Introduction to Inequalities

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

Intermediate20 minLesson

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.}

Try it now

Which symbol means 'greater than'?

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.

Interactive Visual

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Practice Problems

16 problems

Problem 1 of 16

Easy

Which symbol means 'greater than'?

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

<

means strictly less than (not equal), while

\leq

includes equality. So

3 < 3

is false, but

3 \leq 3

is true.

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