https://file.aiquickdraw.com/mathbot/file/1726863058151kfkg0jqo.jpg

To graph the region of feasibility based on the given inequalities:

1. **Inequality 1: \( y \geq 3x - 1 \)**

   - This inequality represents a line \( y = 3x - 1 \) with a slope of 3 and a y-intercept of \(-1\).
   - The region above this line (including the line itself) is feasible.

2. **Inequality 2: \( y \leq 2x + 1 \)**

   - This inequality represents a line \( y = 2x + 1 \) with a slope of 2 and a y-intercept of 1.
   - The region below this line (including the line itself) is feasible.

3. **Inequality 3: \( x \geq 0 \)**

   - This inequality restricts the region to the right of the y-axis (including the y-axis itself).

### Steps to Graph:

1. **Plot the Line \( y = 3x - 1 \):**
   - Start at the y-intercept \((0, -1)\).
   - Use the slope (rise over run) to find another point: from \((0, -1)\), move up 3 units and right 1 unit to plot the next point at \((1, 2)\).
   - Draw the line through these points.
   - Shade the region above this line.

2. **Plot the Line \( y = 2x + 1 \):**
   - Start at the y-intercept \((0, 1)\).
   - Use the slope (2) to find another point: from \((0, 1)\), move up 2 units and right 1 unit to plot the next point at \((1, 3)\).
   - Draw the line through these points.
   - Shade the region below this line.

3. **Mark the \( x \geq 0 \) Region:**
   - Shade the region to the right of the y-axis.

4. **Identify the Feasible Region:**
   - The feasible region is where the shaded regions from all three inequalities overlap.
   - This will be a bounded region on the graph where all three conditions are satisfied.

### Would you like a graph drawn for this region? If so, I can generate an image for you. 

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**Questions for further exploration:**

1. How do we determine the intersection points of the lines \( y = 3x - 1 \) and \( y = 2x + 1 \)?
2. What changes in the feasible region if \( y \leq 3x - 1 \) instead of \( y \geq 3x - 1 \)?
3. How does the slope of the lines affect the feasible region?
4. How would you graph the inequalities if \( x \) had an upper bound, such as \( x \leq 4 \)?
5. What would be the feasible region if the inequality \( y \geq 3x - 1 \) were replaced with \( y \leq 3x - 1 \)?

**Tip:** When graphing inequalities, always start by plotting the boundary lines first, then determine which side of the line to shade based on the inequality sign.

Graph the region of feasibility: y ≥ 3x - 1, y ≤ 2x + 1, x ≥ 0.

Learn how to graph the feasible region defined by y ≥ 3x - 1, y ≤ 2x + 1, and x ≥ 0. This problem involves graphing linear inequalities and finding the overlapping region of feasibility. Suitable for students in Grades 9-12.

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