Chapter Two of Inequalities: Graphing Inequalities on a Number Line

Now that we know how to write and interpret inequalities, let’s learn how to graph them on a number line. This helps us visualize the solution set—all the possible values that satisfy the inequality.

2.1 What Is a Solution Set?

A solution set is the group of numbers that make an inequality true. For example:

The inequality x < 5 means that any number less than 5 is a solution—like 4, 0, -2, or even 4.999.

We can represent all those numbers at once on a number line!

2.2 Tools for Graphing

To graph an inequality, we need two things:

• A number line
  
  

• A point showing the boundary value (called the endpoint)
  
  

We also need to know two important rules:

Symbol                                             Type of Point                           Arrow Direction

                < or >                                         Open circle (not included)                   Left for <, Right for >

               ≤ or ≥                                            Closed circle (included)                    Left for ≤, Right for ≥

2.3 Graphing Examples

Example 1: Graph x < 4

• Use an open circle at 4 (because 4 is not included)
  
  

• Shade the number line to the left of 4
  
  

📉 Number line:

← ●———————

   4

Example 2: Graph x ≥ -2

• Use a closed circle at -2 (because it is included)
  
  

• Shade to the right of -2
  
  

📉 Number line:

—●→  

 -2

2.4 Real-Life Example

You must be at least 13 years old to create a social media account.

• “At least” means ≥
  
  

• Let 𝑎 be age
  
  

• Inequality: a ≥ 13
  
  

✅ Graph:

• Closed circle at 13
  
  

• Arrow to the right
  
  

2.5 Common Mistakes to Avoid

Mistake                                                                                                   Correction

Using open circle for ≥ or ≤                                          Always use a closed circle if the value is included

Reversing the arrow direction                                                                                   Think: < is left, > is right

Forgetting to label the number line                                        Always label your key point (like 3, -5, or 0)

2.6 Practice Problems

Draw a number line and graph each inequality:

1. x ≤ 2
   
   

2. x > -3
   
   

3. a ≥ 7
   
   

4. b < 0
   
   

5. y ≥ -5
   
   

2.7 Challenge Problems (Word Problems)

Problem 1:

Tickets for a movie cost no more than $12. Let 𝑥 represent the price of a ticket.
✅ Write and graph the inequality.

Problem 2:

You must be older than 17 to vote in some countries. Let 𝑎 be your age.
✅ Write and graph the inequality.

Problem 3:

A school allows at most 30 students in a class. Let 𝑠 represent the number of students.
✅ Write and graph the inequality.

2.8 Summary

✅ Key Points:

• Use a number line to show all values that satisfy an inequality.
  
  

• Use an open circle for < or >, and a closed circle for ≤ or ≥.
  
  

• Shade left for < and ≤, right for > and ≥.
  
  

• Always label the number and draw the correct direction of the arrow.
  
  

In the next chapter, we’ll learn how to solve one-step inequalities—just like solving equations, but with inequality symbols instead of an equal sign.