To find the solution set of the inequalities 2x + y ≥ 4 and 3x + y ≤ 9, we need to graph these inequalities and determine the region that satisfies both conditions.
1. **Graphing the first inequality (2x + y ≥ 4)**:
- Convert to equality: y = -2x + 4 (this is a line with a negative slope).
- Draw the line on the graph. Since it is ≥, the line will be solid.
- Test a point (like (0,0)): 2(0) + 0 ≥ 4 (false), so the region above the line is included in the solution set.
2. **Graphing the second inequality (3x + y ≤ 9)**:
- Convert to equality: y = -3x + 9 (this is another line with a steeper negative slope).
- Draw this line on the graph. Since it is ≤, the line will be solid.
- Test the same point (0,0): 3(0) + 0 ≤ 9 (true), so the region below the line is included in the solution set.
3. **Finding the intersection region**:
- The solution set is where the region above the first line and below the second line intersect.
- Check points in the overlap to ensure they satisfy both inequalities.
Thus, the solution set includes points (x, y) that satisfy both inequalities simultaneously. This region can be shaded on the graph where the two areas overlap.
In conclusion, the solution set of the given system is represented graphically as the area between the two lines, including the lines themselves where applicable. To express specific solutions, we can also find points that lie within this region.