Final answer:
To graph the feasible region for the given linear programming problem, plot the boundary lines for the inequalities, determine which side of the boundaries satisfies the inequalities, and shade the appropriate regions. Finally, show the feasible region where all the shaded areas overlap in the first quadrant where a and b are non-negative.
Step-by-step explanation:
To graph the feasible region for the problem, we first plot each inequality on a coordinate plane. We start by identifying the boundary lines for the inequalities when the inequalities are turned into equalities: -4a + 3b = 6 and a - b = 4. These lines divide the plane into distinct regions.
The next step is to determine which side of each boundary line satisfies the inequality. We do this by choosing a test point not on the line and substituting it into the inequality. For example, choosing a test point like (0,0) will show that for the first inequality -4(0) + 3(0) ≤ 6, which simplifies to 0 ≤ 6, a true statement, so we shade the region including (0,0). For the second inequality 1(0) - 1(0) ≤ 4, which simplifies to 0 ≤ 4, also true, so we shade the region again including (0,0).
Lastly, since a and b are non-negative, we are limited to the first quadrant of the coordinate plane (where both a and b are greater than or equal to zero). The feasible region is where all shaded areas overlap in the first quadrant. We then identify the vertices of the feasible region.
Graph the lines, perform shading appropriately for each inequality, and finally show the feasible region as the area of overlap in the first quadrant.