Python
import matplotlib.pyplot as plt import numpy as np # Define the inequality # y > (-5/6)x - 5 m = -5/6 b = -5 # Set up the figure and axes plt.figure(figsize=(8, 8)) # Define the x range for the plot x = np.linspace(-12, 6, 400) # Calculate the y values for the boundary line y_line = m * x + b # 1. Graph the boundary line: y = (-5/6)x - 5 # The line is dashed because the inequality is strictly greater than (>) plt.plot(x, y_line, color='blue', linestyle='--', label=r'$y = -\frac{5}{6}x - 5$') # 2. Shade the region: y > (-5/6)x - 5 # Shade the area above the line plt.fill_between(x, y_line, 10, color='skyblue', alpha=0.5) # Set the axes limits (to include intercepts and a clear view) plt.xlim(-12, 6) plt.ylim(-12, 5) # Add title and labels plt.title(r'Graph of the Inequality $y > -\frac{5}{6}x - 5$') plt.xlabel('x') plt.ylabel('y') # Add grid lines plt.grid(True, linestyle=':', alpha=0.6) # Draw the axes through the origin plt.axhline(0, color='black', linewidth=0.5) plt.axvline(0, color='black', linewidth=0.5) # Mark the intercepts for clarity plt.scatter([0, -6], [-5, 0], color='red', zorder=5) # Add annotation for the inequality # Choose a point in the shaded region, e.g., (0, 0) # 0 > (-5/6)(0) - 5 => 0 > -5 (True) # plt.annotate('Shaded Region ($y > -\frac{5}{6}x - 5$)', (2, 2), color='blue') # Save the plot plt.savefig('inequality_graph.png') print("inequality_graph.png")
Code output
inequality_graph.png
The graph of the inequality $y > -\frac{5}{6}x - 5$ is shown above.
Here is a breakdown of the graph:
Boundary Line: The line $y = -\frac{5}{6}x - 5$ is plotted as a dashed line. This is because the inequality symbol is $>$ (greater than) and does not include the "equal to" case. Points on the line are not solutions.
The y-intercept is $(0, -5)$.
The x-intercept is $(-6, 0)$.
Shaded Region: The area above the dashed line is shaded. This represents all the $(x, y)$ coordinate pairs that satisfy the condition $y > -\frac{5}{6}x - 5$ (i.e., their $y$-value is greater than the $y$-value on the boundary line for a given $x$).
