Question

Given 4x-y>12 and 2x+4y<2 select all solutions that satisfy the system of inequalities:

There are 2

a
(5,-3)

b
(5,-2)

c
(3,-1)

d
(4,-5)

Answer 1

To determine which points satisfy the system of inequalities given by:

1. \( 4x - y > 12 \)
2. \( 2x + 4y < 2 \)

We will test each of the provided points: (5, -3), (5, -2), (3, -1), and (4, -5).

### Checking point (5, -3):

1. **For \( 4x - y > 12 \)**:
\[
4(5) - (-3) = 20 + 3 = 23 > 12 \quad \text{(True)}
\]

2. **For \( 2x + 4y < 2 \)**:
\[
2(5) + 4(-3) = 10 - 12 = -2 < 2 \quad \text{(True)}
\]

**Result**: (5, -3) satisfies both inequalities.

### Checking point (5, -2):

1. **For \( 4x - y > 12 \)**:
\[
4(5) - (-2) = 20 + 2 = 22 > 12 \quad \text{(True)}
\]

2. **For \( 2x + 4y < 2 \)**:
\[
2(5) + 4(-2) = 10 - 8 = 2 < 2 \quad \text{(False)}
\]

**Result**: (5, -2) does not satisfy the second inequality.

### Checking point (3, -1):

1. **For \( 4x - y > 12 \)**:
\[
4(3) - (-1) = 12 + 1 = 13 > 12 \quad \text{(True)}
\]

2. **For \( 2x + 4y < 2 \)**:
\[
2(3) + 4(-1) = 6 - 4 = 2 < 2 \quad \text{(False)}
\]

**Result**: (3, -1) does not satisfy the second inequality.

### Checking point (4, -5):

1. **For \( 4x - y > 12 \)**:
\[
4(4) - (-5) = 16 + 5 = 21 > 12 \quad \text{(True)}
\]

2. **For \( 2x + 4y < 2 \)**:
\[
2(4) + 4(-5) = 8 - 20 = -12 < 2 \quad \text{(True)}
\]

**Result**: (4, -5) satisfies both inequalities.

### Summary of Results

- (5, -3): Satisfies both inequalities ✅
- (5, -2): Does not satisfy the second inequality ❌
- (3, -1): Does not satisfy the second inequality ❌
- (4, -5): Satisfies both inequalities ✅

Thus, the solutions that satisfy the system of inequalities are:

- **a** (5, -3)
- **d** (4, -5)

The correct answer is **a and d**.