Introduction to Inequalities

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

Intermediate20 minLesson
Printable worksheet

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.}

Try it now

Which symbol means 'greater than'?

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.

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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≤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

<< means strictly less than (not equal), while ≤\leq includes equality. So 3<33 < 3 is false, but 3≤33 \leq 3 is true.
<< means strictly less than (not equal), while ≤\leq includes equality. So 3<33 < 3 is false, but 3≤33 \leq 3 is true.
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.
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
Today's challengeOne problem, three tries

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