AQA A-Level Mathematics: Linear and Quadratic Inequalities — mark scheme explained

The question

Solve the inequality 4x - 5 < 7 . [Paraphrased for study — not reproduced from any exam paper.]

Mark scheme, decoded

What each mark is really for — in plain English — and the wording trap that loses it.

• S1

  Add 5 to both sides: 4x < 12

• S2

  Divide both sides by 4: x < 3

Model answer

Worked through, with each step tagged to the mark it earns.

1. S1

   Add 5 to both sides: 4x < 12

2. S2

   Divide both sides by 4: x < 3

3. Final answer: x < 3

Common mistakes

• Forgetting to reverse the inequality sign when multiplying or dividing by a negative number. — 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. — 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. — Use a dashed line for strict inequalities ( < or > ) and a solid line for non-strict inequalities (≤ or ≥).
• Incorrectly shading the region when graphing inequalities. — 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. — 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. — 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. — 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. — 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

• Full worked solution (all marking points)2 marks

Related questions

• AQA A-Level Mathematics: Surds and Rationalising the Denominator — mark scheme explained →
• AQA A-Level Mathematics: Simultaneous Equations: Linear and Quadratic — mark scheme explained →