The inequality \( y \leq 2x + 1 \) represents the region of the coordinate plane that includes all points \((x, y)\) where \(y\) is less than or equal to the line defined by \( y = 2x + 1 \). Let’s analyze this inequality step-by-step:

### 1. **The Line \( y = 2x + 1 \):**
   - The equation \( y = 2x + 1 \) is a straight line with:
     - **Slope:** \( 2 \) (rise/run = 2/1, meaning it increases by 2 units vertically for every 1 unit horizontally).
     - **y-Intercept:** \( (0, 1) \), where the line crosses the y-axis.
   - This line divides the plane into two regions.

### 2. **The Inequality \( y \leq 2x + 1 \):**
   - The \( \leq \) symbol means the solution includes the line \( y = 2x + 1 \) itself (solid boundary) and all points **below or on the line**.

### 3. **Graphing the Inequality:**
   - Draw the line \( y = 2x + 1 \).
   - Shade the region **below the line**, as the inequality includes all \( y \)-values less than or equal to \( 2x + 1 \).
   - Use test points, e.g., \((0, 0)\):
     - Substitute into \( y \leq 2x + 1 \): \( 0 \leq 2(0) + 1 \), which is true.
     - This confirms that \((0, 0)\) is in the solution region.

### 4. **Boundary Type:**
   - The boundary \( y = 2x + 1 \) is solid because the inequality includes \( y = 2x + 1 \).

Would you like me to sketch this graph, provide further analysis, or solve a related problem? Let me know!

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### Expanding Thoughts:
1. What happens if the inequality is \( y > 2x + 1 \)?
2. How would the graph change if the slope or y-intercept were different?
3. How can we check whether a point satisfies the inequality?
4. What are some applications of inequalities in real-world scenarios?
5. How do systems of inequalities affect solution regions?

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**Tip:** Test points like \((0, 0)\) or others away from the boundary are quick ways to verify which region satisfies the inequality.

Explore how to graph and solve the inequality y ≤ 2x + 1. This problem covers algebraic concepts such as linear inequalities, slope-intercept form, and solution regions on the coordinate plane. Perfect for students in Grades 8-10 looking to deepen their understanding of inequalities.

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