Solving Inequalities with Interval Method

Have you ever had a feeling you need to solve an inequality like this one ((x+6)^3)/(7-x)^5>0 and you did not know how to start?

The code, used for the Jupyter notebook is here:

import numpy as np
import matplotlib.pyplot as plt

# Function definition
def f(x):
    return ((x + 6)**3 * (x - 4)) / ((7 - x)**7)

# Define x values, avoiding division by zero at x = 7
x_values = np.linspace(-10, 10, 1000)
x_values = x_values[x_values != 7]  # Exclude x=7 to avoid division by zero
y_values = f(x_values)

# Plot the function
plt.figure(figsize=(12, 8))
plt.plot(x_values, y_values, label="(x+6)^3 * (x-4) / (7-x)^7", linewidth=2, color='black')

# Highlight the solution intervals: (-∞, -6) and (4, 7)
plt.fill_betweenx([-200, 200], -10, -6, color='green', alpha=0.3, label="Solution Interval (-∞, -6)")
plt.fill_betweenx([-200, 200], 4, 7, color='blue', alpha=0.3, label="Solution Interval (4, 7)")

# Mark critical points and undefined regions
plt.axvline(-6, color='red', linestyle='--', linewidth=1, label="x=-6 (root, not included)")
plt.axvline(4, color='red', linestyle='--', linewidth=1, label="x=4 (root, not included)")
plt.axvline(7, color='purple', linestyle='--', linewidth=1, label="x=7 (undefined)")
plt.axhline(0, color='black', linestyle='--', linewidth=1, label="y=0 (x-axis)")

# Add titles, labels, and legend
plt.title("Solution of the Inequality (x+6)^3 * (x-4) / (7-x)^7 > 0", fontsize=16)
plt.xlabel("x", fontsize=14)
plt.ylabel("f(x)", fontsize=14)
plt.ylim(-200, 200)
plt.legend(fontsize=12)
plt.grid(True)

# Show the plot
plt.show()

Btw, if after watching the video you are wondering how to write this formula in Jupyter Notebook:

The answer is the following:

1. Change the cell type to Markdown (Shortcut “M”)
2. Paste that:

Solving for:
$$
\frac{(x+6)^3(x-4)}{(7-x)^7} > 0
$$

🙂