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, it is within the solution set for both inequalities
b No, it is in the solution set for y < 6x - 3, but on the line for y > x + 5
c Yes, because both of these inequalities include points on the line as a solution
d No, it is not in the solution set for either inequality

Answer 1

To determine if the point (3, 8) is a solution to the simultaneous inequalities \( y > x + 5 \) and \( y < 6x - 3 \), we need to evaluate both inequalities with \( x = 3 \) and \( y = 8 \).

1. For the first inequality \( y > x + 5 \):

   • Substitute \( x = 3 \): \[ y > 3 + 5 \] \[ y > 8 \] Here, \( 8 > 8 \) is false since it does not satisfy the strict inequality.

2. For the second inequality \( y < 6x - 3 \):

   • Substitute \( x = 3 \): \[ y < 6 \cdot 3 - 3 \] \[ y < 18 - 3 \] \[ y < 15 \] Here, \( 8 < 15 \) is true.

Since the point (3, 8) does not satisfy the first inequality but does satisfy the second, it is not a solution to the simultaneous inequalities.

Thus, the correct response is:

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