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(B) To determine the correct system of inequalities for the shaded region,we analyze the boundary lines shown in the figure:$1$. The line passing thro

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$x+2y \geq 6, 5x+3y \geq 15, x \leq 7, y \leq 6, x, y \geq 0$

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(B) To determine the correct system of inequalities for the shaded region,we analyze the boundary lines shown in the figure:$1$. The line passing through $(0, 3)$ and $(6, 0)$ has the equation $\frac{x}{6} + \frac{y}{3} = 1$,which simplifies to $x + 2y = 6$. Since the shaded region is above this line,the inequality is $x + 2y \geq 6$.$2$. The line passing through $(0, 5)$ and $(3, 0)$ has the equation $\frac{x}{3} + \frac{y}{5} = 1$,which simplifies to $5x + 3y = 15$. Since the shaded region is above this line,the inequality is $5x + 3y \geq 15$.$3$. The vertical line is $x = 7$. Since the shaded region is to the left of this line,the inequality is $x \leq 7$.$4$. The horizontal line is $y = 6$. Since the shaded region is below this line,the inequality is $y \leq 6$.$5$. The region is in the first quadrant,so $x \geq 0$ and $y \geq 0$.Combining these,the system is $x + 2y \geq 6, 5x + 3y \geq 15, x \leq 7, y \leq 6, x, y \geq 0$. This corresponds to option $(B)$.

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

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