Graphing Inequalities on a Number Line

Learn how to represent inequalities visually using open circles, closed circles, and shading on a number line.

Intermediate25 minLesson
Printable worksheet

Definition

Graphing an inequality means showing all solutions on a number line. Unlike an equation with one answer, an inequality has infinitely many solutions!

The Four Inequality Symbols

SymbolMeaningCircle TypeExample
<<less thanOpen ○x<3x < 3
>>greater thanOpen ○x>−2x > -2
≤\leqless than or equal toClosed ●x≤5x \leq 5
≥\geqgreater than or equal toClosed ●x≥−1x \geq -1

Key Rules

  1. 1.Open circle (○): The boundary point is NOT included (<< or >>)
  2. 2.Closed circle (●): The boundary point IS included (≤\leq or ≥\geq)
  3. 3.Shade the arrow in the direction of all solutions

Try it now

What type of circle should be used for x<7x < 7?

Worked Examples

Graph the inequality x<4x < 4 on a number line.
1

Identify the boundary point

The number after the symbol is 4 → Boundary: 4
2

Determine the circle type

The symbol is << (strict inequality, not equal) → Open circle ○
3

Decide shading direction

x<4x < 4 means all numbers LESS than 4 → Shade LEFT (toward smaller numbers)
4

Draw the graph

Draw open circle at 4, shade arrow pointing left → ○←──── at 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 ≤\leq and ≥\geq (included).

Shading the wrong direction

Why it's wrong: Confusion about which way is 'greater' or 'less' on the number line.

Correct: Less than (<<, ≤\leq) → shade LEFT. Greater than (>>, ≥\geq) → shade RIGHT. Remember: right is greater!

Confusing −5<−3-5 < -3 with −5>−3-5 > -3

Why it's wrong: With negative numbers, the digit size can be misleading.

Correct: On a number line, −5-5 is to the LEFT of −3-3, so −5<−3-5 < -3. 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

to

x < 4·(−∞, 4)

Click on numbers to select them. Adjust the range to explore different values.

to

x ≥ -2·[-2, ∞)

Click on numbers to select them. Adjust the range to explore different values.

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

16 problems
Problem 1 of 16
Easy
What type of circle should be used for x<7x < 7?

Why It Matters

Graphing inequalities helps us visualize ranges of values in real life:
  • Height requirements: "You must be at least 120 cm tall" → h≥120h \geq 120
  • Speed limits: "Speed must be less than 50 km/h" → s<50s < 50
  • Temperature ranges: "Keep between 2°C and 8°C" → 2≤t≤82 \leq t \leq 8
  • Age restrictions: "Must be 18 or older" → a≥18a \geq 18
Being able to graph these ranges helps us quickly see which values are allowed!

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 h≥140h \geq 140 and graphed with a closed circle at 140, shading right.

1Try It Yourself

A water slide requires riders to be less than 200 cm tall and at least 120 cm tall.

Graph the height requirement h≥120h \geq 120 on a number line.

Step 1: Choose the right answer

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 t<5t < 5: open circle at 5, shade left.

2Try It Yourself

Frozen food must be stored at −18°C-18°C or colder.

Write and graph this as an inequality.

Step 1: Choose the right answer

Choose the inequality for 'at or below −18-18':

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 s<30s < 30: open circle at 30, shade left.

3Try It Yourself

On a highway, the minimum speed is 60 km/h and maximum is 120 km/h.

Graph the minimum speed requirement s≥60s \geq 60.

Step 1: Choose the right answer

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 ≤\leq and ≥\geq, the boundary IS included
  • 3Shade left for 'less than' (<<, ≤\leq), toward smaller numbers
  • 4Shade right for 'greater than' (>>, ≥\geq), toward larger numbers
  • 5Always test a value from your shaded region to verify your graph is correct

Frequently Asked Questions

Think of the line under ≤\leq and ≥\geq as 'filling in' the circle. No line = open circle. Line underneath = closed (filled) circle.
Think of the line under ≤\leq and ≥\geq as 'filling in' the circle. No line = open circle. Line underneath = closed (filled) circle.
Rewrite it with xx first: 5>x5 > x is the same as x<5x < 5. The number that xx is compared to (5) is your boundary point.
Yes! Pick a number from your shaded region and substitute it into the original inequality. If it makes the inequality true, your graph is correct.

Glossary

Inequality
A mathematical statement comparing two expressions using <<, >>, ≤\leq, or ≥\geq
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
Today's challengeOne problem, three tries

More in This Topic

Introduction to Inequalities

Intermediate20 min

Solving and Graphing Inequalities

Intermediate25 min

One-Step Inequalities

Intermediate25 min

Two-Step Inequalities

Intermediate30 min