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(D) $1$. The shaded region lies in the first quadrant, so the non-negativity constraints are $x \geq 0$ and $y \geq 0$.$2$. The region is bounded by t

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$2x + y \leq 12, x + 2y \leq 12, x + 1.25y \geq 5, x \geq 0, y \geq 0$

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(D) $1$. The shaded region lies in the first quadrant, so the non-negativity constraints are $x \geq 0$ and $y \geq 0$.$2$. The region is bounded by three lines:$(i)$ The line passing through $(0, 12)$ and $(6, 0)$ has the equation $\frac{x}{6} + \frac{y}{12} = 1$, which simplifies to $2x + y = 12$. Since the shaded region is towards the origin, the constraint is $2x + y \leq 12$.(ii) The line passing through $(0, 6)$ and $(12, 0)$ has the equation $\frac{x}{12} + \frac{y}{6} = 1$, which simplifies to $x + 2y = 12$. Since the shaded region is towards the origin, the constraint is $x + 2y \leq 12$.(iii) The line passing through $(0, 4)$ and $(5, 0)$ has the equation $\frac{x}{5} + \frac{y}{4} = 1$, which simplifies to $4x + 5y = 20$, or $x + 1.25y = 5$. Since the shaded region is away from the origin, the constraint is $x + 1.25y \geq 5$.$3$. Combining these, the system of constraints is $2x + y \leq 12, x + 2y \leq 12, x + 1.25y \geq 5, x \geq 0, y \geq 0$. This matches option $D$.

In the following figure, the shaded region represents the system of constraints:

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