Question

Given the system of inequalties:

x-y<3

-x-y≥4

Select ALL solutions

a
(-4,0)

b
(0,0)

c
(-5,-1)

d
(1,0)

Answer 1

To solve the system of inequalities, we will first rewrite them:

1. \( x - y < 3 \) can be rewritten as \( y > x - 3 \).
2. \( -x - y \geq 4 \) can be rewritten as \( y \leq -x - 4 \).

Now, we can analyze each of the provided options to see if they satisfy both inequalities.

**Option a: (-4, 0)**
- Check \( x - y < 3 \):
\[
-4 - 0 < 3 \quad \text{(True, since } -4 < 3\text{)}
\]
- Check \( -x - y \geq 4 \):
\[
-(-4) - 0 \geq 4 \implies 4 \geq 4 \quad \text{(True)}
\]
- (-4, 0) is a solution.

**Option b: (0, 0)**
- Check \( x - y < 3 \):
\[
0 - 0 < 3 \quad \text{(True, since } 0 < 3\text{)}
\]
- Check \( -x - y \geq 4 \):
\[
-0 - 0 \geq 4 \implies 0 \geq 4 \quad \text{(False)}
\]
- (0, 0) is not a solution.

**Option c: (-5, -1)**
- Check \( x - y < 3 \):
\[
-5 - (-1) < 3 \implies -5 + 1 < 3 \implies -4 < 3 \quad \text{(True)}
\]
- Check \( -x - y \geq 4 \):
\[
-(-5) - (-1) \geq 4 \implies 5 + 1 \geq 4 \implies 6 \geq 4 \quad \text{(True)}
\]
- (-5, -1) is a solution.

**Option d: (1, 0)**
- Check \( x - y < 3 \):
\[
1 - 0 < 3 \quad \text{(True, since } 1 < 3\text{)}
\]
- Check \( -x - y \geq 4 \):
\[
-1 - 0 \geq 4 \implies -1 \geq 4 \quad \text{(False)}
\]
- (1, 0) is not a solution.

**Final Answers:** The solutions that satisfy both inequalities are:
- a) (-4, 0)
- c) (-5, -1)

So the correct options are a and c.