1. The problem requires identifying the shaded region that satisfies the inequalities: $$y \leq x, \quad x \leq 4, \quad y \geq 1$$ 2. Let's analyze each inequality: - The inequality $$y \leq x$$ represents points on or below the line $$y = x$$. - The inequality $$x \leq 4$$ represents points to the left or on the vertical line $$x = 4$$. - The inequality $$y \geq 1$$ represents points above or on the horizontal line $$y = 1$$. 3. Combining these, the region must be inside the area bounded by these three lines: - Below or on the diagonal line $$y = x$$. - Left or on the vertical line $$x = 4$$. - Above or on the horizontal line $$y = 1$$. 4. The intersection of these inequalities forms a triangular region with vertices at: - The intersection of $$y = 1$$ and $$x = 4$$ is point $$(4,1)$$. - The intersection of $$y = x$$ and $$x = 4$$ is point $$(4,4)$$. - The intersection of $$y = x$$ and $$y = 1$$ is point $$(1,1)$$. 5. Therefore, the shaded region is the triangle bounded by points $$(1,1), (4,1), (4,4)$$. 6. This region satisfies all three inequalities. Final answer: The shaded region is the triangle bounded by $$y = 1$$, $$x = 4$$, and $$y = x$$, inside the lines as described. This corresponds to the given graphic description for region R.