with the inequalities determining whether the boundary lines are included or excluded.