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(A) To determine the system of linear inequalities,we identify the boundary lines of the shaded region:$1$. The line passing through $(0, 25)$ and $(5

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$x+2y \geq 50, 2x+y \leq 100, 2x-y \leq 0, x, y \geq 0$

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(A) To determine the system of linear inequalities,we identify the boundary lines of the shaded region:$1$. The line passing through $(0, 25)$ and $(50, 0)$ has the equation $\frac{x}{50} + \frac{y}{25} = 1$,which simplifies to $x + 2y = 50$. Since the shaded region is above this line,the inequality is $x + 2y \geq 50$.$2$. The line passing through $(0, 100)$ and $(50, 0)$ has the equation $\frac{x}{50} + \frac{y}{100} = 1$,which simplifies to $2x + y = 100$. Since the shaded region is below this line,the inequality is $2x + y \leq 100$.$3$. The line passing through $(0, 0)$ and $(10, 20)$ has the equation $y = 2x$,which simplifies to $2x - y = 0$. Since the shaded region is to the left of this line (e.g.,test point $(0, 10)$ gives $2(0) - 10 = -10 \leq 0$),the inequality is $2x - y \leq 0$.$4$. The region is in the first quadrant,so $x, y \geq 0$.Thus,the correct set of inequalities is $x + 2y \geq 50, 2x + y \leq 100, 2x - y \leq 0, x, y \geq 0$.

The shaded region in the following figure is the solution set of the inequations:

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