A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Linear and Quadratic Inequalities — mark scheme explained
The short answer
In this section, we will explore how to solve linear and quadratic inequalities in a single variable. We will also learn how to interpret these inequalities graphically and express their solutions using set notation or logical operators.
The question
Solve the inequality 4x - 5 < 7 .
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For solving inequalities, marks are typically awarded for each step in the solution process. For graphical representation, marks are given for correctly drawing the line/parabola and shading the appropriate region. For expressing solutions, marks are given for correct use of set notation or logical operators.
What the command words demand
- Solve
- Find the solution to the given inequality.
- Graph
- Draw a graph of the inequality, showing the correct line and shaded region.
- Express
- Write the solution using set notation or logical operators.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 2-3 minutes per mark. For example, a 4-mark question should take approximately 8-12 minutes to complete.
- Add 5 to both sides: 4x < 121 mark
- Divide both sides by 4: x < 31 mark
Final answer: x < 3
Work through every step correctly and you earn all 2 marks.
Another worked example
Solve the inequality 2x + 1 ≥ 5 .
- Subtract 1 from both sides: 2x ≥ 41 mark
- Divide both sides by 2: x ≥ 21 mark
Final answer: x ≥ 2
Work through every step correctly and you earn all 2 marks.
Common mistakes
Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
Why it happens: Students often forget this crucial step, leading to incorrect solutions.
Fix: Always check if you are multiplying or dividing by a negative number and reverse the inequality sign accordingly.
Incorrectly identifying the intervals for quadratic inequalities.
Why it happens: Students may not correctly determine the intervals where the inequality holds, leading to incorrect solutions.
Fix: Find the roots of the corresponding quadratic equation and test points in each interval to ensure you identify the correct intervals.
Using a solid line instead of a dashed line when graphing strict inequalities.
Why it happens: Students may confuse the use of solid and dashed lines, leading to incorrect graphical representations.
Fix: Use a dashed line for strict inequalities ( < or > ) and a solid line for non-strict inequalities (≤ or ≥).
Incorrectly shading the region when graphing inequalities.
Why it happens: Students may shade the wrong region, leading to incorrect graphical solutions.
Fix: Always test a point in each interval to determine which region to shade. For linear inequalities, choose a point not on the line and substitute it into the inequality to see if it holds.
Failing to express solutions using set notation or logical operators correctly.
Why it happens: Students may struggle with the correct use of set notation or logical operators, leading to incorrect expressions of solutions.
Fix: Practice expressing solutions in both forms. For example, x < 2 can be written as {x | x < 2} , and x < 2 or x > 3 can be written as {x | x < 2} ∪ {x | x > 3} .
Incorrectly solving inequalities involving fractions.
Why it happens: Students may struggle with the algebraic manipulation required to solve inequalities involving fractions, leading to incorrect solutions.
Fix: Clear the fractions by multiplying both sides of the inequality by the least common denominator (LCD) and then solve as usual. Be careful when multiplying or dividing by a negative number.
Failing to check for extraneous solutions in inequalities involving absolute values.
Why it happens: Students may not realize that solving inequalities involving absolute values can sometimes lead to extraneous solutions, leading to incorrect answers.
Fix: Always check the solutions by substituting them back into the original inequality to ensure they are valid.
Incorrectly interpreting the graphical representation of inequalities involving parabolas.
Why it happens: Students may struggle with understanding which region to shade when graphing quadratic inequalities, leading to incorrect graphical solutions.
Fix: Test a point in each interval to determine where the inequality holds. For example, test points inside and outside the parabola to see which region satisfies the inequality.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Solve Linear Inequality | Rearrange and solve a linear inequality to find the range of values for x. | 2 |
| Solve Linear Inequality | Rearrange and solve a linear inequality to find the range of values of x. | 2 |
| Solve Quadratic Inequality | Factorise the quadratic, find its critical values, then state the range where the expression is negative. | 3 |
| Solve Quadratic Inequality | Factorise the quadratic and find the range of x values satisfying the inequality. | 3 |
| Graph Linear Inequality | Draw the boundary line and shade the region satisfying the given linear inequality. | 2 |
| Graph Quadratic Inequality | Draw the parabola as a dashed curve and shade the region satisfying the inequality. | 2 |
| Total across these question types | 14 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.