IGCSE Maths · Algebra and Graphs · Core & Extended

Inequalities: Number Lines, Solving and Graphing Regions

Core’s entire syllabus requirement here is reading and representing inequalities on a number line — already touched on back in the Number unit. Everything beyond that — solving inequalities algebraically, graphing them as regions, and listing the inequalities that define a shape — is Extended-only, tagged throughout.

1. Number Line Notation

A quick recap: open circles mark strict inequalities (<, >) because the boundary value itself isn’t included; closed (filled) circles mark inclusive inequalities (⩽, ⩾) because it is.

Reading −3 ⩽ x < 1 −3 −2 −1 0 1 The filled circle at −3 shows it’s included (⩽); the open circle at 1 shows it’s excluded (<). The thick line between them is every value that satisfies the inequality.
Try it — Reading and Writing Inequalities
  1. Write down the inequality shown by a number line with a filled circle at 2 and an open circle at 6, shaded between them.
  2. Describe how you would draw −4 < x ⩽ 0 on a number line.
  3. List all integer values of x that satisfy −2 ⩽ x < 3.
Show answers
  1. 2 ⩽ x < 6
  2. Open circle at −4 (not included), filled circle at 0 (included), shaded between them.
  3. −2, −1, 0, 1, 2

2. Constructing and Solving Linear Inequalities Extended only

Solve exactly like a linear equation — with one crucial extra rule: flip the inequality sign whenever you multiply or divide both sides by a negative number.

Worked example — no sign flip needed Solve 3x < 2x + 4.
3x − 2x < 4 → x < 4
Worked example — sign flip required Solve 5 − 2x > 9.
−2x > 4
Divide both sides by −2, and flip the inequality: x < −2
Worked example — collecting terms on both sides Solve 4x + 3 ⩾ 7x − 6.
3 + 6 ⩾ 7x − 4x → 9 ⩾ 3x
Divide by 3 (positive, no flip needed): 3 ⩾ x, i.e. x ⩽ 3
Try it — Solving Linear Inequalities Extended
  1. Solve 5x + 2 > 17.
  2. Solve 8 − 3x ⩽ 2.
  3. Solve 6x − 5 < 3x + 10.
  4. Solve 2(x − 3) ⩾ x + 1.
Show answers
  1. 5x > 15 → x > 3
  2. −3x ⩽ −6 → divide by −3, flip: x ⩾ 2
  3. 3x < 15 → x < 5
  4. 2x − 6 ⩾ x + 1 → x ⩾ 7

3. Compound (Double) Inequalities Extended only

When a value is trapped between two limits, solve by applying the same operation to all three parts at once.

Worked example Solve −3 ⩽ 3x − 2 < 7.
Add 2 to all three parts: −1 ⩽ 3x < 9
Divide all three parts by 3: −1/3 ⩽ x < 3
Try it — Compound Inequalities Extended
  1. Solve −4 ⩽ 2x + 2 < 10.
  2. Solve 1 < 4x − 3 ⩽ 13.
  3. Solve −6 ⩽ 5 − x < 2 (careful — this one needs a sign flip somewhere).
Show answers
  1. Subtract 2: −6 ⩽ 2x < 8; divide by 2: −3 ⩽ x < 4
  2. Add 3: 4 < 4x ⩽ 16; divide by 4: 1 < x ⩽ 4
  3. Subtract 5: −11 ⩽ −x < −3; divide by −1 and flip both inequalities (and reverse the order): 3 < x ⩽ 11

4. Graphing Linear Inequalities in Two Variables Extended only

Conventions: a broken line for a strict inequality (<, >), a solid line for an inclusive one (⩽, ⩾), and shading over the unwanted region, leaving the wanted region clear — unless a question tells you otherwise.

Two simple examples x y x < 1 1 x y y ⩾ 1 1 Left: x < 1 — a broken (dashed) vertical line at x = 1, with the unwanted region (x ⩾ 1) shaded. Right: y ⩾ 1 — a solid horizontal line at y = 1, with the unwanted region (y < 1) shaded. In both cases, the clear (unshaded) area is where the inequality is actually satisfied.

