AQA · GCSE · Mathematics
Course version: 8300 · Version from Sept 2015
Solving equations and inequalities: solve linear inequalities in one variable represent the solution set on a number line
A linear inequality in one variable compares expressions using <, >, ≤ or ≥.
- 3 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
A linear inequality in one variable compares expressions using <, >, ≤ or ≥. To solve it, rearrange similarly to a linear equation: collect variable terms on one side and constants on the other, then divide by the coefficient. Crucially, multiplying or dividing both sides by a negative number reverses the inequality sign. For example, solving 3x + 2 < 11 gives 3x < 9, so x < 3. Solving 5 − 2x ≥ 1 gives −2x ≥ −4, so x ≤ 2. The solution set is then represented on a number line: use an open circle for a strict inequality (<, >) and a filled circle for inclusive inequalities (≤, ≥). Draw an arrow or line from the circle indicating all values in the direction of the solution.
What earns the marks
- Rearranges the inequality correctly by adding or subtracting the same quantity from both sides.
- Divides or multiplies both sides by the coefficient of the variable, reversing the inequality sign when the multiplier or divisor is negative.
- States the solution set correctly, for example x < 3 or x ≤ 2.
- Draws a number line with an open circle for < or > and a filled circle for ≤ or ≥.
- Indicates the correct direction of the solution set with an arrow or line extending from the circle.
Common mistakes and how to put them right
Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
Put it right: Reverse the sign whenever the multiplier or divisor is negative.
Using a filled circle for a strict inequality such as x < 3.
Put it right: Use an open circle for strict inequalities and a filled circle only for ≤ or ≥.
Drawing the arrow in the wrong direction after solving, for example showing x > 3 when the solution is x < 3.
Put it right: Check the final inequality before drawing the arrow.
Treating the inequality as an equation and giving a single value rather than a solution set.
Put it right: State the range of values and show it on the number line.
Exam tips
- Always check whether the coefficient of the variable is negative before dividing, because this determines whether the inequality sign reverses.
- Substitute a test value from your solution set back into the original inequality to confirm the direction is correct.
- Label the number line clearly and use the correct circle type: open for strict inequalities and filled for inclusive inequalities.
Try the lesson
The start of the Solving equations and inequalities: solve linear inequalities in one variable represent the solution set on a number line lesson, made from checked teaching notes for this specification point.
Have a go first
Solve the inequality x + 4 < 9.
Listen
A linear inequality in one variable compares expressions using , ≤ or ≥. To solve it, rearrange similarly to a linear equation: collect variable terms on one side and constants on the other, then divide by the coefficient. Crucially, multiplying or dividing both sides by a negative number reverses the inequality sign. For example, solving 3x + 2 < 11 gives 3x < 9, so x < 3. Solving 5 − 2x ≥ 1 gives −2x ≥ −4, so x ≤ 2. The solution set is then represented on a number line: use an open circle for a strict inequality () and a filled circle for inclusive inequalities (≤, ≥). Draw an arrow or line from the circle indicating all values in the direction of the solution. A strong answer earns marks for these points. Rearranges the inequality correctly by adding or subtracting the same quantity from both sides. Divides or multiplies both sides by the coefficient of the variable, reversing the inequality sign when the multiplier or divisor is negative. States the solution set correctly, for example x < 3 or x ≤ 2. Draws a number line with an open circle for and a filled circle for ≤ or ≥. A common slip: Forgetting to reverse the inequality sign when multiplying or dividing by a negative number. Instead, reverse the sign whenever the multiplier or divisor is negative. Exam tip: Always check whether the coefficient of the variable is negative before dividing, because this determines whether the inequality sign reverses.
Key words
Say what each word means before you turn the card.
Inequality notation and direction
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Representing solution sets on a number line
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Checking solutions by substitution
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See every definition
- Inequality notation and direction
- A linear inequality in one variable compares expressions using <, >, ≤ or ≥, and multiplying or dividing both sides by a negative number reverses the inequality sign.
- Representing solution sets on a number line
- Use an open circle for a strict inequality and a filled circle for inclusive inequalities, then draw an arrow or line from the circle indicating all values in the direction of the solution.
- Checking solutions by substitution
- Substitute a test value from your solution set back into the original inequality to confirm the direction is correct.
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