Question

Given the system of inequalities, select all solutions that apply.

x-y≤4

3x+6y<8

There are 3



a
(0,-5)

b
(0,-2)

c
(5,0)

d
(0,0)

Answer 1

To determine which of these points satisfy the given system of inequalities, we need to check each point against the inequalities:

1. \( x - y \leq 4 \)
2. \( 3x + 6y < 8 \)

Let's evaluate each point:

### Point a: (0, -5)
1. For \( x - y \):
\[
0 - (-5) \leq 4 \implies 0 + 5 \leq 4 \implies 5 \leq 4 \quad \text{(False)}
\]
2. For \( 3x + 6y \):
\[
3(0) + 6(-5) < 8 \implies 0 - 30 < 8 \implies -30 < 8 \quad \text{(True)}
\]

**Result**: Not a solution (fails the first inequality).

### Point b: (0, -2)
1. For \( x - y \):
\[
0 - (-2) \leq 4 \implies 0 + 2 \leq 4 \implies 2 \leq 4 \quad \text{(True)}
\]
2. For \( 3x + 6y \):
\[
3(0) + 6(-2) < 8 \implies 0 - 12 < 8 \implies -12 < 8 \quad \text{(True)}
\]

**Result**: This point is a solution.

### Point c: (5, 0)
1. For \( x - y \):
\[
5 - 0 \leq 4 \implies 5 \leq 4 \quad \text{(False)}
\]
2. For \( 3x + 6y \):
\[
3(5) + 6(0) < 8 \implies 15 + 0 < 8 \implies 15 < 8 \quad \text{(False)}
\]

**Result**: Not a solution (fails both inequalities).

### Point d: (0, 0)
1. For \( x - y \):
\[
0 - 0 \leq 4 \implies 0 \leq 4 \quad \text{(True)}
\]
2. For \( 3x + 6y \):
\[
3(0) + 6(0) < 8 \implies 0 < 8 \quad \text{(True)}
\]

**Result**: This point is a solution.

### Summary of Results

- Point a: Not a solution
- Point b: Solution
- Point c: Not a solution
- Point d: Solution

Thus, the points that satisfy the system of inequalities are **b (0, -2)** and **d (0, 0)**.