Question

Determine whether the points (-3,-2) and (3,2) are in the solution set of the system of inequalities below.

y ≤ (1/2) x + 2
y ≤ -2x - 3
a) Neither of the points is in the solution set.
b) Both points are in the solution set.
c) The point (-3,-2) is not in the solution set, and the point (3,2) is in the solution set.
d) The point (-3,-2) is in the solution set, and the point (3,2) is not in the solution set.

Answer 1

To determine whether the points (-3,-2) and (3,2) are in the solution set of the system of inequalities, we substitute the x and y values of each point into the inequalities and check if the inequalities hold true. The point (-3,-2) is in the solution set, and the point (3,2) is not in the solution set. The correct answer is d) The point (-3,-2) is in the solution set, and the point (3,2) is not in the solution set.

Step-by-step explanation:

To determine whether the points (-3,-2) and (3,2) are in the solution set of the system of inequalities, we substitute the x and y values of each point into the inequalities and check if the inequalities hold true.

For the first inequality y ≤ (1/2)x + 2:

• Substituting (-3,-2) gives: -2 ≤ (1/2)(-3) + 2 => -2 ≤ -1.5 + 2 => -2 ≤ 0.5. This is true.
• Substituting (3,2) gives: 2 ≤ (1/2)(3) + 2 => 2 ≤ 1.5 + 2 => 2 ≤ 3.5. This is true.

For the second inequality y ≤ -2x - 3:

• Substituting (-3,-2) gives: -2 ≤ -2(-3) - 3 => -2 ≤ 6 - 3 => -2 ≤ 3. This is true.
• Substituting (3,2) gives: 2 ≤ -2(3) - 3 => 2 ≤ -6 - 3 => 2 ≤ -9. This is false.

Therefore, the point (-3,-2) is in the solution set, and the point (3,2) is not in the solution set. The correct answer is d) The point (-3,-2) is in the solution set, and the point (3,2) is not in the solution set.