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Teacher Guide: Introduction to Inequalities

Learn the basics of inequalities and how to compare values using inequality symbols.

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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
  • Understand what an inequality represents
  • Identify and use the symbols <<, >>, ≤\leq, ≥\geq, and ≠\neq correctly
  • Compare integers (including negative numbers) using inequalities
  • Translate real-world situations into inequality statements
  • Determine if a given value satisfies an inequality
Prerequisites
  • • Understanding of positive and negative numbers
  • • Familiarity with the number line
  • • Basic understanding of comparing numbers
Discussion Starters
  • 1. Where have you seen greater than or less than symbols before?
  • 2. Why might we need to say 'at least' instead of 'exactly' in real life?
  • 3. Can you think of a rule at home or school that could be written as an inequality?
  • 4. How is an inequality different from an equation?
Common Misconceptions

The bigger digit is always the bigger number (e.g., −9>−2-9 > -2)

Remediation: Use the number line consistently. Show that −9-9 is further LEFT, which means SMALLER. Use temperature comparisons: '−9°C-9°C vs −2°C-2°C - which is colder?'

Thinking ≤\leq and << mean the same thing

Remediation: Use boundary examples: 'If you must be at least 13 to join, can a 13-year-old join?' (Yes with ≥13\geq 13, no with >13> 13)

Differentiation Ideas

For Struggling Students:

  • • Use physical number line manipulatives
  • • Focus only on << and >> before introducing ≤\leq and ≥\geq
  • • Use the 'alligator mouth' mnemonic consistently
  • • Connect every comparison to concrete contexts (temperature, money)

For On-Level Students:

  • • Compare positive and negative integers
  • • Write inequalities from word problems
  • • Determine if values satisfy given inequalities

For Advanced Students:

  • • Introduce compound inequalities (e.g., −3<x<5-3 < x < 5)
  • • Explore how flipping an inequality works
  • • Connect to graphing inequalities on a number line
Standards Alignment
  • 6.NS.C.7 (CCSS.MATH.CONTENT.6.NS.C.7)

    Understand ordering and absolute value of rational numbers

  • 6.EE.B.5 (CCSS.MATH.CONTENT.6.EE.B.5)

    Understand solving an equation or inequality as finding values that make it true

  • 6.EE.B.8 (CCSS.MATH.CONTENT.6.EE.B.8)

    Write an inequality to represent a constraint or condition in a real-world problem

Lesson Resources
  • visualInteractive Number Line

    Students place numbers and see inequality relationships

  • activitySymbol Sorting Game

    Match inequality symbols to their meanings

  • worksheetReal-World Inequalities

    Write inequalities for everyday situations

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

An inequality is a mathematical statement that compares two values that are not necessarily equal.
We use these symbols to show relationships:
SymbolMeaningExample
<<Less than3<53 < 5 (3 is less than 5)
>>Greater than7>27 > 2 (7 is greater than 2)
≤\leqLess than or equal tox≤4x \leq 4 (x is at most 4)
≥\geqGreater than or equal toy≥1y \geq 1 (y is at least 1)
≠\neqNot equal to5≠35 \neq 3 (5 is not equal to 3)
If a<b, then a is to the LEFT of b on the number line.\text{If } a < b \text{, then } a \text{ is to the LEFT of } b \text{ on the number line.}

Worked Examples

Is the statement 8>58 > 5 true or false?
1

Identify the symbol

The symbol >> means 'greater than' → Reading: '8 is greater than 5'
2

Check the relationship

Is 8 larger than 5? Yes! → 8 is indeed larger
3

Determine truth value

Since 8 IS greater than 5, the statement is TRUE

Common Mistakes

Confusing << and >> symbols

Why it's wrong: The symbols look similar and it's easy to mix them up.

Correct: The symbol 'points to' the smaller value. Think of it as an alligator mouth that always wants to eat the bigger number: 3<53 < 5 (mouth opens toward 5).

Thinking −10>−2-10 > -2 because 10>210 > 2

Why it's wrong: With negative numbers, the larger digit doesn't mean a larger value.

Correct: Use the number line: −10-10 is further LEFT than −2-2, so −10<−2-10 < -2. Think temperature: −10°C-10°C is colder than −2°C-2°C.

Confusing ≤\leq with 'less than'

Why it's wrong: Students forget the 'or equal to' part.

Correct: ≤\leq means 'less than OR equal to'. So 5≤55 \leq 5 is TRUE because 5 equals 5.

Why It Matters

Inequalities describe real-world situations where things aren't exactly equal:
  • Speed limits: You must drive at most 50 km/h (v≤50v \leq 50)
  • Age requirements: You must be at least 18 to vote (age≥18\text{age} \geq 18)
  • Budget constraints: You can spend less than 100 euros (cost<100\text{cost} < 100)
  • Temperature: It's warmer than yesterday (Ttoday>TyesterdayT_{\text{today}} > T_{\text{yesterday}})
Unlike equations (which have one solution), inequalities often have many solutions!

Real World Applications

Speed Limits and Traffic Rules

Traffic laws use inequalities to set maximum and minimum speeds.

Example:

A highway speed limit of 120 km/h means your speed vv must satisfy v≤120v \leq 120.

1Try It Yourself

A school zone has a maximum speed of 30 km/h. A driver is going 35 km/h.

Is the driver breaking the speed limit? Write an inequality to show why.

Step 1: Choose the right answer

If speed must be at most 30, which rule is it?

Shopping on a Budget

When shopping, you need to ensure your total stays within your budget.

Example:

If you have 50 euros, the total cost cc must satisfy c≤50c \leq 50.

2Try It Yourself

You have 40 euros and want to buy a book for 15 euros and a game.

What's the maximum price you can pay for the game?

Step 1: Choose the right answer

If book + game must be at most 40, which inequality fits?

Key Takeaways

  • 1Inequalities compare values using <<, >>, ≤\leq, ≥\geq, and ≠\neq
  • 2The symbols << and >> always 'point to' the smaller value
  • 3On a number line, smaller values are to the LEFT
  • 4≤\leq means 'less than OR equal to' and ≥\geq means 'greater than OR equal to'
  • 5Unlike equations, inequalities can have many solutions

Frequently Asked Questions

What's the difference between << and ≤\leq?

<< means strictly less than (not equal), while ≤\leq includes equality. So 3<33 < 3 is false, but 3≤33 \leq 3 is true.

How do I remember which symbol is which?

Think of the symbol as an alligator's mouth that always opens toward the bigger number. Or remember: the small end points to the small number.

Can an inequality have no solutions?

Yes! For example, x>5x > 5 AND x<3x < 3 has no solution because no number is both greater than 5 and less than 3.

Glossary

Inequality
A mathematical statement comparing two values that may not be equal, using symbols like <<, >>, ≤\leq, ≥\geq
Less than (<<)
A symbol showing that the left value is smaller than the right value
Greater than (>>)
A symbol showing that the left value is larger than the right value
Less than or equal to (≤\leq)
A symbol meaning the left value is smaller than OR the same as the right value
Greater than or equal to (≥\geq)
A symbol meaning the left value is larger than OR the same as the right value
Solution set
All values that make an inequality true

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