Learn number line inequalities
A number-line inequality graph uses an endpoint and a ray to show every value in its solution set. Open circles exclude the endpoint, while closed circles include it.
You will read and draw inequality solutions on a number line.
The idea in plain English
A number-line graph shows every number that makes an inequality true.
The circle tells whether the endpoint counts. The arrow tells which direction the solutions continue.
How to recognise this kind of question
- There is a horizontal number line.
- One point has an open or closed circle.
- A ray is shaded left or right.
Quick decision: Open means not included; closed means included. Left means less; right means greater.
The method
- Locate the boundary value on the number line.
- Use an open circle for < or > and a closed circle for ≤ or ≥.
- Follow a left ray for values less than the boundary and a right ray for values greater than it.
Worked examples
A graph has a closed circle at −2 and a ray to the right.
- The closed circle includes −2.
- The right ray represents values greater than −2.
Answer: x ≥ −2
Graph x ≥ −2.
- ≥ includes −2, so place a closed circle at −2.
- Greater values are to the right, so shade right.
Answer: Closed circle at −2; shade right
Your turn—with help
Graph y < 4.
Need a hint?
Does < include 4? Which side contains smaller numbers?
Show the answer and steps
- Place an open circle at 4 because 4 is not included.
- Shade left toward values smaller than 4.
Answer: Open circle at 4; shade left
Watch out for this mistake
Incorrect: x > 2 uses a closed circle
Why it fails: The symbol > does not include the endpoint 2.
Do this instead: Use an open circle at 2 and shade right.
Quick check
Try these without looking. Then open each answer to check your thinking.
A closed circle at 3 shaded right
Need a hint?
Closed includes 3; right is greater.
Show the answer and steps
- Choose ≥ because the endpoint counts and values are greater.
Answer: x ≥ 3
An open circle at −1 shaded left
Need a hint?
Open excludes −1; left is less.
Show the answer and steps
- Choose < because the endpoint does not count.
Answer: x < −1
Graph n ≤ 5
Need a hint?
≤ includes 5 and means smaller values.
Show the answer and steps
- Close the circle at 5.
- Shade left.
Answer: Closed circle at 5; shade left
Common mistakes
- Confusing open and closed endpoints.
- Reading a left-pointing ray as greater than.
- Reversing the inequality when the variable is written first.
Practice tips
- Say 'included' or 'not included' when checking the endpoint.
- Test one number located on the shaded ray.