Inequality regions and boundary lines
Use half-planes to show all points that satisfy several linear inequalities.
Extended only.
Before you begin
- Plot coordinates and draw a straight line from two points.
- Interpret strict and inclusive inequalities.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which equation represents a vertical line through x = 2?
What you will learn
- Draw solid or broken inequality boundaries.
- Use a test point to identify the permitted half-plane.
- Describe a region with simultaneous inequalities.
Draw the equality as the boundary
For y > 2x + 1, first draw the line y = 2x + 1 through points such as (0, 1) and (1, 3). Use a broken line because the equality is excluded. For y ≥ 2x + 1, use a solid line because the boundary points are allowed. Horizontal and vertical inequalities use y = k and x = k as their boundaries.
Substitute a convenient point that is not on the boundary to decide which half-plane is permitted. For (0, 0), the inequality 0 > 1 is false, so the permitted side is the side opposite the origin. A point on the boundary cannot distinguish the two half-planes.
Show the unwanted region according to the question
In the Cambridge convention, shade the unwanted half-plane unless the question directs otherwise. For y ≥ 2x + 1, shade below the line and leave the region above unshaded; the solid boundary remains permitted. State which convention you are following so a shaded graph is not misread.
For x ≥ 0, y ≥ 0 and x + y ≤ 6, the allowed region lies in the first quadrant below the solid line x + y = 6. The three boundary intersections form a triangle with vertices (0, 0), (6, 0) and (0, 6). The region includes interior points as well as its boundaries.
Read a region back into inequalities
If a region is to the right of x = 1, above y = 2 and below x + y = 7, its inequalities are x ≥ 1, y ≥ 2 and x + y ≤ 7 when all lines are solid. Change an inclusive sign to a strict sign when its boundary is broken. Test an interior point such as (2, 3) against every condition.
A pair must satisfy every inequality to belong to the common region. The fact that it satisfies two of three is insufficient. If a problem restricts the coordinates to integers, check those discrete points after drawing the continuous region. This lesson describes regions; it does not introduce linear-programming optimisation.
Pause and explain
For y < x + 1, which boundary and side are permitted?
Worked example
Describe the region x ≥ 0, y ≥ 0, x + y ≤ 6, and decide whether (2, 4) and (3, 4) belong.
Show the worked solution
- Draw both axes and the line joining (6, 0) to (0, 6), all solid.
- The permitted region is the first-quadrant triangle on or below x + y = 6; shade outside it.
- (2, 4) is on the included boundary. (3, 4) fails because 3 + 4 = 7 > 6.
Answer Triangle with vertices (0, 0), (6, 0), (0, 6); (2, 4) belongs, (3, 4) does not
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.
Draw the region x < 3 and y ≥ 1, using unwanted-region shading.
Give me a hint
The two boundaries are one vertical and one horizontal line.
Compare my reasoning
- Use a broken vertical line x = 3 and a solid horizontal line y = 1.
- Allowed points are left of x = 3 and on or above y = 1.
- Shade the right and lower unwanted regions; (2, 2) checks the remaining overlap.
Left of x = 3, on or above y = 1; x boundary excluded
Look for these in your work
- Chose each line style from the sign.
- Left the common allowed region unshaded.
A region is right of solid x = 1, above broken y = 2, and below solid x + y = 8. Write its inequalities.
Give me a hint
Solid includes equality; broken excludes it.
Compare my reasoning
- The right-hand condition is x ≥ 1.
- The height condition is y > 2.
- The upper boundary gives x + y ≤ 8; all three conditions must hold.
x ≥ 1, y > 2, x + y ≤ 8
Look for these in your work
- Interpreted the permitted side of each boundary.
- Recorded strictness independently for each line.
A learner chooses non-negative integer counts x of maths tasks and y of reading tasks. A session allows at most six tasks and requires at least two reading tasks. Describe the region and list possible y when x = 3.
Give me a hint
Combine the total limit with positivity and the reading minimum.
Compare my reasoning
- The constraints are x ≥ 0, y ≥ 2 and x + y ≤ 6, with integer counts.
- At x = 3, the total limit gives y ≤ 3.
- Combined with y ≥ 2, the choices are y = 2 or 3.
x ≥ 0, y ≥ 2, x + y ≤ 6; at x = 3, y = 2 or 3
Look for these in your work
- Translated all three requirements.
- Applied the integer restriction after intersecting the conditions.
Common mistakes
- Using a solid line for a strict condition.
- Shading the allowed side when the question asks for unwanted-region shading.
- Checking only one inequality for a point.
What can you explain now?
Does (2, 3) satisfy x > 1, y ≥ 2 and x + y < 5?
Compare with the explanation
No
The first two conditions hold, but 2 + 3 = 5 is excluded by the final strict inequality.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Draw a region with one strict boundary. Give a point inside, outside and on that boundary, then justify whether each is allowed.
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources