To graph the system of inequalities provided, we will first graph each inequality separately and then identify the overlapping region where all inequalities intersect. Here's how you can graph the system of inequalities:
1. Graph the inequality x₁ + x₂ ≤ 40:
- To graph x₁ + x₂ ≤ 40, we first plot the line x₁ + x₂ = 40. To do this, find the x and y-intercepts by setting x₁ = 0 and solving for x₂, and then setting x₂ = 0 and solving for x₁. Connect these points to draw the line.
- Since the inequality is ≤ (less than or equal to), the region below or on the line x₁ + x₂ = 40 is shaded. This region represents all the points that satisfy the inequality.
2. Graph the inequality 3x₁ + x₂ ≤ 84:
- For 3x₁ + x₂ ≤ 84, plot the line 3x₁ + x₂ = 84 using a similar method as above to find the intercepts and draw the line.
- Shade the region below or on the line 3x₁ + x₂ = 84 since it is also ≤ (less than or equal to) inequality.
3. Identify the overlapping region:
- The overlapping region of the shaded areas from both inequalities is the solution to the system of inequalities x₁ + x₂ ≤ 40 and 3x₁ + x₂ ≤ 84.
- This overlapping region will be the area where the shadings from both inequalities intersect. The points within this region satisfy both inequalities simultaneously.
4. Include the constraint x₁, x₂ ≥ 0:
- Since x₁ and x₂ must be greater than or equal to 0, the final solution will be limited to the first quadrant (where both x and y are positive).
By following these steps and graphing each inequality separately, then identifying the overlapping region that satisfies all inequalities while considering the constraint x₁, x₂ ≥ 0, you can accurately represent the system of inequalities graphically.
