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To solve the given question, we need to graphically represent the system of linear inequalities and determine the correct graph that satisfies both inequalities.

### System of inequalities:
1. \( x + y < 2 \)
2. \( x + 3y \geq 0 \)

#### Step 1: Analyze each inequality
1. **For \( x + y < 2 \):**
   - Rewrite as \( y < -x + 2 \).
   - The boundary line is \( y = -x + 2 \), which is a straight line with slope \(-1\) and y-intercept \(2\). 
   - The inequality \( y < -x + 2 \) means we shade the region below the line \( y = -x + 2 \).

2. **For \( x + 3y \geq 0 \):**
   - Rewrite as \( y \geq -\frac{1}{3}x \).
   - The boundary line is \( y = -\frac{1}{3}x \), which is a straight line with slope \(-\frac{1}{3}\) and passes through the origin.
   - The inequality \( y \geq -\frac{1}{3}x \) means we shade the region above or on the line \( y = -\frac{1}{3}x \).

#### Step 2: Identify the overlapping region
To satisfy both inequalities:
- The solution lies **below the line** \( y = -x + 2 \) and **above or on the line** \( y = -\frac{1}{3}x \).

#### Step 3: Match with the options
Looking at the provided graphs:
- **Option (a)** correctly shows the intersection of the shaded regions below \( y = -x + 2 \) and above \( y = -\frac{1}{3}x \).

### Correct Answer:
**Option (a)**

Would you like a detailed explanation of how to graph each inequality or any further clarifications?  

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### Related Questions:
1. How do you find the intersection points of two linear inequalities?
2. What is the difference between a strict inequality (\(<, >\)) and a non-strict inequality (\( \leq, \geq \)) in graphing?
3. How do slopes of the lines influence the orientation of the shaded regions?
4. How can you verify which region satisfies an inequality using a test point?
5. How are boundary lines represented differently for strict and non-strict inequalities?

### Tip:
Always test a point (like \( (0,0) \), unless it lies on a boundary) to confirm the correct region for shading in an inequality.

Graph the system of linear inequalities. x + y < 2 and x + 3y ≥ 0

Learn how to graph the system of linear inequalities x + y < 2 and x + 3y ≥ 0. This problem involves graphing the boundary lines and shading the appropriate regions to identify the solution set. Suitable for high school students studying Algebra and Inequalities.

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