If feasible region is as shown in the figure, then the related inequalities are}
Show Hint
Logic Tip: You don't always need to derive the equations from scratch. Pick a point clearly inside the shaded region from the graph (like $(1.5, 2.5)$ or $(2,3)$) and plug it directly into the options. The correct option will satisfy ALL inequalities.
Concept:
To determine the system of inequalities from a feasible region graph, identify the equations of the boundary lines and then test a point inside the shaded region to determine the correct inequality sign ($\le$ or $\ge$) for each line.
Step 1: Determine the equation and inequality for the first line.
From the visual context (and standard LP problems of this type), one boundary line has $x$-intercept $(4,0)$ and $y$-intercept $(0,3)$.
Using the intercept form $\frac{x}{a} + \frac{y}{b} = 1$:
$$\frac{x}{4} + \frac{y}{3} = 1$$
Multiply by 12:
$$3x + 4y = 12$$
The shaded region lies above/away from the origin with respect to this line. Testing a point in the region (e.g., $x=1, y=3$) gives $3(1)+4(3) = 15 \ge 12$.
Thus, the inequality is:
$$3x + 4y \ge 12$$
Step 2: Determine the equation and inequality for the second line.
Another boundary line passes through the origin $(0,0)$ and points like $(3,3)$. This is the line $y = x$, or $y - x = 0$.
The shaded region lies above this line. Testing a point in the region (e.g., $x=1, y=3$) gives $3 - 1 = 2 \ge 0$.
Thus, the inequality is:
$$y - x \ge 0$$
Step 3: Determine the constraints for the horizontal line and axes.
The top boundary of the region is a horizontal line passing through $y=3$. The region lies below it.
Thus, the inequality is:
$$y \le 3$$
The region is entirely within the first quadrant, which gives the non-negativity constraints:
$$x \ge 0, y \ge 0$$
Combining all conditions yields option (A).
