How to use this lesson
1. Complete the warm-up questions from memory
2. Study the worked examples carefully
3. Try the practice problems: Green → Amber → Red
4. Check your answers and complete the mastery checklist
2. Study the worked examples carefully
3. Try the practice problems: Green → Amber → Red
4. Check your answers and complete the mastery checklist
🔥
Warm-Up: Retrieval Practice⏱ Quick Recall
These questions bridge Unit 4 and prepare you for inequalities.Q1 What does x > 3 mean?
Q2 Solve: 4x = 20
Q3 True or False: Dividing by a negative reverses the inequality sign.
Q4 🔄 RECALL Expand: 3(x − 4)
Lesson Progress
✓ Warm-up — complete
→ Worked examples & key concepts
· Tiered practice (Green → Amber → Red)
· Mistake Detective & Examiner Lens
· Mastery checklist
💡
The Big Idea: Boundaries on a Number LineAn inequality describes a range of values, not just one answer.
< less than
> greater than
≤ less than or equal
≥ greater than or equal
Key rule: Solve like an equation, BUT if you multiply or divide by a negative number, REVERSE the inequality sign.
Open circle = NOT included (<, >). Filled circle = included (≤, ≥).
📖
Core DefinitionsInequality
A comparison using <, >, ≤, ≥
Strict Inequality
Uses < or > (boundary not included)
Inclusive Inequality
Uses ≤ or ≥ (boundary is included)
Solution Set
All values that satisfy the inequality
Number Line
Visual display using circles and arrows
Integer Solutions
Whole numbers that satisfy the inequality
⚠️ Critical Rule
When multiplying or dividing by a negative number, REVERSE the inequality sign.✏️
Worked ExamplesWorked Example 01One-Step (Positive Coefficient)
Solve: 5x ≤ 20
1
Divide both sides by 5:
x ≤ 4✓ Answer
x ≤ 4
Closed circle at 4, arrow to the left.
Worked Example 02One-Step (Negative Coefficient)CRITICAL
Solve: −3x > 9
1
Divide both sides by −3 — REVERSE the sign:
x < −3✓ Answer
x < −3
⚠️ Dividing by negative reversed > to <. Open circle at −3, arrow left.
Worked Example 03Two-Step (Positive)
Solve: 2x + 6 < 14
1
Subtract 6:
2x < 82
Divide by 2:
x < 4✓ Answer
x < 4
Open circle at 4, arrow left.
Worked Example 04Two-Step (Negative Coefficient)KEY SKILL
Solve: −5x + 10 < 0
1
Subtract 10:
−5x < −102
Divide by −5, REVERSE:
x > 2✓ Answer
x > 2
Worked Example 05Integer Solutions
Solve: 2 < x ≤ 6 (list integer solutions)
1
x must be greater than 2 (not including 2, because <)
2
x must be less than or equal to 6 (including 6, because ≤)
Open at 2, closed at 6. Integers: 3, 4, 5, 6
✓ Integer Solutions
3, 4, 5, 6
📐 Part 2 — Solving Compound Inequalities
Sometimes you need to solve an inequality with three parts. Apply the same operation to all parts at once.Worked Example 06Solving a Compound InequalityNEW
Solve: −1 < 2x + 3 ≤ 9
1
Subtract 3 from all three parts:
−4 < 2x ≤ 62
Divide all three parts by 2:
−2 < x ≤ 3✓ Solution
−2 < x ≤ 3
💡 Integer solutions: −1, 0, 1, 2, 3 (not −2 because < means strictly greater than)
Worked Example 07Compound Inequality (Harder)
Solve: 1 ≤ 3x − 2 < 13 and list integer solutions
1
Add 2 to all parts:
3 ≤ 3x < 152
Divide by 3:
1 ≤ x < 5✓ Solution
1 ≤ x < 5 → integers: 1, 2, 3, 4
✅
Method Checklist📋 Solving Inequalities
✓
Solve like an equation✓
Reverse sign when dividing/multiplying by negative✓
Represent solution on number line✓
Open circle for < or >✓
Closed circle for ≤ or ≥🧮
Arithmetic Support−10 ÷ −2 = 5
Dividing by negative reverses the sign
Open circle = NOT included (<, >)
Closed circle = IS included (≤, ≥)
💪
Tiered Practice🔓 How It Works
Tap any card to reveal the full working and answer.0 / 13
Problems revealed
