1. **State the problem:** We are asked to graph the solution regions of three systems of linear inequalities. 2. **Part a:** The inequalities are: $$2x + y \leq 8$$ $$x + y \leq 6$$ $$x \geq 0$$ $$y \geq 0$$ - These inequalities represent half-planes bounded by lines: - Line 1: $2x + y = 8$ (intercepts: $x=4$, $y=8$) - Line 2: $x + y = 6$ (intercepts: $x=6$, $y=6$) - And the coordinate axes (due to $x,y \geq 0$). - The solution region is the intersection of these half-planes in the first quadrant. 3. **Part b:** The inequalities are: $$3x + 6y \geq 18$$ $$6x + 3y \leq 18$$ $$x \geq 0$$ $$y \geq 0$$ - Convert to standard line forms: - $3x + 6y = 18$ or $x + 2y = 6$ - $6x + 3y = 18$ or $2x + y = 6$ - Inequalities specify: - $x + 2y \geq 6$ - $2x + y \leq 6$ - Along with $x,y \geq 0$, the solution is the intersection of these regions. 4. **Part c:** The inequalities are: $$x + y \leq 6$$ $$2x + 3y \geq 12$$ $$x \geq 2$$ $$y \geq 0$$ - Lines: - $x + y = 6$ - $2x + 3y = 12$ - Vertical line $x = 2$ - The solution is the intersection of: - $x + y \leq 6$ - $2x + 3y \geq 12$ - $x \geq 2$, $y \geq 0$ **Summary:** Each region is bounded by the lines given and constrained to the first quadrant ($x,y \geq 0$). A precise graph requires plotting these lines and shading the regions that satisfy the inequalities. Because the user asked for graphs, here is the combined Desmos LaTeX commands for graphing the boundaries: