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We are given the system of inequalities:

1. \( x + y < 3 \)
2. \( 2x - y \geq -6 \)

We need to check which of the points \( A, B, C, D, E, F, G \) satisfy both inequalities.

Let’s break it down:

### For each point:
We need to plug in the coordinates of each point into the inequalities and check if they satisfy both conditions.

### Points:
- \( A (4, -4) \)
- \( B (2, 6) \)
- \( C (-2, 6) \)
- \( D (-4, 0) \)
- \( E (4, 2) \)
- \( F (6, -8) \)
- \( G (-6, -4) \)

#### 1. Inequality: \( x + y < 3 \)
#### 2. Inequality: \( 2x - y \geq -6 \)

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### Checking for each point:

#### 1. Point A (4, -4):
- \( x + y = 4 + (-4) = 0 \) → Satisfies \( x + y < 3 \)
- \( 2x - y = 2(4) - (-4) = 8 + 4 = 12 \) → Satisfies \( 2x - y \geq -6 \)
  
**Point A satisfies both inequalities.**

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#### 2. Point B (2, 6):
- \( x + y = 2 + 6 = 8 \) → Does not satisfy \( x + y < 3 \)

**Point B does not satisfy the first inequality.**

---

#### 3. Point C (-2, 6):
- \( x + y = -2 + 6 = 4 \) → Does not satisfy \( x + y < 3 \)

**Point C does not satisfy the first inequality.**

---

#### 4. Point D (-4, 0):
- \( x + y = -4 + 0 = -4 \) → Satisfies \( x + y < 3 \)
- \( 2x - y = 2(-4) - 0 = -8 \) → Does not satisfy \( 2x - y \geq -6 \)

**Point D does not satisfy the second inequality.**

---

#### 5. Point E (4, 2):
- \( x + y = 4 + 2 = 6 \) → Does not satisfy \( x + y < 3 \)

**Point E does not satisfy the first inequality.**

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#### 6. Point F (6, -8):
- \( x + y = 6 + (-8) = -2 \) → Satisfies \( x + y < 3 \)
- \( 2x - y = 2(6) - (-8) = 12 + 8 = 20 \) → Satisfies \( 2x - y \geq -6 \)

**Point F satisfies both inequalities.**

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#### 7. Point G (-6, -4):
- \( x + y = -6 + (-4) = -10 \) → Satisfies \( x + y < 3 \)
- \( 2x - y = 2(-6) - (-4) = -12 + 4 = -8 \) → Does not satisfy \( 2x - y \geq -6 \)

**Point G does not satisfy the second inequality.**

---

### Conclusion:
The points that satisfy both inequalities are **A** and **F**.

The correct answer is **B. A, F, G**.

Would you like further details or clarifications?

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Here are 5 questions for further practice:
1. How do you graph a system of inequalities?
2. What is the difference between a solution to an inequality and an equation?
3. How would the solution set change if one of the inequalities was non-strict (e.g., \( x + y \leq 3 \))?
4. Can a point be a solution to one inequality but not to the system as a whole?
5. How do you test whether a point is on the boundary of the solution set?

**Tip:** When solving systems of inequalities graphically, always check which regions satisfy both inequalities by shading the appropriate areas for each.

Given the system of inequalities: x + y < 3, 2x - y > -6, which of the following points are solutions to this system of inequalities?

Analyze the system of inequalities x + y < 3 and 2x - y > -6, identifying which points satisfy both conditions. This algebra problem requires understanding of inequalities and coordinate geometry. Suitable for Grades 8-10.

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