Question

Given the system of inequalities:

x+y<3

-x+y≥4

select ALL solutions

a
(-5,0)

b
(-10,1)

c
(-2,5)

d
(0,3)

Answer 1

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

1. \( x + y < 3 \)
2. \( -x + y \geq 4 \)

**a. Point (-5, 0)**
- For \( x + y < 3 \):
\(-5 + 0 < 3 \) ⟹ \(-5 < 3\) → **True**

- For \(-x + y \geq 4\):
\(-(-5) + 0 \geq 4\) ⟹ \(5 \geq 4\) → **True**

**Solution Point (-5, 0) is valid.**

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**b. Point (-10, 1)**
- For \( x + y < 3 \):
\(-10 + 1 < 3\) ⟹ \(-9 < 3\) → **True**

- For \(-x + y \geq 4\):
\(-(-10) + 1 \geq 4\) ⟹ \(10 + 1 \geq 4\) ⟹ \(11 \geq 4\) → **True**

**Solution Point (-10, 1) is valid.**

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**c. Point (-2, 5)**
- For \( x + y < 3 \):
\(-2 + 5 < 3\) ⟹ \(3 < 3\) → **False**

- For \(-x + y \geq 4\):
\(-(-2) + 5 \geq 4\) ⟹ \(2 + 5 \geq 4\) ⟹ \(7 \geq 4\) → **True**

**Solution Point (-2, 5) is NOT valid** (because it fails the first inequality).

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**d. Point (0, 3)**
- For \( x + y < 3 \):
\(0 + 3 < 3\) ⟹ \(3 < 3\) → **False**

- For \(-x + y \geq 4\):
\(-0 + 3 \geq 4\) ⟹ \(3 \geq 4\) → **False**

**Solution Point (0, 3) is NOT valid.**

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### Summary of Valid Solutions:
The solutions that satisfy both inequalities are:
- **a. (-5, 0)**
- **b. (-10, 1)**

Thus, the correct answers are:
**a and b.**