numpy as np import matplotlib.pyplot as plt

Define the x range

x = np.linspace(-10, 10, 400)

Define the equations

y1 = (1/3) * x - 2 y2 = -x + 2

Plot the inequalities

plt.figure(figsize=(10, 6)) plt.plot(x, y1, label=r'$y \leq \frac{1}{3}x - 2$', color='blue') plt.plot(x, y2, label=r'$y \leq -x + 2$', color='red')

Fill the solution region

plt.fill_between(x, np.minimum(y1, y2), -10, where=(np.minimum(y1, y2) > -10), color="gray", alpha=0.5)

Set limits and labels

plt.xlim(-10, 10) plt.ylim(-10, 10) 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.title("Solution for the System of Inequalities") plt.xlabel("x-axis") plt.ylabel("y-axis") plt.legend(loc="upper left") plt.show()Here is the sketch of the solution for the given system of inequalities. The shaded region represents the solution area where both inequalities are satisfied simultaneously. Let me know if you need a deeper explanation or further assistance.

Related Questions:

1. How do we determine the intersection points of the lines in such systems?
2. What happens if the inequalities were reversed to "greater than" instead of "less than"?
3. How can you verify a specific point lies within the solution region?
4. What are the implications of having parallel lines in a system of inequalities?
5. How do you solve a system of inequalities with three or more inequalities?

Tip:

Always test a point (like (0, 0)) to check if it satisfies the inequalities, which helps identify the correct shading region.