Inequalities on a number line
Translate a constraint and show an entire set of possible values.
Before you begin
- Comparing numbers
- Reading a number line
Translate the boundary precisely
An inequality describes a range of values. A temperature below 5 °C is t < 5; a temperature above 5 °C is t > 5. The boundary five is excluded in both statements. Words such as at least and at most include the boundary and use ≥ or ≤. Decide whether the quantity can take any real value or only values such as whole-number counts.
Show the whole solution set
On a number line, use an open circle for an excluded boundary and a filled circle for an included one. Shade the direction of all allowed values: right for greater values and left for smaller values. An arrow shows that the interval continues. For a continuous quantity, there are infinitely many values in an interval, not just the labelled integers.
Worked example
A sample must be stored below 4 °C. Write the inequality and describe its number line.
Show the worked solution
- Let t be the temperature in degrees Celsius.
- Below means strictly less than, so write t < 4.
- Draw an open circle at 4 and shade to the left; 4 itself does not meet the condition.
t < 4; open circle at 4, shading left.
Try it yourself
Which value satisfies x > 2.5?
Common mistakes
- At most and less than differ at the boundary.
- Check any restrictions imposed by the physical context.
What can you explain now?
A container can hold at most 12 L. Write a constraint for its volume v, including that volume cannot be negative.
Compare with the explanation
0 ≤ v ≤ 12.
At most includes 12, and physical volume is nonnegative. Use filled boundary circles at zero and twelve and shade between them.
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Original StudyForA content. AI-assisted checks completed; subject-teacher review is still pending. These lessons are not endorsed by an exam board.
Common Core mathematics standards ↗