Inequalities on graphs and set notation

An inequality in x and y describes a whole region of a graph. You find it by drawing the boundary line and choosing a side.

Practise this lesson free (6 questions)

Draw the boundary

Draw the line as if the inequality were an equation.
Use a solid line for ≤ or ≥ (points on the line are included) and a dashed line for < or > (they are not).
To describe a region, say which side of each boundary line it is on, for example y < 2x + 1.

Choose the side

Test a point that is not on the line, such as (0, 0).
For y < 2x + 1: 0 < 2 × 0 + 1 is true, so the side containing (0, 0) is the region.
Shade the region (or the unwanted side) and label it R.

Set notation and integer points

Solutions can be written in set notation: x > 3 becomes {x : x > 3}.
To find integer points in a region, list the points with whole-number coordinates that fit every inequality.
x > 0, y > 0, x + y < 5 contains 6 integer points: (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (3, 1).
1 2 3 4 5 6 1 2 3 4 5 6 x y x + y = 5 dashed: boundarynot included
The region x > 0, y > 0, x + y < 5 contains 6 integer points.
Worked example

Does the point (1, 2) lie in the region y > 2x − 1?

  1. Substitute x = 1: 2x − 1 = 1
  2. Is 2 > 1? Yes.

Answer: Yes, (1, 2) is in the region

Key idea

Solid line for ≤ and ≥, dashed for < and >. Test a point like (0, 0) to pick the side. {x : x > 3} means all x greater than 3.

Key words
Inequality
A statement using <, >, ≤ or ≥ to compare two quantities.
Equation
A statement with an equals sign that is true for particular values of the letter, such as 3x + 1 = 10.
Integer
A whole number: positive, negative or zero.

All GCSE and International GCSE Mathematics key terms and definitions →

Quick check

Try two quick questions

See if it has sunk in. Answers are marked straight away.

Extra practice

2 more questions

Try each one, then tap to check.

  1. Which point lies in the region y < x + 3?

    1. (1, 5)
    2. (−2, 2)
    3. (2, 5)
    4. (0, 2)
    Show the answer

    Answer: (0, 2)

    At (0, 2): 2 < 0 + 3 is true.

  2. Which boundary should be drawn as a solid line?

    1. y ≥ 2x − 1
    2. x < 4
    3. y > 2x − 1
    4. y < 2x − 1
    Show the answer

    Answer: y ≥ 2x − 1

    ≥ includes the line, so it is solid.

Check you have got it

Answer 6 quick questions with instant marking. If you get one wrong, GCSE-ready shows you why and gives you another go. It is free, and you do not need an account.

Practise this lesson free (6 questions)All GCSE and International GCSE Mathematics lessons

Video for this lesson coming soon. Subscribe on YouTube

Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

Related lessons