Solve and show linear and compound inequalities
Read strict and inclusive intervals on a number line · Reverse an inequality when multiplying or dividing by a negative · Solve a compound inequality and apply integer restrictions
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Linear equations
- Order symbols
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Read strict and inclusive intervals on a number line
- Reverse an inequality when multiplying or dividing by a negative
- Solve a compound inequality and apply integer restrictions
Intervals and endpoints
An open circle excludes an endpoint (< or >); a filled circle includes it (≤ or ≥). Shade all real numbers satisfying the condition. A compound statement such as −2 ≤ x < 3 means both bounds hold: include −2 but exclude 3.
Order-preserving and order-reversing steps
Adding the same amount to both sides preserves order. Multiplication or division by a positive number also preserves it; a negative number reverses it. For −2x > 6, division by −2 gives x < −3.
In a compound inequality, perform the same addition or positive scaling on every part. If a problem counts whole objects, first solve the real inequality then keep the allowed integers.
Sources for this lesson
Pause and explain
Solving −2x > 6 gives which inequality?
Worked example
Solve −5 ≤ 2x−1 < 7 and describe its number line.
Show the worked solution
- Add 1 to every part: −4 ≤ 2x < 8.
- Divide by positive 2: −2 ≤ x < 4.
- Shade between a filled circle at −2 and an open circle at 4.
Answer −2 ≤ x < 4.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Solve 3x+2 ≤ 11 and describe the endpoint.
Give me a hint
Undo addition, then positive multiplication.
Compare my reasoning
- 3x ≤ 9, so x ≤ 3.
- Use a filled circle at 3 and shade left.
x ≤ 3; filled endpoint, shaded left.
Look for these in your work
- Retain equality
- Show all real solutions
Solve 5−2x < 13.
Give me a hint
After subtracting 5, divide by a negative.
Compare my reasoning
- −2x < 8.
- x > −4; use an open endpoint at −4.
x > −4.
Look for these in your work
- Reverse the inequality
- Exclude equality
Solve 1 < 2n+1 ≤ 9 for integer n.
Give me a hint
Solve the real interval first.
Compare my reasoning
- 0 < 2n ≤ 8, so 0 < n ≤ 4.
- Integer solutions are 1, 2, 3, 4.
n = 1, 2, 3 or 4.
Look for these in your work
- Treat all parts consistently
- Apply the integer context
Common mistakes
- Reverse the sign for negative division.
- Strict and inclusive endpoints are different.
What can you explain now?
Solving −2x > 6 gives which inequality?
Compare with the explanation
x < −3.
Dividing by a negative reverses order.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Solving −2x > 6 gives which inequality?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.