Question

Given the system of inequalities, select all solutions that satisfy the system. There are 2

4x-4y>12

2x+2y≥6

a
(3,0)

b
(5,-2)

c
(6,0)

d
(5,2)

Answer 1

To determine which points satisfy the given system of inequalities, we will examine each point against both inequalities:

1. **Inequality 1:** \(4x - 4y > 12\)
- This can be rewritten as \(x - y > 3\) by dividing the entire inequality by 4.

2. **Inequality 2:** \(2x + 2y \geq 6\)
- This simplifies to \(x + y \geq 3\) by dividing the entire inequality by 2.

Now, we will evaluate each option:

### Option a: \( (3, 0) \)
- For \(4x - 4y > 12\):
\[
4(3) - 4(0) = 12 \quad \text{(not satisfied; we need > 12)}
\]
- For \(2x + 2y \geq 6\):
\[
2(3) + 2(0) = 6 \quad \text{(satisfied; we have \(\geq 6\))}
\]
- Not a solution.

### Option b: \( (5, -2) \)
- For \(4x - 4y > 12\):
\[
4(5) - 4(-2) = 20 + 8 = 28 \quad \text{(satisfied; we have > 12)}
\]
- For \(2x + 2y \geq 6\):
\[
2(5) + 2(-2) = 10 - 4 = 6 \quad \text{(satisfied; we have \(\geq 6\))}
\]
- This is a solution.

### Option c: \( (6, 0) \)
- For \(4x - 4y > 12\):
\[
4(6) - 4(0) = 24 \quad \text{(satisfied; we have > 12)}
\]
- For \(2x + 2y \geq 6\):
\[
2(6) + 2(0) = 12 \quad \text{(satisfied; we have \(\geq 6\))}
\]
- This is a solution.

### Option d: \( (5, 2) \)
- For \(4x - 4y > 12\):
\[
4(5) - 4(2) = 20 - 8 = 12 \quad \text{(not satisfied; we need > 12)}
\]
- For \(2x + 2y \geq 6\):
\[
2(5) + 2(2) = 10 + 4 = 14 \quad \text{(satisfied; we have \(\geq 6\))}
\]
- Not a solution.

### Summary:
From the evaluations above, the points that satisfy both inequalities are:
- **Option b:** \( (5, -2) \)
- **Option c:** \( (6, 0) \)

Thus, the answers are **b (5, -2)** and **c (6, 0)**.