To graph the compound inequality $x \geq 3$ or $x < 1$ on a number line, here is the step-by-step process:

1. Understand each part of the inequality:

   • $x \geq 3$: This means that $x$ is greater than or equal to 3. The graph will start at 3 (solid circle at 3 because of the "equal to") and continue to the right.
   • $x < 1$: This means that $x$ is less than 1. The graph will be to the left of 1 (open circle at 1 because it is not "equal to").

2. Combine using "or":

   • "Or" means the graph will include all points that satisfy either inequality. So, we graph both parts together.

3. Graphing:

   • Draw a solid circle at $x = 3$, then shade to the right.
   • Draw an open circle at $x = 1$, then shade to the left.

Let me create this graph for you!import matplotlib.pyplot as plt import numpy as np

Create the number line

x = np.linspace(-2, 5, 400)

Plot the segments

plt.figure(figsize=(10, 2)) plt.axhline(0, color='black', linewidth=0.5) # Number line

Shade x >= 3

plt.plot(x[x >= 3], [0]*len(x[x >= 3]), color='blue', linewidth=2, label="x >= 3") plt.scatter(3, 0, color='blue', s=100, edgecolor='black', zorder=5) # Solid dot at 3

Shade x < 1

plt.plot(x[x < 1], [0]*len(x[x < 1]), color='red', linewidth=2, label="x < 1") plt.scatter(1, 0, color='white', s=100, edgecolor='black', zorder=5) # Open circle at 1

Formatting the graph

plt.yticks([]) plt.xticks(range(-2, 6)) plt.ylim(-1, 1) plt.xlim(-2, 5) plt.legend(loc='upper right') plt.title("Graph of x >= 3 or x < 1") plt.show()/tmp/ipykernel_12/3875470736.py:13: UserWarning: You passed a edgecolor/edgecolors ('black') for an unfilled marker ('x'). Matplotlib is ignoring the edgecolor in favor of the facecolor. This behavior may change in the future. plt.scatter(3, 0, color='blue', s=100, edgecolor='black', zorder=5) # Solid dot at 3 /tmp/ipykernel_12/3875470736.py:17: UserWarning: You passed a edgecolor/edgecolors ('black') for an unfilled marker ('x'). Matplotlib is ignoring the edgecolor in favor of the facecolor. This behavior may change in the future. plt.scatter(1, 0, color='white', s=100, edgecolor='black', zorder=5) # Open circle at 1

<<ImageDisplayed>>successHere is the graph for the compound inequality $x \geq 3$ or $x < 1$.

• The blue line represents $x \geq 3$, starting at 3 (solid dot) and extending to the right.
• The red line represents $x < 1$, with an open circle at 1 and shading to the left.

Do you have any questions or need further clarification?

Here are five related questions to expand on this:

1. What does "and" mean in compound inequalities compared to "or"?
2. How would the graph change if the inequality was $x > 3$ instead of $x \geq 3$?
3. Can you explain how to write the solution to this inequality in interval notation?
4. How would this graph look if it combined $x \geq 1$ and $x < 3$ with "and"?
5. How can inequalities be solved when dealing with more complex equations?

Tip: Always remember that open and closed circles represent strict (< or >) and inclusive (≤ or ≥) inequalities, respectively, when graphing.