Question

Given the system of inequalities:

2x-2y>2

2x+2y≤6

Select all solutions

a
(0,0)

b
(2,3)

c
(2,0)

d
(2,-2)

Answer 1

To determine which points are solutions to the given system of inequalities, we'll analyze each point against the two inequalities:

1. **Inequalities:**
- \(2x - 2y > 2\) (or simplified, \(x - y > 1\))
- \(2x + 2y \leq 6\) (or simplified, \(x + y \leq 3\))

2. **Evaluating each option:**

**Point (0, 0):**
- For \(2x - 2y > 2\):
\[2(0) - 2(0) > 2 \implies 0 > 2\] (False)
- For \(2x + 2y \leq 6\):
\[2(0) + 2(0) \leq 6 \implies 0 \leq 6\] (True)

This point is not a solution, since it fails the first inequality.

**Point (2, 3):**
- For \(2x - 2y > 2\):
\[2(2) - 2(3) > 2 \implies 4 - 6 > 2 \implies -2 > 2\] (False)
- For \(2x + 2y \leq 6\):
\[2(2) + 2(3) \leq 6 \implies 4 + 6 \leq 6 \implies 10 \leq 6\] (False)

This point is not a solution.

**Point (2, 0):**
- For \(2x - 2y > 2\):
\[2(2) - 2(0) > 2 \implies 4 - 0 > 2 \implies 4 > 2\] (True)
- For \(2x + 2y \leq 6\):
\[2(2) + 2(0) \leq 6 \implies 4 + 0 \leq 6 \implies 4 \leq 6\] (True)

This point is a solution.

**Point (2, -2):**
- For \(2x - 2y > 2\):
\[2(2) - 2(-2) > 2 \implies 4 + 4 > 2 \implies 8 > 2\] (True)
- For \(2x + 2y \leq 6\):
\[2(2) + 2(-2) \leq 6 \implies 4 - 4 \leq 6 \implies 0 \leq 6\] (True)

This point is also a solution.

3. **Summary of results:**
- (0, 0): Not a solution
- (2, 3): Not a solution
- (2, 0): Solution
- (2, -2): Solution

**Final answer:**
The solutions are **c (2, 0)** and **d (2, -2)**.