Pearson Edexcel · A-Level · Mathematics
Course version: 9MA0 · Version from Mar 2026
Solve linear and quadratic inequalities in a single variable and interpret…
Full specification wording
Algebra and functions: 2.5 Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions. Express solutions through correct use of ‘and’ and ‘or’, or through set notation. Represent linear and quadratic inequalities such as y > x + 1 and y > ax² + bx + c graphically.
An inequality is solved like an equation, with one extra rule: multiplying or dividing by a negative number reverses the inequality sign.
Checked against the Pearson Edexcel 9MA0 specification (opens in a new tab) on .
- Point 5 of 11
- 3 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
An inequality is solved like an equation, with one extra rule: multiplying or dividing by a negative number reverses the inequality sign. Linear inequalities with brackets are expanded first; those with fractions are cleared by multiplying by the positive lowest common denominator. A quadratic inequality is solved by finding the critical values where the expression equals zero, then testing intervals or sketching the parabola. For a > 0, the solution of ax² + bx + c > 0 lies outside the roots, written with ‘or’, while ax² + bx + c < 0 lies between them, written with ‘and’. Set notation may be used instead, for example {x : x < −1} ∪ {x : x > 3}. Graphically, y > x + 1 is the half-plane above the line y = x + 1, and y > ax² + bx + c is the region above the parabola, with a dashed boundary when the inequality is strict.
What earns the marks
- Solves a linear inequality, including one with brackets or fractions, and reverses the sign correctly when multiplying or dividing by a negative number.
- Finds the critical values of a quadratic inequality by solving the associated quadratic equation.
- Selects the correct region by testing a value or sketching the parabola, then writes the answer with ‘and’ or ‘or’ as appropriate.
- Expresses the solution set in set notation, using the correct symbols for union or intersection.
- Shades or describes the correct half-plane or region for a linear or quadratic inequality in two variables, using a dashed boundary for strict inequalities.
Common mistakes and how to put them right
Forgetting to reverse the inequality sign when dividing by a negative number.
Put it right by reversing the sign whenever the multiplier or divisor is negative.
Writing a quadratic solution as a single interval when the parabola opens upwards and the inequality is ‘greater than’.
Put it right by using two intervals joined by ‘or’.
Using a solid boundary line for a strict inequality.
Put it right by drawing a dashed line or curve for > and <.
Exam tips
- Sketch the line or parabola and mark the critical values before writing the final answer.
- Match the wording to the region: ‘and’ for between the roots, ‘or’ for outside the roots when a > 0.
- Check the direction of the inequality by testing a convenient value such as x = 0.
Try the lesson
The start of the Algebra and functions: 2.5 Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions. Express solutions through correct use of ‘and’ and ‘or’, or through set notation. Represent linear and quadratic inequalities such as y > x + 1 and y > ax² + bx + c graphically. lesson, made from checked teaching notes for this specification point.
Have a go first
Solve the inequality 3x − 5 < 7.
Listen
An inequality is solved like an equation, with one extra rule: multiplying or dividing by a negative number reverses the inequality sign. Linear inequalities with brackets are expanded first; those with fractions are cleared by multiplying by the positive lowest common denominator. A quadratic inequality is solved by finding the critical values where the expression equals zero, then testing intervals or sketching the parabola. For a > 0, the solution of ax² + bx + c > 0 lies outside the roots, written with ‘or’, while ax² + bx + c < 0 lies between them, written with ‘and’. Set notation may be used instead, for example {x : x 3}. Graphically, y > x + 1 is the half-plane above the line y = x + 1, and y > ax² + bx + c is the region above the parabola, with a dashed boundary when the inequality is strict. A strong answer earns marks for these points. Solves a linear inequality, including one with brackets or fractions, and reverses the sign correctly when multiplying or dividing by a negative number. Finds the critical values of a quadratic inequality by solving the associated quadratic equation. Selects the correct region by testing a value or sketching the parabola, then writes the answer with ‘and’ or ‘or’ as appropriate. Expresses the solution set in set notation, using the correct symbols for union or intersection. A common slip: Forgetting to reverse the inequality sign when dividing by a negative number. Instead, reversing the sign whenever the multiplier or divisor is negative. Exam tip: Sketch the line or parabola and mark the critical values before writing the final answer.
Key words
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Linear inequalities and sign reversal
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Quadratic inequalities and critical values
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Set notation and interval language
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See every definition
- Linear inequalities and sign reversal
- Solved like an equation, but multiplying or dividing by a negative number reverses the inequality sign; brackets are expanded first and fractions cleared by multiplying by the positive lowest common denominator.
- Quadratic inequalities and critical values
- Solved by finding the critical values where the expression equals zero, then testing intervals or sketching the parabola; for a > 0, greater than lies outside the roots and less than lies between them.
- Set notation and interval language
- Solutions may be written with ‘and’ or ‘or’, or in set notation such as {x : x < −1} ∪ {x : x > 3}, using the correct symbols for union or intersection.
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Test your understanding of Algebra and functions: 2.5 Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions. Express solutions through correct use of ‘and’ and ‘or’, or through set notation. Represent linear and quadratic inequalities such as y > x + 1 and y > ax² + bx + c graphically.
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