One-Variable Inequalities - Sixth Grade
Complete Notes & Formulas
1. What are Inequalities?
Definition
An inequality is a mathematical statement
comparing two expressions that are NOT equal
Shows a RANGE of possible values
Inequality vs Equation
| Equation | Inequality |
|---|---|
| Uses = sign | Uses <, >, ≤, or ≥ |
| Has ONE solution | Has MANY solutions |
| Example: x = 5 | Example: x > 5 |
2. Inequality Symbols
The Four Main Symbols
| Symbol | Meaning | Example | Read As |
|---|---|---|---|
| < | Less than | x < 5 | x is less than 5 |
| > | Greater than | x > 5 | x is greater than 5 |
| ≤ | Less than or equal to | x ≤ 5 | x is at most 5 |
| ≥ | Greater than or equal to | x ≥ 5 | x is at least 5 |
Memory Trick: The symbol "opens" toward the LARGER number! Think of it as a hungry alligator eating the bigger number!
3. Solutions to Inequalities
What is a Solution?
A solution is ANY value that makes
the inequality TRUE
Inequalities usually have INFINITELY MANY solutions!
Example: Is x = 7 a solution to x > 5?
Test: Substitute x = 7
7 > 5 ?
Yes! 7 is greater than 5
Answer: YES, x = 7 is a solution ✓
Other Solutions to x > 5
• x = 6 ✓ (works)
• x = 10 ✓ (works)
• x = 100 ✓ (works)
• x = 5.1 ✓ (works)
• x = 5 ✗ (does NOT work - not greater than)
• x = 3 ✗ (does NOT work)
4. Graphing Inequalities on Number Lines
Circle Types
OPEN CIRCLE ○ for < and >
(Does NOT include the number)
CLOSED CIRCLE ● for ≤ and ≥
(DOES include the number)
Direction of Arrow
| Inequality | Circle Type | Arrow Direction |
|---|---|---|
| x < 5 | Open ○ | ← Left |
| x > 5 | Open ○ | Right → |
| x ≤ 5 | Closed ● | ← Left |
| x ≥ 5 | Closed ● | Right → |
Visual Examples
x > 3
x ≤ 2
5. Writing Inequalities from Number Lines
Steps
Step 1: Find the critical number (where the circle is)
Step 2: Check if circle is open or closed
Step 3: Check which direction the arrow points
Step 4: Write the inequality
Decision Chart
| Circle | Arrow | Symbol | Example (at 5) |
|---|---|---|---|
| Open ○ | Left ← | < | x < 5 |
| Open ○ | Right → | > | x > 5 |
| Closed ● | Left ← | ≤ | x ≤ 5 |
| Closed ● | Right → | ≥ | x ≥ 5 |
6. Solving One-Step Inequalities
General Rules (Same as Equations!)
Addition: Subtract same number from both sides
Subtraction: Add same number to both sides
Multiplication: Divide both sides by same number
Division: Multiply both sides by same number
⚠️ SPECIAL RULE FOR MULTIPLICATION/DIVISION ⚠️
When multiplying or dividing by a NEGATIVE number
FLIP THE INEQUALITY SIGN!
< becomes >
> becomes <
Example 1: Solve x + 7 < 15
x + 7 < 15
x + 7 − 7 < 15 − 7 (Subtract 7)
x < 8
Answer: x < 8
Example 2: Solve 3x ≥ 12
3x ≥ 12
3x ÷ 3 ≥ 12 ÷ 3 (Divide by 3 - positive, no flip)
x ≥ 4
Answer: x ≥ 4
Example 3: Solve −2x > 10 (WITH NEGATIVE!)
−2x > 10
−2x ÷ (−2) ? 10 ÷ (−2) (Divide by negative - FLIP!)
x < −5 (Sign flipped from > to <)
Answer: x < −5
7. Writing Inequalities from Word Problems
Key Phrases
| Phrase | Symbol |
|---|---|
| less than, fewer than, below | < |
| greater than, more than, above | > |
| at most, no more than, maximum | ≤ |
| at least, no less than, minimum | ≥ |
Example 1: "A number is at least 12"
• "A number" → x
• "at least" → ≥
• "12" → 12
Inequality: x ≥ 12
Example 2: "The sum of x and 5 is less than 20"
• "sum of x and 5" → x + 5
• "is less than" → <
• "20" → 20
Inequality: x + 5 < 20
8. One-Step Inequality Word Problems
Steps
Step 1: Identify the variable
Step 2: Write the inequality
Step 3: Solve the inequality
Step 4: Graph the solution (if asked)
Step 5: Check your answer makes sense
Example: Movie Theater Problem
Problem: Maria has $45. Movie tickets cost $12 each. How many tickets can she buy?
Step 1: Let t = number of tickets
Step 2: Cost must be at most $45
12t ≤ 45
Step 3: Solve
12t ÷ 12 ≤ 45 ÷ 12
t ≤ 3.75
Step 4: Since can't buy 0.75 tickets
Answer: She can buy at most 3 tickets
Quick Reference: Inequalities
| Symbol | Circle on Graph | Example | Meaning |
|---|---|---|---|
| < | Open ○ | x < 5 | Less than 5 |
| > | Open ○ | x > 5 | Greater than 5 |
| ≤ | Closed ● | x ≤ 5 | At most 5 |
| ≥ | Closed ● | x ≥ 5 | At least 5 |
💡 Important Tips to Remember
✓ Inequalities have many solutions (range of values)
✓ Open circle ○ for < and > (does NOT include)
✓ Closed circle ● for ≤ and ≥ (DOES include)
✓ Arrow points to all solutions
✓ FLIP inequality when multiply/divide by negative!
✓ "At most" means ≤
✓ "At least" means ≥
✓ Test a value to check your solution
✓ Graph goes left for < or ≤
✓ Graph goes right for > or ≥
🧠 Memory Tricks & Strategies
Alligator Trick:
"The alligator eats the bigger number - its mouth opens toward the larger side!"
Circle Types:
"If there's an equal bar, the circle is closed - it includes that number!"
Flipping Sign:
"Negative times or divide? The inequality must flip to survive!"
At Least vs At Most:
"At LEAST means you can have that OR MORE (≥). At MOST means that OR LESS (≤)"
Arrow Direction:
"Greater goes right, less goes left - that's the way to graph it best!"
Master One-Variable Inequalities! < > ≤ ≥
Remember: Don't forget to flip when dividing/multiplying by negatives!