3.7PureStretch

Regions

Sometimes a solution is a 2D area rather than a set of x-values. Shading regions shows where several inequalities are all satisfied at once — the foundation of linear programming you will meet later.

25 min Video by Zeeshan Zamurred Equations and Inequalities

Edexcel AS level Maths: 3.7 RegionsWatch the full walkthrough before the notes below.

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What you'll be able to do

Draw boundary lines and curves for inequalities
Decide which side of a boundary to shade
Use solid and dashed lines correctly
Identify the region satisfying several inequalities

Boundaries: solid or dashed

First draw the boundary by replacing the inequality with “=”. Use a $solid$ line if the inequality is $\leq$ or $\geq$ (boundary included) and a $dashed$ line if it is $<$ or $>$ (boundary excluded).

\leq, \geq\Rightarrow solid line; <, >\Rightarrow dashed line

The line style records whether the boundary itself counts.

Which side to shade

Pick a test point not on the line (the origin is easiest if it is not on the boundary). If it satisfies the inequality, shade its side; if not, shade the other side.

1Test the origin

(0, 0)

: is

0 > 2 (0) + 1

? That is

0 > 1

, which is false.

2So the origin is NOT in the region — shade the other side (above the line).

Answershade above the (dashed) line

Tip — The origin is the fastest test point — provided the boundary does not pass through it.

Combining inequalities

With several inequalities, shade each one and the solution is the area satisfying $all$ of them — the overlap. It is often clearer to shade out the unwanted regions, leaving the required area clean.

Formula recap

\leq, \geq\to solid, <, >\to dashed

Line style rule.

test the origin (0, 0)

Fastest way to pick a side.

required region = overlap of all

Where every inequality holds.

Common mistakes to avoid

Using a solid line for a strict inequality (< or >).

Strict inequalities use a dashed line; the boundary is not included.

Shading the wrong side without testing.

Always test a point to confirm which side satisfies the inequality.

Key takeaways

Draw boundaries with “=”; solid for ≤/≥, dashed for </>.
Test a point (often the origin) to choose the side to shade.
The solution to several inequalities is the overlapping region.

Test yourself

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