x > 3 · x < −1 · x ≥ 5 · x ≤ 0 Greater than · Less than · At least · At most
Solving
2x ≤ 10 → x ≤ 5 Divide both sides (same direction)
Flip Rule ⚠️
−3x > 12 → x < −4 Divide by negative → flip the symbol!
Which symbol fits? Click to reveal! 🍬
What you'll learn:
Inequality symbols: <, >, ≤, ≥ and their meanings
Showing inequalities on a number line (open/closed circles)
Solving one-step inequalities: x + 3 > 7, 2x ≤ 10
Solving two-step inequalities: 3x − 2 > 10
Flipping the inequality when dividing by a negative number
Integer solutions: list all integers satisfying −2 < x ≤ 5
Writing inequalities from word problems
📖 Learn: Inequalities
Part 1: Inequality Symbols
An inequality compares two values and tells us one is bigger, smaller, or equal to the other.
Symbol
Meaning
Example
Read as
<
Less than
x < 5
"x is less than 5"
>
Greater than
x > −2
"x is greater than −2"
≤
Less than or equal to
x ≤ 8
"x is at most 8"
≥
Greater than or equal to
x ≥ 0
"x is at least 0"
💡 Memory trick: the open end of the symbol always points to the bigger number. Think of it as a hungry mouth eating the larger value: 3 < 7 — the mouth opens toward 7.
Part 2: Number Lines
We show inequalities on a number line using circles and arrows:
Open circle ○ — the value is not included (strict inequality: < or >)
Closed/filled circle ● — the value is included (≤ or ≥)
The arrow points in the direction of all values that satisfy the inequality
Part 3: Solving One-Step Inequalities
Solve an inequality the same way you solve an equation — do the same operation to both sides.
Solve x + 3 > 7: subtract 3 from both sides → x > 4
Solve 2x ≤ 10: divide both sides by 2 → x ≤ 5
Solve x − 4 ≥ −1: add 4 to both sides → x ≥ 3
💡 The inequality symbol stays the same as long as you add, subtract, multiply or divide by a positive number.
Part 4: Solving Two-Step Inequalities + The Flip Rule
Use two steps — same as solving two-step equations.
Solve 3x − 2 > 10: add 2 → 3x > 12 → divide by 3 → x > 4
Solve 2x + 5 ≤ 11: subtract 5 → 2x ≤ 6 → divide by 2 → x ≤ 3
⚠️ THE FLIP RULE
When you multiply or divide both sides by a negative number, you must flip the inequality symbol.
Example: −2x < 8 Divide by −2: x > −4 (symbol flips from < to >!)
Example: −3x ≥ 15 Divide by −3: x ≤ −5 (symbol flips from ≥ to ≤!)
💡 Think: if −1 < 3, multiply both sides by −1 → 1 > −3 (the order reverses on the number line!)
Part 5: Integer Solutions & Word Problems
Sometimes we need integer (whole number) solutions only.
Find all integers satisfying −2 < x ≤ 5 x can be: −1, 0, 1, 2, 3, 4, 5 (−2 is excluded because it's strict <; 5 is included because of ≤)
Find all integers satisfying 1 ≤ x < 6 x can be: 1, 2, 3, 4, 5
Word Problems → Inequalities:
"A bag must weigh more than 2 kg but no more than 10 kg." → 2 < w ≤ 10
"You need at least 40 marks to pass." → m ≥ 40
"The roller coaster requires riders to be under 130 cm." → h < 130
💡 Key words: more than → > · less than → < · at least / minimum → ≥ · at most / maximum → ≤
💡 Worked Examples
Example 1: Equation vs Inequality — Side by Side
See how the approach is similar, but the solution is different!
⚖️ Equation: 2x + 3 = 11
Step 1: Subtract 3 from both sides 2x = 8
Step 2: Divide both sides by 2 x = 4
Solution: x = 4 (one value)
📐 Inequality: 2x + 3 > 11
Step 1: Subtract 3 from both sides 2x > 8
Step 2: Divide both sides by 2 x > 4
Solution: x > 4 (infinite values: 5, 6, 7, …)
Key difference: an equation gives one answer; an inequality gives a range of answers.
Example 2: One-Step Inequality (with number line)
Solve x + 3 > 7 and show on a number line.
Step 1: Subtract 3 from both sides: x > 4
Solution: x > 4 — open circle at 4, arrow pointing right
Example 3: Two-Step Inequality
Solve 3x − 2 > 10
Step 1: Add 2 to both sides: 3x > 12
Step 2: Divide both sides by 3: x > 4
Solution: x > 4
Check: try x = 5 → 3(5) − 2 = 13 > 10 ✓ try x = 3 → 3(3) − 2 = 7, not > 10 ✗
Example 4: Negative Division — The Flip!
Solve −2x < 8
Step 1: Divide both sides by −2
⚠️ Dividing by −2 (a negative) → flip the symbol: < becomes >
Step 2: x > −4
Solution: x > −4
Verify with x = 0: −2(0) = 0, and 0 < 8 ✓ 0 > −4 ✓
📊 Interactive Number Line Plotter
Enter an inequality and watch it appear on the number line!
Examples: x > 3 · x <= -1 · x >= -2 · x < 5
🍬 Integer Solution Clicker
Click all the integers that satisfy the inequality shown. Correct ones turn green, wrong ones flash red!
Loading…
Quick-plot these inequalities:
✏️ Exercise 1: Write the Inequality
Look at each number line and write the inequality it represents. Use <, >, ≤, or ≥.
✏️ Exercise 2: Draw the Inequality
For each inequality, choose the correct number line representation by clicking the right option.
✏️ Exercise 3: Solve One-Step Inequalities
Solve each inequality. Write your answer in the form x > a, x < a, x ≥ a, or x ≤ a (e.g. "x > 4").
✏️ Exercise 4: Two-Step Inequalities & The Flip Rule
Solve each inequality step by step. Watch for the flip rule when dividing by a negative!
⚠️ The Flip Rule — Step Builder
🚨 Dividing by a NEGATIVE — the symbol FLIPS! < → >
✏️ Exercise 5: Integer Solutions & Word Problems
Part A: List the integers. Part B: Write an inequality for each word problem.
📝 Practice Questions
🏆 Challenge Questions
Harder problems — take your time and show your working!