1. **State the problem:** We need to determine which graph corresponds to the system of inequalities: $$y \geq 3x + 2$$ $$y \leq -2x - 5$$ 2. **Understand the inequalities:** - The first inequality $y \geq 3x + 2$ represents the region above or on the line $y = 3x + 2$. - The second inequality $y \leq -2x - 5$ represents the region below or on the line $y = -2x - 5$. 3. **Analyze the lines:** - Line 1: $y = 3x + 2$ has a positive slope (3) and y-intercept at 2. - Line 2: $y = -2x - 5$ has a negative slope (-2) and y-intercept at -5. 4. **Find the intersection point:** Set $3x + 2 = -2x - 5$ to find where the lines cross. $$3x + 2 = -2x - 5$$ $$3x + 2 + 2x + 5 = 0$$ $$5x + 7 = 0$$ $$5x = -7$$ $$x = -\frac{7}{5}$$ Substitute back to find $y$: $$y = 3\left(-\frac{7}{5}\right) + 2 = -\frac{21}{5} + 2 = -\frac{21}{5} + \frac{10}{5} = -\frac{11}{5}$$ So the intersection point is $$\left(-\frac{7}{5}, -\frac{11}{5}\right)$$. 5. **Determine the shaded region:** - For $y \geq 3x + 2$, the region is above the line with positive slope and y-intercept 2. - For $y \leq -2x - 5$, the region is below the line with negative slope and y-intercept -5. 6. **Match with the graphs:** - The first line crosses the y-axis above zero (at 2). - The second line crosses the y-axis below zero (at -5). - The shaded region is the intersection of the area above the first line and below the second line. 7. **Conclusion:** - Image 1 shows a shaded triangular region to the left of the intersection, bounded by lines crossing y-axis above and below zero, matching the description. - Image 2 shows shading in the bottom-left quadrant, which does not match. - Image 3 shows shading in the top-right quadrant but the lines do not match the intercepts. - Image 4 shows shading above the x-axis but the lines' positions do not match the intercepts. **Final answer:** The system corresponds to **Image 1**.