Final answer:
To graph the system of inequalities, graph each inequality separately, determine the overlapped region, and determine if the system is infeasible, has alternate optimal solutions, or is unbounded.
Step-by-step explanation:
To graph the system of inequalities, we will first graph the individual inequalities and then determine the region that satisfies all of them.
- Graph the inequality 4y ≤ 21 by first rewriting it in slope-intercept form as y ≤ (21/4). Draw a dashed horizontal line at y = (21/4), and shade the region below the line to represent the inequality.
- Graph the inequality 6x + 5y ≤ 3. Rewrite it in slope-intercept form as y ≤ (-6/5)x + (3/5). Draw a dashed line with a slope of (-6/5) passing through the y-intercept at (0, 3/5). Shade the region below the line to represent the inequality.
- Graph the inequality 3x + y > 5. Rewrite it in slope-intercept form as y > -3x + 5. Draw a solid line with a slope of -3 passing through the y-intercept at (0, 5). Shade the region above the line to represent the inequality.
After graphing all three inequalities, the region where the shaded regions overlap represents the feasible solutions to the system of inequalities. If the overlapped region is empty, then the system is infeasible. If the overlapped region is a line segment or a point, then there are alternate optimal solutions. If the overlapped region extends infinitely, then the system is unbounded.