Solving the Inequalities 6x y>=18, x 4y>=12, 2x y>=10

Step 1: Simplify the Inequalities
To solve the inequalities, we need to simplify them first. We can do this by dividing both sides of the inequality by the coefficient of the variable.

- 6x y>=18 becomes x y>=3
- x 4y>=12 becomes x y>=3
- 2x y>=10 becomes x y>=5

Step 2: Plot the Inequalities on a Graph
We can plot each inequality on a graph by finding the x and y intercepts and drawing a line that passes through them. Then, we need to determine which side of the line is the solution set.

- x y>=3: Plot the line y=3/x. The solution set is above the line because any point above the line will have a y value greater than 3/x.
- x 4y>=12: Plot the line y=x/4+3. The solution set is below the line because any point below the line will have a y value less than x/4+3.
- x y>=5: Plot the line y=5/x. The solution set is above the line because any point above the line will have a y value greater than 5/x.

Step 3: Determine the Intersection of the Solution Sets
To find the solution set of the system of inequalities, we need to determine the intersection of the solution sets of each inequality.

- The solution set for x y>=3 is above the line y=3/x.
- The solution set for x 4y>=12 is below the line y=x/4+3.
- The solution set for x y>=5 is above the line y=5/x.

The only region that satisfies all three inequalities is the area above the line y=3/x and below the line y=x/4+3. Therefore, the solution set is the shaded region between these two lines.

Conclusion
The solution set of the system of inequalities 6x y>=18, x 4y>=12, 2x y>=10 is the shaded region between the lines y=3/x and y=x/4+3.