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 Equations and Inequalities

Edexcel AS level Maths: 3.7 RegionsCurated by Kepler, not produced by us. Watch it 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 $\textbf{solid}$ 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).

Answer

shade 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.

Formula recap

\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.

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

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

The inequality $y \ge x + 2$ uses which boundary line?
- Solid
- Dashed
- No line
- Dotted and solid
Show answer and explanation
Answer: Solid
≥ includes the boundary, so the line is solid.
The inequality $y < 3x$ uses which boundary line?
- Dashed
- Solid
- No line
- Double line
Show answer and explanation
Answer: Dashed
Strict < excludes the boundary, so the line is dashed.
Testing $(0,0)$ in $y > x + 1$ gives $0 > 1$ , which is false. You should:
- Shade the side away from the origin
- Shade the origin’s side
- Shade both sides
- Not shade anything
Show answer and explanation
Answer: Shade the side away from the origin
The origin fails the inequality, so the region is on the other side.
The best first test point is usually:
- The origin (if not on the line)
- A point on the line
- The y-intercept
- A random large number
Show answer and explanation
Answer: The origin (if not on the line)
(0,0) gives the simplest arithmetic, unless the boundary passes through it.
With three inequalities, the required region is:
- Where all three are satisfied
- Where any one is satisfied
- The largest single region
- The boundary lines only
Show answer and explanation
Answer: Where all three are satisfied
The solution is the overlap satisfying every inequality.
Why might you shade the UNWANTED regions instead?
- It leaves the required region clear
- It is the only correct method
- It changes the answer
- To use a solid line
Show answer and explanation
Answer: It leaves the required region clear
Shading out unwanted areas leaves the solution region clean and easy to read.

Next step

Check your understanding

Work through questions on Regions. Use the feedback to decide what needs another look.

2 past-paper questions · 10 marks

Practise this topic

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.