1. 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 in the solution set for y≤6x−3, but on the line for y>x+5.
C) Yes, it is within the solution set for both inequalities.
D) No, it is not in the solution set for either inequality.

2. Which point is a solution to the simultaneous inequalities y < 1/2x and y > 1/3x + 3?
A) (25,12)
B) (2,2)
C) (10,5)
D) (3,6)

Answer 1

1. To determine if the point (3, 8) is a solution to the simultaneous inequalities y > x + 5 and y ≤ 6x − 3, we will evaluate each inequality separately.

• For the first inequality: \( y > x + 5 \) Substituting the point (3, 8): \( 8 > 3 + 5 \) \( 8 > 8 \) This is false (the point is not greater than, it is equal to).

• For the second inequality: \( y ≤ 6x - 3 \) Substituting the point (3, 8): \( 8 ≤ 6(3) - 3 \) \( 8 ≤ 18 - 3 \) \( 8 ≤ 15 \) This is true.

Since the point (3, 8) does not satisfy the first inequality but satisfies the second, the correct response is: B) No, it is in the solution set for y ≤ 6x − 3, but on the line for y > x + 5.

2. Now, we will evaluate which point is a solution to the simultaneous inequalities y < 1/2x and y > 1/3x + 3.

• A) For the point (25, 12):

  • \( 12 < \frac{1}{2}(25) \): \( 12 < 12.5 \) (true)
  • \( 12 > \frac{1}{3}(25) + 3 \): \( 12 > \frac{25}{3} + 3 \), which evaluates to \( 12 > 8.33 \) (true).

• B) For the point (2, 2):

  • \( 2 < \frac{1}{2}(2) \): \( 2 < 1 \) (false).

• C) For the point (10, 5):

  • \( 5 < \frac{1}{2}(10) \): \( 5 < 5 \) (false).

• D) For the point (3, 6):

  • \( 6 < \frac{1}{2}(3) \): \( 6 < 1.5 \) (false).

Out of all the points, only option A) (25, 12) satisfies both inequalities. The correct answer is A) (25,12).