Question

Select all solutions to the system of inequalities:

-5x-y≥15

4x-2y≤4

There are 2

a
(0,0)

b
(-5,0)

c
(-3,-1)

d
(-3,-4)

Answer 1

To determine which points satisfy the given system of inequalities, we will analyze them one by one.

1. **Inequality 1: -5x - y ≥ 15**
- We can rewrite it as \( y ≤ -5x - 15 \).

2. **Inequality 2: 4x - 2y ≤ 4**
- We can rewrite it as \( 2y ≥ 4x - 4 \), or \( y ≥ 2x - 2 \).

Now, we will test each of the given points (a, b, c, d) against both inequalities.

### Point a: (0, 0)

1. For -5x - y ≥ 15:
- \(-5(0) - 0 = 0\)
- \(0 ≥ 15\) (False)

2. For 4x - 2y ≤ 4:
- \(4(0) - 2(0) = 0\)
- \(0 ≤ 4\) (True)

**This point does not satisfy the first inequality.**

### Point b: (-5, 0)

1. For -5x - y ≥ 15:
- \(-5(-5) - 0 = 25\)
- \(25 ≥ 15\) (True)

2. For 4x - 2y ≤ 4:
- \(4(-5) - 2(0) = -20\)
- \(-20 ≤ 4\) (True)

**This point satisfies both inequalities.**

### Point c: (-3, -1)

1. For -5x - y ≥ 15:
- \(-5(-3) - (-1) = 15 + 1 = 16\)
- \(16 ≥ 15\) (True)

2. For 4x - 2y ≤ 4:
- \(4(-3) - 2(-1) = -12 + 2 = -10\)
- \(-10 ≤ 4\) (True)

**This point satisfies both inequalities.**

### Point d: (-3, -4)

1. For -5x - y ≥ 15:
- \(-5(-3) - (-4) = 15 + 4 = 19\)
- \(19 ≥ 15\) (True)

2. For 4x - 2y ≤ 4:
- \(4(-3) - 2(-4) = -12 + 8 = -4\)
- \(-4 ≤ 4\) (True)

**This point satisfies both inequalities.**

### Summary of points that satisfy the system of inequalities:

- Point b: (-5, 0) – ✅ Satisfies both
- Point c: (-3, -1) – ✅ Satisfies both
- Point d: (-3, -4) – ✅ Satisfies both

Only points **b (-5, 0)**, **c (-3, -1)**, and **d (-3, -4)** are solutions to the system.

Thus, the solution set includes points:

**(-5, 0), (-3, -1), and (-3, -4).**