Teacher Guide: Graphing Inequalities on a Number Line
Learn how to represent inequalities visually using open circles, closed circles, and shading on a number line.
Use this lesson with your class
Free, no student accounts needed.
Share with students
Students open the lesson and practise with instant feedback.
Class quiz
10 questions on Inequalities. Students join with a name, you see everyone's score.
For Teachers
- Graph simple inequalities using open and closed circles
- Determine shading direction based on the inequality symbol
- Write inequalities from number line graphs
- Connect inequality graphs to real-world contexts
- • Understanding of the number line
- • Knowledge of negative numbers and their positions
- • Familiarity with inequality symbols (, , , )
- 1. Where have you seen number ranges used in real life (signs, labels, requirements)?
- 2. Why do you think we use different circles for different inequality symbols?
- 3. If a graph shows an open circle at 10 with shading to the right, could the value be exactly 10?
- 4. How would you graph 'any number except 5'? Is that possible with one inequality?
The arrow should always point right
Open and closed circles are just stylistic choices
Negative numbers flip the shading direction
For Struggling Students:
- • Use color coding: red for open circles, blue for closed
- • Provide a reference card with all four symbols and their circle types
- • Start with whole number boundaries only (no negatives)
For On-Level Students:
- • Graph inequalities with negative boundaries
- • Write inequalities from given graphs
- • Solve one-step inequalities and graph solutions
For Advanced Students:
- • Introduce compound inequalities (e.g., )
- • Graph inequalities with fractions or decimals as boundaries
- • Create real-world scenarios for given inequality graphs
- 6.EE.B.8 (CCSS.MATH.CONTENT.6.EE.B.8)
Write an inequality of the form or to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities have infinitely many solutions; represent solutions on number line diagrams.
- 7.EE.B.4b (CCSS.MATH.CONTENT.7.EE.B.4.B)
Solve word problems leading to inequalities of the form or and graph the solution set.
- visualInteractive Number Line
Students practice placing circles and shading on a digital number line
- activityReal-World Inequality Match
Match scenarios to their inequality graphs
- worksheetGraph and Write Practice
Practice graphing from inequalities and writing inequalities from graphs
Lesson Content
Everything students see: definition, examples, common mistakes, applications. Tap to open.
Lesson Content
Everything students see: definition, examples, common mistakes, applications. Tap to open.
Definition
The Four Inequality Symbols
| Symbol | Meaning | Circle Type | Example |
|---|---|---|---|
| less than | Open ○ | ||
| greater than | Open ○ | ||
| less than or equal to | Closed ● | ||
| greater than or equal to | Closed ● |
Key Rules
Worked Examples
Graph the inequality on a number line.
Identify the boundary point
The number after the symbol is 4 → Boundary: 4
Determine the circle type
The symbol is (strict inequality, not equal) → Open circle ○
Decide shading direction
means all numbers LESS than 4 → Shade LEFT (toward smaller numbers)
Draw the graph
Draw open circle at 4, shade arrow pointing left → ○←———— at 4
Answer: Open circle at 4, arrow pointing left. Solutions include 3, 2, 1, 0, -1, and all numbers less than 4.
Common Mistakes
Using a closed circle for or
Why it's wrong: Students forget that and are strict inequalities that do NOT include the boundary point.
Correct: Use open circle for and (not included). Use closed circle for and (included).
Shading the wrong direction
Why it's wrong: Confusion about which way is 'greater' or 'less' on the number line.
Correct: Less than (, ) → shade LEFT. Greater than (, ) → shade RIGHT. Remember: right is greater!
Confusing with
Why it's wrong: With negative numbers, the digit size can be misleading.
Correct: On a number line, is to the LEFT of , so . Further left = smaller.
Forgetting to flip the inequality when multiplying by negative
Why it's wrong: When solving inequalities, multiplying or dividing by a negative reverses the direction.
Correct: If you multiply or divide by a negative number, flip the inequality symbol!
Why It Matters
- Height requirements: "You must be at least 120 cm tall" →
- Speed limits: "Speed must be less than 50 km/h" →
- Temperature ranges: "Keep between 2°C and 8°C" →
- Age restrictions: "Must be 18 or older" →
Real World Applications
Theme Park Height Requirements
Theme parks use inequalities to set minimum heights for rides.
Example:
A roller coaster requires riders to be at least 140 cm tall. This is written as and graphed with a closed circle at 140, shading right.
A water slide requires riders to be less than 200 cm tall and at least 120 cm tall.
Graph the height requirement on a number line.
Step 1: Write the mathematical expression
What type of circle goes at 120?
Temperature Safety for Food
Food safety guidelines use inequalities to show safe storage temperatures.
Example:
Refrigerated food should be kept below 5°C. This is : open circle at 5, shade left.
Frozen food must be stored at or colder.
Write and graph this as an inequality.
Step 1: Write the mathematical expression
Write the inequality for 'at or below ':
Speed Limits
Traffic laws use inequalities to define legal speeds.
Example:
In a school zone, speed must be less than 30 km/h. This is : open circle at 30, shade left.
On a highway, the minimum speed is 60 km/h and maximum is 120 km/h.
Graph the minimum speed requirement .
Step 1: Write the mathematical expression
Is driving exactly 60 km/h legal?
Key Takeaways
- 1Use an open circle (○) for and — the boundary is NOT included
- 2Use a closed circle (●) for and — the boundary IS included
- 3Shade left for 'less than' (, ) — toward smaller numbers
- 4Shade right for 'greater than' (, ) — toward larger numbers
- 5Always test a value from your shaded region to verify your graph is correct
Frequently Asked Questions
How do I remember which circle to use?
What if the inequality is written 'backwards' like ?
Can I check if my graph is correct?
Glossary
- Inequality
- A mathematical statement comparing two expressions using , , , or
- Boundary point
- The number where the inequality changes from true to false (the circle location)
- Open circle
- An unfilled circle (○) showing that the point is NOT included in the solution
- Closed circle
- A filled circle (●) showing that the point IS included in the solution
- Solution set
- All values that make the inequality true