Here is the sketch of the feasible region for the given system of inequalities. The shaded region represents the area that satisfies all the inequalities simultaneously:

1. $5x + 2y \geq 10$ (blue line)
2. $x + 3y \geq 6$ (green line)
3. $3x + 4y \geq 12$ (red line)
4. $x \geq 0$ (right of the y-axis)
5. $y \geq 0$ (above the x-axis)

The feasible region is the gray area where all the conditions are met.

Would you like more details or have any questions?

Here are 5 related questions you might want to explore:

1. What are the vertices of the feasible region?
2. How can we calculate the area of the feasible region?
3. Can we determine the maximum or minimum value of a function within this region?
4. How does changing the inequality signs affect the feasible region?
5. What are some real-world applications of solving such systems?

Tip: When solving systems of inequalities, graphing is a powerful visual tool to understand constraints and optimize solutions.