Here is the graph of the system of linear inequalities:

1. $4x + y \leq 8$ (blue line is the boundary)
2. $x \geq -4$ (red dashed line is the boundary)
3. $y \leq 3$ (purple dashed line is the boundary)

The shaded green region represents the solution to the system where all three inequalities are satisfied.

Do you have any questions or would you like more details on any part of the solution?

Here are some related questions:

1. How would the solution region change if the inequality $4x + y \leq 8$ became $4x + y \geq 8$?
2. What happens if you replace $x \geq -4$ with $x \leq -4$?
3. Can you solve this system algebraically rather than graphically?
4. What is the intersection point of the boundary lines?
5. How can we test whether a point lies in the solution region?

Tip: When solving systems of inequalities, always start by graphing each boundary line and shading the region where the inequality holds.