B: Algebra and functions: 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.

The idea

Solve linear inequalities by expanding brackets and collecting terms; reverse the sign if multiplying or dividing by a negative. For quadratics, rearrange to make one side zero. Find critical values and sketch the parabola to identify the correct regions. If the x² coefficient is positive, < 0 lies between the roots and > 0 outside. If there are no real roots, the expression is always positive or always negative. Here 'fractions' means numerical fractions, e.g. (x+1)/2 > (2x−3)/3; clear these by multiplying by the positive common denominator, not by algebraic denominators. Express solutions using 'and', 'or', or set notation like {x : x < −1} ∪ {x : x > 3}. Graphically, represent strict inequalities with dashed lines/curves and shade the required region.

What earns the marks

• Expands brackets and collects like terms to solve linear inequalities, reversing the inequality sign when dividing by a negative.
• Solves linear inequalities containing numerical fractions by multiplying by a positive common denominator, e.g. (x+1)/2 > (2x−3)/3 gives 3(x+1) > 2(2x−3).
• Finds critical values for quadratic inequalities and uses a sketch to select the correct region, accounting for the sign of the x² coefficient.
• Identifies that a quadratic inequality with no real roots is either satisfied by all real values or no real values.
• Expresses solution sets accurately using 'and' for bounded intervals, 'or' for disjoint intervals, or formal set notation.
• Draws a dashed boundary for strict inequalities (<, >) or a solid boundary for inclusive inequalities (≤, ≥) and shades the correct region.

Common mistakes and how to put them right

• Forgetting to reverse the inequality sign when dividing by a negative number.

  Put it right by checking the final range with a test value.

• Assuming < 0 always means 'between the roots' for quadratics without checking if the x² coefficient is positive.

  Put it right by sketching the specific parabola.

• Multiplying an inequality by an algebraic denominator without squaring it, which loses valid negative regions.

  Put it right by multiplying by the denominator squared and noting the denominator ≠ 0. This technique is beyond this specification and is not required here.

• Writing disjoint quadratic solutions as a single continuous interval (e.g., −2 > x > 2).

  Put it right by separating them with 'or' or using union set notation.

Exam tips

• Always sketch the quadratic graph to decide between 'and' and 'or' before writing the final answer, especially if the x² coefficient is negative.
• When solving inequalities with numerical fractions, multiply by the positive common denominator and show each step clearly.
• For graphical inequalities, clearly label the boundary as dashed or solid and use a test point like (0, 0) to confirm which side to shade.