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., $-9 > -2$ )

Remediation: Use the number line consistently. Show that $-9$ is further LEFT, which means SMALLER. Use temperature comparisons: ' $-9°C$ vs $-2°C$ - 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 $\geq 13$ , no with $> 13$ )

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., $-3 < x < 5$ )
• 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:

Symbol	Meaning	Example
$<$	Less than	$3 < 5$ (3 is less than 5)
$>$	Greater than	$7 > 2$ (7 is greater than 2)
$\leq$	Less than or equal to	$x \leq 4$ (x is at most 4)
$\geq$	Greater than or equal to	$y \geq 1$ (y is at least 1)
$\neq$	Not equal to	$5 \neq 3$ (5 is not equal to 3)

\text{If } a < b \text{, then } a \text{ is to the LEFT of } b \text{ on the number line.}

Worked Examples

Is the statement $8 > 5$ true or false?

Identify the symbol

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

Check the relationship

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

Determine truth value

Since 8 IS greater than 5, the statement is TRUE

Answer: True. The statement $8 > 5$ correctly shows that 8 is greater than 5.

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: $3 < 5$ (mouth opens toward 5).

Thinking $-10 > -2$ because $10 > 2$

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

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

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 $5 \leq 5$ 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 ( $v \leq 50$ )
Age requirements: You must be at least 18 to vote ( $\text{age} \geq 18$ )
Budget constraints: You can spend less than 100 euros ( $\text{cost} < 100$ )
Temperature: It's warmer than yesterday ( $T_{\text{today}} > T_{\text{yesterday}}$ )

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 $v$ must satisfy $v \leq 120$ .

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: Write the mathematical expression

If speed must be at most 30, write the rule:

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 $c$ must satisfy $c \leq 50$ .

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: Write the mathematical expression

If book + game must be at most 40:

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 < 3

is false, but

3 \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 > 5

AND

x < 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