import numpy as np import matplotlib.pyplot as plt

Define the grid for the graph

x = np.linspace(-10, 10, 400) y = np.linspace(-10, 10, 400) X, Y = np.meshgrid(x, y)

Define the inequalities

ineq1 = (X - 4Y >= -8) # x - 4y >= -8 ineq2 = (2X + Y <= 4) # 2x + y <= 4

Plot the inequalities

plt.figure(figsize=(8, 8)) plt.contourf(X, Y, ineq1 & ineq2, levels=1, colors=['lightblue'], alpha=0.5)

Add boundary lines for the inequalities

plt.plot(x, (x + 8) / 4, label='x - 4y = -8', color='blue') # Line for ineq1 plt.plot(x, 4 - 2*x, label='2x + y = 4', color='green') # Line for ineq2

Formatting the graph

plt.axhline(0, color='black',linewidth=0.5) plt.axvline(0, color='black',linewidth=0.5) plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5) plt.xlim(-10, 10) plt.ylim(-10, 10) plt.legend() plt.title('Graph of the System of Inequalities') plt.xlabel('x') plt.ylabel('y') plt.show() <<ImageDisplayed>>successHere is the graph of the solution set for the given system of inequalities:

1. The shaded region (light blue) represents the solution where both inequalities are satisfied.
2. The boundary lines:
   • $x - 4y = -8$ (blue line)
   • $2x + y = 4$ (green line)

The area satisfying both constraints lies at their intersection, including regions on and below the boundaries.

Would you like a step-by-step explanation, or do you have any specific questions about this?

Follow-up questions to consider:

1. How can we verify a specific point lies within the solution region?
2. What changes in the graph if one of the inequalities becomes strict (e.g., $>$ or $<$)?
3. What happens if we add another inequality to the system?
4. Can we express the solution set algebraically without graphing?
5. How do the slope and intercept of the boundary lines affect the shaded region?

Tip:

Always test boundary and nearby points to ensure the shading direction is correct for each inequality!