How many points having integer coordinates ar | vedclass

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(B) The feasible region is defined by the inequalities $3x + 4y \leq 12$,$x \geq 0$,and $y \geq 1$.We need to find integer pairs $(x, y)$ that satisfy

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$6$

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(B) The feasible region is defined by the inequalities $3x + 4y \leq 12$,$x \geq 0$,and $y \geq 1$.We need to find integer pairs $(x, y)$ that satisfy these conditions.Since $y \geq 1$,we test integer values for $y$:$1$. If $y = 1$: The inequality becomes $3x + 4(1) \leq 12$,which simplifies to $3x \leq 8$,so $x \leq 2.66$. Since $x \geq 0$,the possible integer values for $x$ are $0, 1, 2$. The points are $(0, 1), (1, 1), (2, 1)$.$2$. If $y = 2$: The inequality becomes $3x + 4(2) \leq 12$,which simplifies to $3x \leq 4$,so $x \leq 1.33$. Since $x \geq 0$,the possible integer values for $x$ are $0, 1$. The points are $(0, 2), (1, 2)$.$3$. If $y = 3$: The inequality becomes $3x + 4(3) \leq 12$,which simplifies to $3x \leq 0$,so $x \leq 0$. Since $x \geq 0$,the only integer value for $x$ is $0$. The point is $(0, 3)$.$4$. If $y > 3$: For $y = 4$,$3x + 16 \leq 12$ implies $3x \leq -4$,which has no non-negative integer solution for $x$.Summing these up,the points are $(0, 1), (1, 1), (2, 1), (0, 2), (1, 2), (0, 3)$.There are $6$ such points.

How many points having integer coordinates are there in the feasible region of the inequalities $3x + 4y \leq 12$,$x \geq 0$,and $y \geq 1$?

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