Determine if the point (3,8) is a solution to the simultaneous inequalities y>x+5 and y≤6x−3 . Select the response that gives the correct answer as well as the correct reasoning.

Question

A: Yes, because both of these inequalities include points on the line as a solution

B: No, it is not in the solution set for either inequality

C: Yes it is within the solution set for both inequalities

D: No, it is in the solution set for y is less than or equal to 6x -3, but on the line for y > x + 5

Answer 1

To determine if the point (3,8) is a solution to the simultaneous inequalities $y>x+5$ and $y\le 6x-3$, we will evaluate whether the point satisfies both inequalities.

1. Evaluate $y>x+5$: $$y=8,x=38>3+58>8\text{(This is false since 8 is not greater than 8.)}$$ Therefore, the point (3,8) does not satisfy $y>x+5$.

2. Evaluate $y\le 6x-3$: $$y=8,x=38\le 6\cdot 3-38\le 18-38\le 15\text{(This is true.)}$$ Therefore, the point (3,8) satisfies $y\le 6x-3$.

Since the point (3,8) does not satisfy the first inequality $y>x+5$ but does satisfy the second inequality $y\le 6x-3$, the correct response is:

D: No, it is in the solution set for y ≤ 6x - 3, but on the line for y > x + 5.