import numpy as np import matplotlib.pyplot as plt

Create a grid of points

x = np.linspace(-1, 3, 400) y = np.linspace(-1, 3, 400) X, Y = np.meshgrid(x, y)

Define the inequalities

ineq1 = X**2 - Y >= 0 # x^2 - y >= 0 ineq2 = X + Y < 2 # x + y < 2 ineq3 = X - Y < 2 # x - y < 2

Combine inequalities

solution = ineq1 & ineq2 & ineq3

Plotting

plt.figure(figsize=(8, 8)) plt.contourf(X, Y, solution, levels=[0, 1], colors=["lightblue"], alpha=0.7) plt.axhline(0, color='black', linewidth=0.5) plt.axvline(0, color='black', linewidth=0.5) plt.xlabel('x') plt.ylabel('y') plt.title('Solution Set for the System of Inequalities') plt.grid(True) plt.show() <<ImageDisplayed>>successHere is the graph of the solution set for the given system of inequalities. The shaded region represents all points $(x, y)$ that satisfy all three inequalities simultaneously:

1. $x^2 - y \geq 0$
2. $x + y < 2$
3. $x - y < 2$

Let me know if you would like further clarifications or have additional questions!

Related Questions:

1. How do we algebraically find the vertices of the shaded region in the graph?
2. What would happen if one inequality was replaced, e.g., $x^2 - y > 0$?
3. How does changing $x + y < 2$ to $x + y \leq 2$ alter the solution?
4. What are real-world scenarios where such inequalities might be used?
5. How can we test if a specific point $(x, y)$ lies in the solution set?

Tip: Always check the boundary conditions when graphing inequalities to ensure accuracy!