The test point method

Not sure which side of a line to shade? Pick any point not on the line — the origin (0, 0) is usually easiest — and substitute it into the inequality. If it’s true, that point’s side is the wanted region; if false, shade that side.

Worked example — a region defined by three inequalities Show the region that satisfies x ⩾ 0, y ⩾ 0, and x + y ⩽ 6.
Test the origin in x + y ⩽ 6: 0 + 0 = 0 ⩽ 6 — true, so the origin’s side of the line x + y = 6 is the wanted side. x y wanted region 6 6 The shaded triangle here is the wanted region (highlighted for clarity in this diagram) — bounded by the y-axis (x ⩾ 0), the x-axis (y ⩾ 0), and the solid line x + y = 6. On your own exam diagrams, remember the convention is the other way round: shade the unwanted area outside the region, unless the question says otherwise.
Try it — Graphing Regions Extended
  1. Describe the line style and shading needed to show x ⩽ 4.
  2. Describe the line style and shading needed to show y > −2.
  3. Use the test point (0,0) to decide which side of the line x + y = 3 satisfies x + y < 3.
Show answers
  1. A solid vertical line at x = 4, with the unwanted region (x > 4) shaded.
  2. A broken horizontal line at y = −2, with the unwanted region (y ⩽ −2) shaded.
  3. At (0,0): 0 + 0 = 0 < 3 — true, so the origin's side (the side containing the origin) satisfies the inequality.

5. Listing Inequalities That Define a Region Extended only

This is the reverse of graphing — given a shaded/highlighted region and its boundary lines, write down every inequality needed to describe it exactly, including whether each boundary is included (solid) or not (broken).

Worked example — using the triangle from Section 4 Write down the three inequalities that define the triangular region bounded by the x-axis, the y-axis, and the solid line x + y = 6, shown in the diagram above.
The region sits above the x-axis: y ⩾ 0
The region sits to the right of the y-axis: x ⩾ 0
The region sits on the origin’s side of the solid line x + y = 6: x + y ⩽ 6
Try it — Listing Inequalities Extended
  1. A region is bounded by a solid vertical line at x = 5 (region to the left), a solid horizontal line at y = 2 (region above), and the origin is not in the region. Write down the two inequalities.
  2. A triangular region is bounded by x = 0, y = 0, and the broken line x + y = 4, with the region below/left of that line (not including the line itself). Write down the three inequalities.
Show answers
  1. x ⩽ 5 and y ⩾ 2
  2. x ⩾ 0, y ⩾ 0, and x + y < 4
Common mistakes to watch for
  • Forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number.
  • Using a solid line for a strict inequality, or a broken line for an inclusive one — check the symbol carefully (⩽/⩾ = solid, </> = broken).
  • Shading the wanted region instead of the unwanted region (or vice versa, if a question specifically asks to shade the wanted region — always re-read what’s being asked).
  • In a compound inequality, only applying an operation to one part instead of all three.
  • Reversing the direction of a compound inequality incorrectly after multiplying/dividing by a negative — both inequality symbols must flip, and the order of the expression effectively reverses too.

6. Quick-Fire Challenge Round

True or False?
  1. An open circle on a number line means the value is included.
  2. Dividing both sides of an inequality by −4 requires flipping the inequality sign.
  3. A strict inequality (< or >) is drawn with a solid line on a graph.
  4. The origin is always a safe choice for the test point method.
Show answers
  1. False — an open circle means the value is not included; a filled circle means it is.
  2. True.
  3. False — strict inequalities use a broken (dashed) line; inclusive inequalities use a solid line.
  4. False — it’s usually the easiest choice, but only if the line doesn’t pass through the origin itself; otherwise pick a different point.
Spot the Error
  1. “Solve −3x > 12: dividing both sides by −3 gives x > −4.”
  2. “To show y ⩽ 2 on a graph, I drew a broken line at y = 2.”
  3. “Solve −5 ⩽ 2x < 9: dividing by 2 (positive, no flip) gives −5 ⩽ x < 9.”
Show answers
  1. Dividing by a negative number requires flipping the inequality: −3x > 12 → x < −4, not x > −4.
  2. ⩽ is an inclusive inequality, so it needs a solid line, not a broken one.
  3. Every part of the compound inequality must be divided by 2, including both bounds: −5/2 ⩽ x < 9/2, not −5 ⩽ x < 9 (the bounds weren't actually divided).

