Solve linear inequalities and represent solutions on a number line; solve quadratic inequalities (Higher tier); and recognise and sketch the graphs of quadratic, cubic, reciprocal and exponential functions.

What this dot point is asking

OCR references A11, A12 and A22 cover solving and representing inequalities, quadratic inequalities at Higher tier, and recognising the graphs of quadratic, cubic, reciprocal and exponential functions. Inequalities extend equation-solving to ranges of values, and graph recognition tests whether you can link an equation to its shape. Both appear across the tiers, and the number-line representation is a frequent non-calculator question.

Solving and representing inequalities

An inequality compares two expressions and is solved much like an equation.

So $3x - 1 < 11$ gives $3x < 12$, then $x < 4$ (open circle at $4$, arrow left). But $-2x \ge 8$ gives $x \le -4$, with the sign flipped because of the division by $-2$. For a double inequality such as $-1 \le 2x < 6$, operate on all three parts at once to get $-\tfrac{1}{2} \le x < 3$, then list integer solutions if asked.

Quadratic inequalities (Higher)

A quadratic inequality is solved through its graph.

To solve $x^2 - x - 6 > 0$, first solve $x^2 - x - 6 = 0$, giving roots $x = 3$ and $x = -2$. Sketch the U-shaped parabola crossing at these roots. The expression is positive (above the axis) outside the roots, so the solution is $x < -2$ or $x > 3$. A "less than" inequality would give the region between the roots, $-2 < x < 3$. Sketching the parabola and reading off the regions is far more reliable than guessing.

The rule of thumb is: for a positive (U-shaped) quadratic, "greater than zero" means the two outer regions, written as two separate inequalities joined by "or", while "less than zero" means the single region between the roots, written as one double inequality. Watch the strictness of the sign: $\ge$ and $\le$ include the roots, while $>$ and $<$ exclude them. Writing the solution set with the correct connector and the correct inclusivity is what earns the final mark.

Recognising non-linear graphs

Each family of function has a characteristic shape.

Why this matters

Inequalities model real constraints ("at least", "no more than", "up to"), and OCR sets contextual questions where forming the inequality is the AO3 challenge. Graph recognition supports curve sketching, solving equations graphically, and interpreting rates of change. Knowing the four shapes lets you check a plotted graph for plausibility and answer "which graph could represent $y = \tfrac{k}{x}$?" instantly.