Linear Inequalities in Two Variables · Form 4
Linear Inequalities in Two Variables: Revision Notes
A tight revision summary of Linear Inequalities in Two Variables for SPM Mathematics, the key ideas, the formulae the exam gives you, and what to focus on before the paper.
The big idea
Instead of a line, an inequality like y ≥ 2x + 1 describes a whole region of the graph. This chapter teaches you to draw that region, combine several inequalities into one feasible area, and read real constraints, budgets, limits, minimums, into mathematical form.
Key ideas to revise
- Solid vs dashed boundary. Use a solid line for ≤ or ≥ (the boundary is included) and a dashed line for < or > (it is not).
- Which side to shade. Test a point not on the line (often the origin) to decide which side satisfies the inequality.
- The feasible region. When several inequalities are shaded together, the region satisfying all of them is where the answer lives.
One mini-example for each key idea
- Boundary type: for y ≥ 2x + 1 the sign is ≥, so points on the line count, rule the boundary y = 2x + 1 as a SOLID line. Result: solid.
- Which side to shade: for y < x + 3, test the origin (0, 0): 0 < 0 + 3 gives 0 < 3, which is true, so shade the side of the line that contains (0, 0).
- The feasible region: with x ≥ 0, y ≥ 1 and x + y ≤ 5 all shaded together, test (1, 2): 1 ≥ 0 is true, 2 ≥ 1 is true, and 1 + 2 = 3 ≤ 5 is true, so (1, 2) lies in the region satisfying all three.
Your pre-paper checklist for this chapter
- Re-derive only the one rule you truly own: ≤ or ≥ → solid line, < or > → dashed line. Jot it in the margin before you draw a single boundary.
- Know what the exam hands you versus what you must supply: it gives the inequalities (or the words to form them) and the grid; you supply the ruled lines, the shading, and a labelled region R.
- For every boundary, find two easy points, usually the x-intercept (set y = 0) and the y-intercept (set x = 0), plot both, then rule a straight line; a freehand wobble costs accuracy marks.
- After shading, pick ONE point clearly inside your final region and confirm it satisfies every inequality; if it fails even one, you have shaded a wrong side and must fix it now.
- The one habit that saves marks: label the overlap R and keep it visually clean, so the marker sees at a glance exactly which area you mean.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How much of my revision time should go on drawing versus algebra?
Most marks here reward an accurate graph, so give the larger share to ruling straight boundaries and clean, consistent shading. Keep a smaller slice for rearranging an inequality to make y the subject, because one sign slip there quietly ruins an otherwise perfect diagram.
Practise both, but weight the drawing.
Should I shade the region I want, or the region I don't want?
Either works, provided you are consistent and state your convention. Many students shade the unwanted side of each line, so the required region is left blank and the overlap is easy to see.
Whichever you pick, add a short key and clearly label the final region R so the marker is never left guessing.
What is the fastest reliable order to draw three inequalities?
Draw all three boundary lines first, deciding solid or dashed for each as you go, before shading anything. Then shade line by line, testing the origin each time it is off the line.
Finish by reading the overlap, testing one inside point, and labelling it R. Lines first, shading second stops most muddled diagrams.