1. The problem involves understanding and graphing the system of inequalities: $$y \leq x + 2$$ $$x + y \geq 4$$ $$y \leq 5$$ $$y \geq 4$$ $$x + y \geq 5$$ 2. Let's analyze each inequality and the corresponding graph description: - For $y \leq x + 2$, the graph is a right triangle with hypotenuse from $(0,5)$ to $(5,0)$, shaded above the line $y = -x + 5$. This suggests the inequality $y \leq x + 2$ is related to the region above $y = -x + 5$. - For $x + y \geq 4$, the graph is a polygon starting at $(0,2)$, shaded above the line $y = x - 2$. - For $y \leq 5$, the graph is a rectangle shaded between $y=4$ and $y=5$. 3. Let's rewrite the inequalities to understand the regions: - $y \leq x + 2$ is a region below or on the line $y = x + 2$. - $x + y \geq 4$ can be rewritten as $y \geq 4 - x$. - $y \leq 5$ is the region below or on the line $y=5$. - $y \geq 4$ is the region above or on the line $y=4$. - $x + y \geq 5$ can be rewritten as $y \geq 5 - x$. 4. The graphs described correspond to these inequalities: - Graph a) describes the region above $y = -x + 5$, which is equivalent to $x + y \geq 5$. - Graph b) describes the region above $y = x - 2$, which corresponds to $x + y \geq 4$. - Graph c) describes the horizontal strip $4 \leq y \leq 5$. 5. The system of inequalities defines a feasible region bounded by these lines: - $y = x + 2$ - $x + y = 4$ - $y = 5$ - $y = 4$ - $x + y = 5$ 6. To graph these, plot the lines and shade the regions satisfying the inequalities: - For $y \leq x + 2$, shade below the line. - For $x + y \geq 4$, shade above the line. - For $y \leq 5$ and $y \geq 4$, shade between these horizontal lines. - For $x + y \geq 5$, shade above the line. 7. The intersection of these regions forms the solution set. Final answer: The system of inequalities defines a polygonal region bounded by the lines $y = x + 2$, $x + y = 4$, $y = 5$, $y = 4$, and $x + y = 5$ with shading as described.