Question

Given the system of inequalities:

2x+y>2

2x+2y≤6

Select all solutions

a
(2,0)

b
(3,0)

c
(6,0)

d
(0,0)

e
(0,4)

Answer 1

To determine which of the options are solutions to the system of inequalities, we will test each pair (x, y) against the inequalities:

1. \(2x + y > 2\)
2. \(2x + 2y \leq 6\)

Let's evaluate each option:

**a. (2, 0)**
- For \(2x + y > 2\):
\(2(2) + 0 > 2\)
\(4 > 2\) → True

- For \(2x + 2y \leq 6\):
\(2(2) + 2(0) \leq 6\)
\(4 \leq 6\) → True

**Conclusion for (2, 0): True**

---

**b. (3, 0)**
- For \(2x + y > 2\):
\(2(3) + 0 > 2\)
\(6 > 2\) → True

- For \(2x + 2y \leq 6\):
\(2(3) + 2(0) \leq 6\)
\(6 \leq 6\) → True

**Conclusion for (3, 0): True**

---

**c. (6, 0)**
- For \(2x + y > 2\):
\(2(6) + 0 > 2\)
\(12 > 2\) → True

- For \(2x + 2y \leq 6\):
\(2(6) + 2(0) \leq 6\)
\(12 \leq 6\) → False

**Conclusion for (6, 0): False**

---

**d. (0, 0)**
- For \(2x + y > 2\):
\(2(0) + 0 > 2\)
\(0 > 2\) → False

- For \(2x + 2y \leq 6\):
\(2(0) + 2(0) \leq 6\)
\(0 \leq 6\) → True

**Conclusion for (0, 0): False**

---

**e. (0, 4)**
- For \(2x + y > 2\):
\(2(0) + 4 > 2\)
\(4 > 2\) → True

- For \(2x + 2y \leq 6\):
\(2(0) + 2(4) \leq 6\)
\(8 \leq 6\) → False

**Conclusion for (0, 4): False**

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### Final results:
- Solutions that satisfy both inequalities are:
- a: (2, 0) - **True**
- b: (3, 0) - **True**
- c: (6, 0) - **False**
- d: (0, 0) - **False**
- e: (0, 4) - **False**

**The solutions to the system of inequalities are a (2, 0) and b (3, 0).**