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.(1 point)

Question

Responses

No, it is in the solution set for y≤6x−3, but on the line for y>x+5.
No, it is in the solution set for y less-than-or-equal-to 6 x minus 3 , but on the line for y greater than x plus 5 .

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

Yes, it is within the solution set for both inequalities.
Yes, it is within the solution set for both inequalities.

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

Answer 1

To determine if the point (3, 8) is a solution to the simultaneous inequalities, we will substitute x = 3 and y = 8 into both inequalities and see if the point satisfies each one.

1. For the inequality \( y > x + 5 \): \[ 8 > 3 + 5 \implies 8 > 8 \] This inequality is not satisfied because 8 is not greater than 8.

2. For the inequality \( y \leq 6x - 3 \): \[ 8 \leq 6(3) - 3 \implies 8 \leq 18 - 3 \implies 8 \leq 15 \] This inequality is satisfied because 8 is indeed less than or equal to 15.

Now, let's summarize:

• The point (3, 8) does not satisfy the first inequality \( y > x + 5 \).
• The point (3, 8) does satisfy the second inequality \( y \leq 6x - 3 \).

Thus, the correct response is: No, it is in the solution set for \( y \leq 6x - 3 \), but on the line for \( y > x + 5 \).