matplotlib.pyplot as plt import numpy as np

Define the inequalities

x = np.linspace(-5, 5, 400) y1 = 6 - x # y ≤ 6 - x y2 = 4 * x + 1 # y ≤ 4x + 1 y3 = 1 # y > 1

Plot the region

plt.figure(figsize=(8, 8)) plt.plot(x, y1, label="y = 6 - x", color="blue") plt.plot(x, y2, label="y = 4x + 1", color="green") plt.axhline(y3, label="y = 1", color="red")

Fill the solution region

plt.fill_between(x, y3, np.minimum(y1, y2), where=(y3 < np.minimum(y1, y2)), color='gray', alpha=0.5)

Configure the plot

plt.xlim(-3, 3) plt.ylim(0, 7) plt.xlabel('x') plt.ylabel('y') plt.axhline(0, color='black', linewidth=0.5, linestyle="--") plt.axvline(0, color='black', linewidth=0.5, linestyle="--") plt.grid(True, linestyle="--", alpha=0.6) plt.legend() plt.title("Solution Region for the System of Inequalities")

Display the plot

plt.show()Here is the graph of the solution region for the given system of inequalities. The shaded region represents the area where all three inequalities are satisfied. The corners of the solution region are formed at the points of intersection of the boundaries:

1. The intersection of $y = 6 - x$ and $y = 4x + 1$.
2. The intersection of $y = 6 - x$ and $y = 1$.
3. The intersection of $y = 4x + 1$ and $y = 1$.

Calculation of Intersection Points:

1. Solve $6 - x = 4x + 1$: $x = 1, \quad y = 5$ Point: $(1, 5)$

2. Solve $6 - x = 1$: $x = 5, \quad y = 1$ Point: $(5, 1)$

3. Solve $4x + 1 = 1$: $x = 0, \quad y = 1$ Point: $(0, 1)$

Thus, the corners of the region are:

• $(1, 5)$
• $(5, 1)$
• $(0, 1)$

Let me know if you'd like further clarification or help with a related topic!

Related Questions:

1. How do you determine the intersection points of linear equations algebraically?
2. What is the significance of shading in graphs of inequalities?
3. How do you verify which side of the line to shade for a given inequality?
4. Can a solution region be unbounded? If so, how does it appear?
5. What are some real-world applications of systems of inequalities?

Tip:

Always test a point (like $(0, 0)$) in each inequality to confirm the correct shading direction!