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(D) To determine the linear constraints for the shaded region,we analyze each boundary line:$1$. The line $x+2y=6$ has the shaded region on the side a

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

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(D) To determine the linear constraints for the shaded region,we analyze each boundary line:$1$. The line $x+2y=6$ has the shaded region on the side away from the origin (since the point $(7, 6)$ satisfies $7+12=19 \geq 6$). Thus,the constraint is $x+2y \geq 6$.$2$. The line $5x+3y=15$ has the shaded region on the side away from the origin (since the point $(7, 6)$ satisfies $35+18=53 \geq 15$). Thus,the constraint is $5x+3y \geq 15$.$3$. The vertical line $x=7$ has the shaded region to its left,so $x \leq 7$.$4$. The horizontal line $y=6$ has the shaded region below it,so $y \leq 6$.$5$. The region is in the first quadrant,so $x, y \geq 0$.Combining these,the constraints are $x+2y \geq 6, 5x+3y \geq 15, x \leq 7, y \leq 6, x, y \geq 0$.

For the following shaded region,the linear constraints are:

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