1. **State the problem:** We need to determine which system of inequalities matches the graph described. 2. **Analyze the graph:** - The red line has a positive slope and passes through points approximately $(0,-2)$ and $(4,2)$. - The blue line has a negative slope and passes through points approximately $(0,12)$ and $(12,0)$. - The shaded region is above the red line and below the blue line. 3. **Find equations of the lines:** - For the red line, slope $m = \frac{2 - (-2)}{4 - 0} = \frac{4}{4} = 1$. - Equation using point-slope form with point $(0,-2)$: $$y = 1 \cdot x - 2 = x - 2$$ - For the blue line, slope $m = \frac{0 - 12}{12 - 0} = \frac{-12}{12} = -1$. - Equation using point-slope form with point $(0,12)$: $$y = -1 \cdot x + 12 = -x + 12$$ 4. **Determine inequalities:** - The shaded region is above the red line, so $y \geq x - 2$. - The shaded region is below the blue line, so $y \leq -x + 12$. 5. **Compare with given options:** - None of the options exactly match $y \geq x - 2$ and $y \leq -x + 12$. 6. **Check closest option:** - The first option has $y \leq -3x - 4$ and $y \geq -x + 6$ (slopes do not match). - The second option has $y \leq 5x - 10$ and $y \geq -x - 1$ (slopes do not match). - The third option has $y \leq -3x - 10$ and $y \geq x + 8$ (slopes do not match). - The fourth option has $y \leq 4x - 1$ and $y \geq -x - 5$ (slopes do not match). 7. **Conclusion:** None of the provided systems exactly match the graph described. **Final answer:** The system of inequalities matching the graph is: $$\boxed{y \geq x - 2 \quad \text{and} \quad y \leq -x + 12}$$