Linear Inequalities in Two Variables · Form 4
What the common region of a system means
When several inequalities apply at once, the common region is the overlap where every condition holds together, the set of all points that satisfy all of the inequalities at the same time.
The overlap of every inequality
Each inequality shades its own region, and the common region is only where all of those shaded areas overlap. A point inside it satisfies every inequality at the same time, so it stands for a situation that meets all the conditions, for example x tables and y chairs that fit both the budget and the room.
Any point outside fails at least one inequality and does not belong to the common region.
Why it matters
Real situations arrive with several limits at once, money, time, seats, and it is hard to hold them all in your head. The common region gathers every point that breaks none of them, turning a long list of worded conditions into a single picture you can see.
Reading a worded problem into inequalities and then into one shaded region is the heart of what this chapter is asking you to understand.
Hidden conditions and a common error
Often x ≥ 0 and y ≥ 0 are hidden conditions, because a count such as the number of buses cannot be negative, keeping the region in the first quadrant. A common error is treating any one shaded area as the answer, when the common region is only the part shared by all of them; if the inequalities never overlap, there is no common region at all.
When you can only have whole objects, remember that only points with whole-number coordinates make real sense.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I find the common region when there are two or three inequalities at once?
Draw every boundary line on the same axes, shade (or leave unshaded) according to each inequality, and find where all the required sides overlap. The common region is the single area satisfying every inequality at once, often the examiner asks you to shade only this overlap, leaving the rest of the graph blank.
Does it matter if the common region looks small or oddly shaped?
No, the common region can be a small triangle, an unbounded strip, or even a single point; its shape depends entirely on the inequalities given. What matters in the exam is that every boundary and shading direction is correct, since the region's shape is simply a consequence of those conditions.
What is a common mistake when there are several lines and inequalities?
Students often mix up which shading belongs to which line, especially when lines cross close together, leading to a wrongly shaped common region. Label each line clearly, for example with its equation, as you draw it, and shade one inequality at a time before combining them.