1. The problem asks to find the three inequalities that define the shaded triangular region bounded by the lines shown on the graph. 2. From the graph description, the boundaries are: - A vertical line at $x=5$ - A horizontal line at $y=3$ - A slanting line passing through points $(1,1)$ and $(5,7)$ 3. The inequalities must describe the region inside the triangle formed by these lines. 4. For the vertical line $x=5$, the shaded region is to the left, so the inequality is: $$x \leq 5$$ 5. For the horizontal line $y=3$, the shaded region is above, so the inequality is: $$y \geq 3$$ 6. For the slanting line through $(1,1)$ and $(5,7)$, first find its equation: The slope $m = \frac{7-1}{5-1} = \frac{6}{4} = \frac{3}{2}$ Equation in point-slope form: $$y - 1 = \frac{3}{2}(x - 1)$$ Simplify: $$y = \frac{3}{2}x - \frac{3}{2} + 1 = \frac{3}{2}x - \frac{1}{2}$$ 7. The shaded region is below this line, so the inequality is: $$y \leq \frac{3}{2}x - \frac{1}{2}$$ 8. Therefore, the three inequalities defining the shaded region are: $$x \leq 5$$ $$y \geq 3$$ $$y \leq \frac{3}{2}x - \frac{1}{2}$$ 9. The inequalities you wrote, $5 \leq 6$, $3 \leq 2$, and $5 \geq 2$, are not related to the graph or the shaded region and are incorrect. Final answer: $$x \leq 5, \quad y \geq 3, \quad y \leq \frac{3}{2}x - \frac{1}{2}$$