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To determine the correct graph that represents the solution set for the given system of inequalities, let’s analyze each inequality separately and identify the region where both inequalities overlap.

### Step 1: Rewrite the inequalities in slope-intercept form
The system of inequalities is:
1. \( x + 2y < -2 \)
2. \( y - x < 3 \)

**Rewrite Inequality 1:**
\[
x + 2y < -2
\]
Subtract \( x \) from both sides:
\[
2y < -x - 2
\]
Divide by 2:
\[
y < -\frac{1}{2}x - 1
\]
This inequality represents a line with a slope of \(-\frac{1}{2}\) and a y-intercept of \(-1\). Since the inequality is \(<\), we shade the region below this line.

**Rewrite Inequality 2:**
\[
y - x < 3
\]
Add \( x \) to both sides:
\[
y < x + 3
\]
This inequality represents a line with a slope of \(1\) and a y-intercept of \(3\). Since the inequality is \(<\), we shade the region below this line.

### Step 2: Determine the Solution Region
The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap.

1. **For \(y < -\frac{1}{2}x - 1\)**: Shade below the line with slope \(-\frac{1}{2}\) and y-intercept \(-1\).
2. **For \(y < x + 3\)**: Shade below the line with slope \(1\) and y-intercept \(3\).

### Step 3: Identify the Correct Graph
In the provided options:
- Look for the graph where the shaded region is below both lines, intersecting where both conditions are true.
- Check the orientation of the lines:
  - The line with slope \(-\frac{1}{2}\) should go downwards to the right.
  - The line with slope \(1\) should go upwards to the right.

### Solution
After analyzing, the correct answer appears to be the **third graph** (from top to bottom), where both inequalities are satisfied and the correct overlapping shaded region is shown.

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Would you like a more detailed breakdown of each graph or have any questions?

### Related Questions:
1. How do you determine which side of a line to shade for inequalities?
2. What does it mean for a system of inequalities to have no solution?
3. How can you tell if a point satisfies both inequalities in a system?
4. How does the slope of a line affect the direction of shading in inequalities?
5. What steps can you take to graph inequalities without a calculator?

### Tip:
For any inequality in the form \( y < mx + b \), always shade below the line, and for \( y > mx + b \), shade above the line.

Which graph best represents the solution set for this system of inequalities?

x + 2y < -2
y - x < 3

Understand how to determine the solution set for the system of inequalities x + 2y < -2 and y - x < 3 by graphing. This problem involves graphing each inequality, identifying shading regions, and finding the intersection area where both inequalities hold true. Suitable for students in Grades 8-10.

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