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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.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Inequalities. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • 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
Prerequisites
  • • Understanding of the number line
  • • Knowledge of negative numbers and their positions
  • • Familiarity with inequality symbols (<<, >>, ≤\leq, ≥\geq)
Discussion Starters
  • 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?
Common Misconceptions

The arrow should always point right

Remediation: Emphasize that the direction depends on the symbol: < and ≤ go LEFT (less), > and ≥ go RIGHT (greater). Use hand motions.

Open and closed circles are just stylistic choices

Remediation: Use concrete examples: 'x<5x < 5: Can xx be 5? No! So we leave the circle empty to show 5 is not allowed.'

Negative numbers flip the shading direction

Remediation: The direction ONLY depends on the symbol, not the boundary value. x>−3x > -3 still shades RIGHT because > means greater.

Differentiation Ideas

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., −2<x≤5-2 < x \leq 5)
  • • Graph inequalities with fractions or decimals as boundaries
  • • Create real-world scenarios for given inequality graphs
Standards Alignment
  • 6.EE.B.8 (CCSS.MATH.CONTENT.6.EE.B.8)

    Write an inequality of the form x>cx > c or x<cx < c 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 px+q>rpx + q > r or px+q<rpx + q < r and graph the solution set.

Lesson Resources
  • 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.

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

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!

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

How do I remember which circle to use?

Think of the line under ≤\leq and ≥\geq as 'filling in' the circle. No line = open circle. Line underneath = closed (filled) circle.

What if the inequality is written 'backwards' like 5>x5 > x?

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.

Can I check if my graph is correct?

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

More in This Topic

Introduction to Inequalities

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Solving and Graphing Inequalities

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One-Step Inequalities

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Two-Step Inequalities

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