Tap problems to reveal working and answers
🟢 Green — Core Skills
🟢 Green · TYPE YOUR ANSWER
Solve: 3x > 12
✍️ Type Your Answer
Solve: 2x > 10
Divide both sides by 2: x > 5
✓ x > 5
Green · Q1
Solve: 4x ≥ 16
Divide both sides by 4:
4x ÷ 4 ≥ 16 ÷ 4
✓ x ≥ 4
Tap to reveal answer
Green · Q2
Solve: x − 3 > 5
Add 3 to both sides:
x − 3 + 3 > 5 + 3
✓ x > 8
Tap to reveal answer
Green · Q3
Solve: −2x < 8
Divide by −2, REVERSE sign:
x > −4
✓ x > −4
Tap to reveal answer
🟡 Amber — Multi-Step
🟡 Amber · TYPE YOUR ANSWER
Solve: 5x − 2 ≤ 13
Amber · Q4
Solve: 3x + 4 ≤ 19
Step 1: Subtract 4 → 3x ≤ 15
Step 2: Divide by 3 → x ≤ 5
✓ x ≤ 5
Tap to reveal answer
Amber · Q5
Solve: 6x − 5 > 13
Step 1: Add 5 → 6x > 18
Step 2: Divide by 6 → x > 3
✓ x > 3
Tap to reveal answer
Amber · Q6
Solve: −4x + 12 ≥ 0
Step 1: Subtract 12 → −4x ≥ −12
Step 2: Divide by −4, REVERSE → x ≤ 3
✓ x ≤ 3
Tap to reveal answer
Amber · Q7
List integers: 5 < x ≤ 10
x > 5 (not equal) and x ≤ 10 (can equal)
Integers: 6, 7, 8, 9, 10
✓ 6, 7, 8, 9, 10
Tap to reveal answer
🔴 Red — Exam Style
🔴 Red · TYPE YOUR ANSWER
Solve: −4x < 20
Red · 2 marks · Q8
Solve fully: −2x + 8 ≥ 4
Step 1: Subtract 8 → −2x ≥ −4
Step 2: Divide by −2, REVERSE → x ≤ 2
✓ x ≤ 2
Tap to reveal answer
Red · 2 marks · Q9
Solve fully: 7x − 3 < 18
Step 1: Add 3 → 7x < 21
Step 2: Divide by 7 → x < 3
✓ x < 3
Tap to reveal answer
Red · 2 marks · Q10
Solve and represent: −3x ≤ 9
Divide by −3, REVERSE sign:
x ≥ −3
Closed circle at −3, arrow right
✓ x ≥ −3
Tap to reveal answer
🔴 Compound Inequalities
Red · 2 marks · Q11
Solve: 3 < 2x + 1 ≤ 11
Subtract 1 from all parts: 2 < 2x ≤ 10
Divide by 2: 1 < x ≤ 5
Integers: 2, 3, 4, 5
✓ 1 < x ≤ 5
Tap to reveal answer
Red · 3 marks · Q12
Solve: −5 ≤ 4x − 1 < 11 List the integer solutions.
Add 1: −4 ≤ 4x < 12
Divide by 4: −1 ≤ x < 3
Integers: −1, 0, 1, 2
✓ −1 ≤ x < 3 Integers: −1, 0, 1, 2
Tap to reveal answer
🔍
Mistake Detective⚠️ Sign reversal errors are extremely common
Study each one carefully.❌ Wrong
−4x ≤ 12
−4x ÷ −4 ≤ 12 ÷ −4
x ≤ −3
Did NOT reverse the sign when dividing by negative!
✅ Correct
−4x ≤ 12
−4x ÷ −4 ≥ 12 ÷ −4
x ≥ −3
❌ Wrong
x ≤ 5 shown with open circle ○
Inclusive inequality (≤) must use closed circle
✅ Correct
x ≤ 5 shown with closed circle ●
Closed = value IS included
🎓
Examiner Lens🎓 To Gain Full Marks
- Show full balance steps (both sides of inequality)
- Write “reverse sign” clearly when dividing by a negative
- Represent correctly with circle and arrow on number line
- Include number line when asked in the question
- State whether circle is open or closed
🏆
Mastery Checklist🏆 Unit 5 Mastery
✓
I can solve one-step inequalities confidently
✓
I can solve two-step inequalities with positive coefficients
✓
I reverse the inequality sign when dividing/multiplying by negative
✓
I can represent solutions correctly on number lines
✓
I can list integer solutions when required
✓
I can solve compound inequalities like −1 < 2x + 3 ≤ 9
Nice work — you’ve completed Unit 5!
5 units remaining. Consistent practice is what turns Grade 3 into Grade 4/5. Keep going.
Ready for Unit 6?
If you can do these 3 things confidently, you're ready to move on:
Your Grade 3→5 Rescue Pack
10 units · Algebra, Graphs & Coordinate Geometry
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👨👩👧 For Parents
Your child has completed Unit 5 of a structured GCSE Foundation Maths revision programme covering Algebra and Graphs — key topics on every Foundation paper.
✔ AQA, Edexcel & OCR aligned
✔ 190+ questions & 70 worked examples
✔ Step-by-step explanations
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