Represent and solve linear inequalities
Solve inequalities, show endpoint inclusion correctly, and graph Extended regions defined by two-variable inequalities.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What endpoint represents x < 4?
What you will learn
- Solve one-variable inequalities
- Represent strict and inclusive endpoints on a number line
- Graph and describe Extended linear-inequality regions
An inequality describes a set of values
Add or subtract equally on both sides. If you multiply or divide by a negative, reverse the sign. < and > are strict; ≤ and ≥ include equality.
Use an open number-line circle for a strict endpoint and a filled circle for an inclusive endpoint. Shade towards values that satisfy the statement.
Two-variable inequalities create regions
Draw the boundary line first: dashed for > or <, solid for ≥ or ≤. Test a point not on the boundary to choose the correct side.
A system defines the overlap of its regions. Extended candidates may use a GDC, but must still interpret boundary inclusion and describe the defining inequalities.
Pause and explain
What happens when dividing an inequality by −3?
Worked example
Solve 3x − 5 ≤ 10 and describe the boundary for y > −x + 2.
Show the worked solution
- Add 5: 3x ≤ 15, so x ≤ 5.
- Use a filled circle at 5 and shade left.
- Draw y = −x + 2 as a dashed boundary; test a point to identify the side above it.
Answer x ≤ 5; dashed line y = −x + 2 with the region above.
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 2x + 1 ≥ 7 and −3x < 12; describe endpoint styles.
Give me a hint
Isolate x and reverse the second sign when dividing by a negative.
Compare my reasoning
- 2x ≥ 6 gives x ≥ 3; equality is included.
- −3x < 12 gives x > −4.
- Use a filled circle at 3 and an open circle at −4.
x ≥ 3; x > −4.
Look for these in your work
- Operations applied on both sides
- Sign reversed
- Endpoint styles match the symbols
For y ≤ x + 2 and y > −x, describe the boundary styles and test (0, 1).
Give me a hint
Use a solid line for ≤, dashed for >, and substitute both coordinates into each inequality.
Compare my reasoning
- y = x + 2 is solid.
- y = −x is dashed.
- At (0, 1), 1 ≤ 2 and 1 > 0, so the point satisfies both.
Solid y = x + 2; dashed y = −x; (0, 1) is in the overlap.
Look for these in your work
- Boundary styles correct
- Both tests evaluated
- Overlap interpreted
A workshop has capacity at most 40 and needs at least 12 student attendees. Let x be adults and y students. Write constraints and test (20, 15).
Give me a hint
Translate the total capacity and minimum separately; attendee counts are non-negative.
Compare my reasoning
- x + y ≤ 40.
- y ≥ 12, x ≥ 0 and y ≥ 0.
- For (20, 15), total 35 ≤ 40 and 15 ≥ 12, so it is feasible.
x + y ≤ 40, y ≥ 12, x ≥ 0, y ≥ 0; (20, 15) is feasible.
Look for these in your work
- Capacity represented
- Minimum represented
- All constraints checked
Common mistakes
- Not reversing the sign after negative division
- Using a filled endpoint for a strict inequality
- Drawing a solid strict boundary
- Choosing a graph region without testing
What can you explain now?
Solve −2x < 8.
Compare with the explanation
x > −4
Divide by −2 and reverse the sign.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
When does the inequality sign reverse, and how do graph boundaries and test points identify a region?
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.