Question

Select all of the ordered pairs that are solutions to the following system of inequalities.
2x+2y≥7
y<3x–5

The ordered pairs:
a. (3,4)
b. (2, -5)
c. (9, -3)
d. (7, 0)

Answer 1

To determine which ordered pairs are solutions to the system of inequalities \(2x + 2y \geq 7\) and \(y < 3x - 5\), we will test each pair one by one.

1. **For the pair (3, 4):**
- Check \(2x + 2y \geq 7\):
\[
2(3) + 2(4) = 6 + 8 = 14 \geq 7 \quad \text{(True)}
\]
- Check \(y < 3x - 5\):
\[
4 < 3(3) - 5 \implies 4 < 9 - 5 \implies 4 < 4 \quad \text{(False)}
\]
- Conclusion: (3, 4) is **not a solution**.

2. **For the pair (2, -5):**
- Check \(2x + 2y \geq 7\):
\[
2(2) + 2(-5) = 4 - 10 = -6 \geq 7 \quad \text{(False)}
\]
- Conclusion: (2, -5) is **not a solution**.

3. **For the pair (9, -3):**
- Check \(2x + 2y \geq 7\):
\[
2(9) + 2(-3) = 18 - 6 = 12 \geq 7 \quad \text{(True)}
\]
- Check \(y < 3x - 5\):
\[
-3 < 3(9) - 5 \implies -3 < 27 - 5 \implies -3 < 22 \quad \text{(True)}
\]
- Conclusion: (9, -3) is a **solution**.

4. **For the pair (7, 0):**
- Check \(2x + 2y \geq 7\):
\[
2(7) + 2(0) = 14 + 0 = 14 \geq 7 \quad \text{(True)}
\]
- Check \(y < 3x - 5\):
\[
0 < 3(7) - 5 \implies 0 < 21 - 5 \implies 0 < 16 \quad \text{(True)}
\]
- Conclusion: (7, 0) is a **solution**.

**Final Results**
The ordered pairs that are solutions to the system of inequalities are:
- c. (9, -3)
- d. (7, 0)