Inequalities on a number line

Objective. Solve one-step inequalities and show solutions on a number line.

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2. Worked example
3. Practice
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Learn: the key idea

Inequalities on a number line show the set of values that satisfy a condition such as x > 3 or x ≤ 5, solved using the same steps as equations but keeping the inequality sign throughout. A one-step inequality like x + 2 < 9 is solved by subtracting 2 from both sides to get x < 7. On a number line, a circle is drawn at the boundary value: an open circle for strict inequalities (< or >) since the boundary is not included, and a filled circle for ≤ or ≥ since the boundary is included, with an arrow showing the direction of all solutions.

Worked example

Solve the inequality 2x − 1 ≥ 9 and describe how to show it on a number line.

1 $Add 1 to both sides: 2x \geq 10$
2 $Divide both sides by 2: x \geq 5$
3 $On the number line, draw a filled circle at 5 since the boundary is included$
4 $Draw an arrow extending to the right to show all values greater than or equal to 5$

Answer: x ≥ 5

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Common mistake. Pupils often use a filled circle for a strict inequality like x > 4, incorrectly including the value 4 itself, when an open circle must be used since 4 does not satisfy x > 4.

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