Graphing Inequalities on a Number Line
Learn how to represent inequalities visually using open circles, closed circles, and shading on a number line.
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
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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!
Interactive Visual
Click on the number line to find the target number. Test your skills!
Interactive Sandbox
Expression Calculator
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Practice Problems
16 problemsWhat type of circle should be used for ?
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
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