7. Mixed Practice — Bringing It All Together

Core & Extended
  1. List the integers that satisfy −5 < x ⩽ 0.
  2. Draw a description of how −1 ⩽ x < 4 would look on a number line.
Show answers
  1. −4, −3, −2, −1, 0
  2. Filled circle at −1, open circle at 4, shaded between them.
Extended-level mixed set Extended only
  1. Solve 7 − 2x ⩽ 1.
  2. Solve −2 < 4x + 6 ⩽ 18.
  3. Describe the line style and shaded region for x + y > 4.
  4. Write down the inequalities defining a region bounded by a solid line at x = 0 (region to the right), a broken line at y = 5 (region below), and the x-axis (region above, solid).
Show answers
  1. −2x ⩽ −6 → x ⩾ 3
  2. Subtract 6: −8 < 4x ⩽ 12; divide by 4: −2 < x ⩽ 3
  3. A broken line at x + y = 4, with the region containing the origin (x + y < 4, the “unwanted” side per convention) shaded, leaving x + y > 4 clear.
  4. x ⩾ 0, y < 5, y ⩾ 0

Practice at Home

A large set again — work through it steadily.

Part A — Number lines (Core & Extended)
  1. List the integers that satisfy −3 ⩽ x < 2.
  2. Describe how 0 < x ⩽ 5 would look on a number line.
  3. Write down the inequality shown by a number line with an open circle at −2 and a filled circle at 3, shaded between.
Part B — Solving linear inequalities Extended
  1. Solve 4x − 7 > 9.
  2. Solve 10 − 3x ⩾ 1.
  3. Solve 5x + 4 < 2x − 5.
  4. Solve 3(x + 2) ⩽ 2(x + 8).
Part C — Compound inequalities Extended
  1. Solve −6 ⩽ 3x + 3 < 15.
  2. Solve 2 < 5x − 3 ⩽ 17.
  3. Solve −4 ⩽ 8 − 2x < 4.
Part D — Graphing and listing regions Extended
  1. Describe the line style and shading for y ⩾ −1.
  2. Describe the line style and shading for x < 3.
  3. Use the test point (0,0) to determine which side of x + 2y = 8 satisfies x + 2y < 8.
  4. A region is bounded by x = 0 (solid, region to the right), y = 0 (solid, region above), and x + y = 8 (solid, region containing the origin). Write down the three inequalities.
Part E — Applied / challenge
  1. A number x satisfies both 2x + 1 > 9 and 3x − 4 < 20. Find the range of possible values of x.
  2. A school hall can seat at most 250 people. If s is the number of students and t is the number of teachers, write an inequality connecting s and t.
  3. Explain, using an example, why the inequality sign must flip when both sides of an inequality are multiplied by a negative number, but not when both sides are multiplied by a positive number.

Key Vocabulary Recap

Strict inequality — < or >, where the boundary value is not included
Inclusive inequality — ⩽ or ⩾, where the boundary value is included
Compound inequality — two inequalities combined, e.g. −3 ⩽ x < 5
Broken line — used on a graph for a strict inequality
Solid line — used on a graph for an inclusive inequality
Test point — a point used to check which side of a line satisfies an inequality

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) section C2.6 (Core subject content) and section E2.6 (Extended subject content) — Inequalities. Core’s requirement is limited to representing and interpreting inequalities on a number line. Extended additionally covers constructing and solving linear inequalities, compound inequalities, graphing linear inequalities in two variables, and listing the inequalities that define a given region — all marked “Extended only” above. Linear programming problems are not included at either tier.

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