To solve the system of inequalities and determine the feasible region for each of the four graphs, we'll go through them one by one. I'll interpret each graph and then outline the solution.

1. Top Left Graph (a)

• Graph Interpretation:
  • Two lines intersect. Based on the typical shading and line patterns, we can infer the inequalities.
  • Let's assume the lines are $y = mx + c$ forms.
• Possible Inequalities:
  • Example inequalities based on the graph:
    1. $y \leq -x + 3$ (a line with a negative slope)
    2. $y \geq 2x - 1$ (a line with a positive slope)
• Feasible Region:
  • The feasible region is the overlapping shaded area, which lies between these two lines.

2. Top Right Graph (b)

• Graph Interpretation:
  • The graph shows a polygonal area where multiple lines intersect.
  • This is likely a more complex system of inequalities.
• Possible Inequalities:
  • Some example inequalities:
    1. $y \geq -\frac{1}{2}x + 2$
    2. $y \leq \frac{3}{2}x + 1$
    3. $y \geq x - 1$
    4. $x \geq 0$ (vertical line)
• Feasible Region:
  • The region where all the inequalities overlap forms the polygonal feasible region.

3. Bottom Left Graph (c)

• Graph Interpretation:
  • The graph shows a triangular or quadrilateral region with sloped lines.
• Possible Inequalities:
  • The inequalities could look something like:
    1. $y \leq x + 2$
    2. $y \geq -2x + 3$
    3. $y \geq 0$ (horizontal line or x-axis)
• Feasible Region:
  • The area where these lines intersect and the inequalities hold true forms the solution region.

4. Bottom Right Graph (d)

• Graph Interpretation:
  • The graph shows a rectangular region, bounded by vertical and horizontal lines.
• Possible Inequalities:
  • The inequalities could be:
    1. $x \geq 1$
    2. $x \leq 4$
    3. $y \geq 2$
    4. $y \leq 5$
• Feasible Region:
  • The feasible region is the rectangle where these inequalities overlap.

Conclusion:

Each graph has a feasible region formed by the intersection of the inequalities, typically shaded to indicate where the solutions to the system of inequalities lie.

Would you like a more detailed step-by-step breakdown for any specific graph or clarification on any points?

Related Questions:

1. How do you algebraically determine the point of intersection for two linear equations?
2. What steps are involved in graphing a system of inequalities by hand?
3. How do you check if a point is within the feasible region?
4. What is the significance of boundary lines being solid or dashed in inequalities?
5. How can you find the area of the feasible region if it's a polygon?

Tip: Always shade lightly when working on paper to ensure the final feasible region is clearly visible after all inequalities have been graphed.