Inequality symbols and the number line revision notes

Edexcel International GCSE Maths (4MA1), chapter 8: Inequalities, specification point 2.8a-f. Free lesson.

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Inequality symbols compare values, and a number line shows which values satisfy an inequality. The circle at each boundary shows whether that boundary value is included.

How to read the four symbols

The pointed end faces the smaller value, while the open end faces the larger value. A line underneath the symbol means the boundary number is included, so it is shown with a filled circle. x < 5 means values below 5. Use an open circle at 5 and an arrow to the left. x ≤ 5 includes 5. Use a filled circle at 5 and an arrow to the left. x > 2 means values above 2. Use an open circle at 2 and an arrow to the right. x ≥ 2 includes 2. Use a filled circle at 2 and an arrow to the right.

How to show one boundary on a number line

First mark the boundary. Then check whether it is included. The direction of the arrow depends on whether the values are greater or less than the boundary.

How to show two boundaries

A compound inequality gives a lower and an upper boundary. Read it as two statements, decide whether each endpoint is included, then join the circles to show the values between them.

How to check circles and negative numbers

A filled circle means the boundary value counts. For a strict inequality, such as x < 4, the boundary does not count, so the circle must be open. On a number line, values further right are always greater, including negative values. For x < 4, use an open circle at 4. Four itself is not less than four. −2 > −5 because −2 is to the right of −5. With negative numbers, the value that looks smaller in digits can be greater.

How to write a compound inequality

Combine the two comparisons in order, placing x between the boundary values. Then use the symbols to choose the circles. “x is greater than −3 and less than or equal to 2” becomes −3 < x ≤ 2. Use an open circle at −3 and a filled circle at 2.