Inequalities | Student Worksheet

The Big Idea

An INEQUALITY compares two expressions that are NOT equal. Unlike equations (which have ONE answer), inequalities have MANY solutions!

Example: x > 5 means x can be 6, 7, 8, 100, or ANY number greater than 5!

The Four Inequality Symbols

Symbol	Meaning	Key Words	Graph
<	Less than	fewer than, below	Open circle ○, shade LEFT
>	Greater than	more than, above	Open circle ○, shade RIGHT
≤	Less than or equal to	at most, no more than	Closed circle ●, shade LEFT
≥	Greater than or equal to	at least, no less than	Closed circle ●, shade RIGHT

Memory Trick: The symbol is like an alligator mouth - it always opens toward the BIGGER side!

Example 1: Understanding Symbols

Open vs. Closed Circles

Open Circle ○

For < and >

Number is NOT included

x > 5 means x can be 5.001, 6, 7... but NOT 5 itself!

Closed Circle ●

For ≤ and ≥

Number IS included

x ≥ 5 means x can be 5, 6, 7, 8... including 5!

Example 2: Solving One-Step Inequalities

Solve inequalities the same way you solve equations - use inverse operations!

Solve: x + 4 > 10

x + 4 > 10
- 4 - 4 (subtract 4 from both sides)
x > 6

Solutions: 7, 8, 9, 10, 11... (any number greater than 6)

Solve: 3y ≤ 15

3y ≤ 15
÷ 3 ÷ 3 (divide both sides by 3)
y ≤ 5

Solutions: 5, 4, 3, 2, 1, 0, -1... (5 and all numbers less than 5)

Example 3: Graphing on a Number Line

Graph: x > 3

←──0──1──2──○──4──5──6──→

Open circle at 3 (3 is NOT included), arrow points RIGHT (toward bigger numbers)

Graph: x ≤ 2

←──-1──0──1──●──3──4──5──→

Closed circle at 2 (2 IS included), arrow points LEFT (toward smaller numbers)

Graph: x ≥ -1

←──-3──-2──●──0──1──2──→

Closed circle at -1 (-1 IS included), arrow points RIGHT (toward bigger numbers)

TRAP ALERT: Common Mistakes!

WRONG: Using a closed circle for x > 5 (5 is NOT included with >)

RIGHT: Use an OPEN circle for > and <. Use a CLOSED circle only for ≥ and ≤.

WRONG: "At least 10" means x > 10

RIGHT: "At least 10" means x ≥ 10 (10 IS included!). "At least" and "at most" ALWAYS include the number.

Your Turn: Practice Problems

1. Write as an inequality: "x is greater than 7"

Answer:

2. Write as an inequality: "y is at most 15"

Answer:

3. Solve: x + 6 < 14

Answer:

4. Solve: 4n ≥ 20

Answer:

5. Graph x > 4 on the number line:

←──1──2──3──4──5──6──7──→

6. Graph y ≤ 2 on the number line:

←──-2──-1──0──1──2──3──4──→

Remember!

< and > use OPEN circles (number NOT included)
≤ and ≥ use CLOSED circles (number IS included)
"At least" means ≥ (greater than or equal to)
"At most" means ≤ (less than or equal to)
Shade LEFT for < and ≤ (toward smaller numbers)
Shade RIGHT for > and ≥ (toward bigger numbers)
Inequalities have MANY solutions, not just one!