B5PureStretch

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.

30 min Algebra and Functions

The big picture

An inequality in two variables describes a region of the plane. Draw each boundary, use a test point to choose the correct side, and keep only the overlap satisfying every condition. A strict inequality excludes its boundary, commonly indicated with a dashed line; a non-strict inequality includes it. For a curved boundary, “above” refers to the y-value at a fixed x.

Edexcel AS level Maths: 3.7 RegionsCurated by Kepler, not produced by us. Watch it, then work through 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 $\textbf{solid}$ line if the inequality is $\le$ or $\ge$ (boundary included) and a $\textbf{dashed}$ line if it is $<$ or $>$ (boundary excluded).

\le, \ge \Rightarrow \text{solid line}; \qquad <, > \Rightarrow \text{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.

Example 1Which side of

y = 2x + 1

satisfies

y > 2x + 1

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 $\textbf{all}$ of them — the overlap. It is often clearer to shade out the unwanted regions, leaving the required area clean.

Intersect inequalities rather than combining answers loosely

An inequality in two variables describes a region of the plane. Draw each boundary, use a test point to choose the correct side, and keep only the overlap satisfying every condition.

A strict inequality excludes its boundary, commonly indicated with a dashed line; a non-strict inequality includes it. For a curved boundary, “above” refers to the y-value at a fixed x.

Example 1Describe the region satisfying

y\geq x^2

and

y\leq4

1Find intersections:

x^2=4

, so x = -2 and x = 2.

2Between these x-values the parabola lies below the horizontal line.

3The required region is

-2\leq x\leq2

and

x^2\leq y\leq4

4Both boundaries are included because the inequalities are non-strict.

AnswerThe closed region above the parabola and below y = 4, including both boundary curves.

Think like an examiner

Write the defining equations and show the substitution before evaluating.
Check the domain, units and interpretation of the final result.

Common misconceptions

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 results to remember

\le, \ge \to \text{solid}, \quad <, > \to \text{dashed}

Line style rule.

\text{test the origin } (0,0)

Fastest way to pick a side.

\text{required region} = \text{overlap of all}

Where every inequality holds.

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

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

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