Solving linear and quadratic inequalities and representing solutions on number lines and graphs, and recognising and sketching the graphs of quadratic, cubic, reciprocal and exponential functions.

What this dot point is asking

Edexcel expects you to solve linear inequalities and, at Higher tier, quadratic inequalities, and to show solutions on number lines. You should also recognise and sketch the standard shapes of quadratic, cubic, reciprocal and exponential graphs. An inequality describes a range of values rather than a single answer, and recognising a graph's shape from its equation is a quick win that supports many other questions.

Solving linear inequalities

A linear inequality is solved with the same steps as an equation, with one extra rule.

To solve $3x + 4 > 19$: subtract $4$ to get $3x > 15$, then divide by $3$ to get $x > 5$. A "double" inequality such as $-1 \le 2x + 3 < 9$ is solved by doing the same operation to all three parts: subtract $3$ ($-4 \le 2x < 6$), then divide by $2$ ($-2 \le x < 3$).

Representing inequalities

On a number line, mark the boundary value and shade the allowed region. Use an open (unfilled) circle for a strict inequality ($<$ or $>$) because the boundary itself is excluded, and a closed (filled) circle for $\le$ or $\ge$ because the boundary is included. An arrow shows the direction of all allowed values. When asked for integer solutions, list only the whole numbers in the range: the integers satisfying $-2 \le x < 3$ are $-2, -1, 0, 1, 2$.

Quadratic inequalities (Higher)

A quadratic inequality is solved by finding where the parabola lies above or below the x-axis.

Recognising other graphs

Knowing a graph's shape from its equation is fast and useful.

• Quadratic $y = ax^2 + bx + c$: a parabola, U-shaped if $a > 0$, n-shaped if $a < 0$.
• Cubic $y = ax^3 + \ldots$: an S-shaped curve that can turn twice.
• Reciprocal $y = \dfrac{k}{x}$: two separate curves in opposite quadrants, never touching the axes (the axes are asymptotes).
• Exponential $y = k a^x$: a curve that rises steeply (growth, $a > 1$) or falls towards zero (decay, $0 < a < 1$), always staying above the x-axis.

These shapes appear in graph-matching questions and in modelling contexts such as compound interest (exponential) and inverse proportion (reciprocal).

Key features to label

When sketching, show the features that earn marks: where the curve crosses the axes, the turning point of a parabola, and any asymptotes. For a quadratic given in factorised form $y = (x - 1)(x + 3)$, the x-intercepts are $1$ and $-3$ (where each bracket is zero), the y-intercept is found by setting $x = 0$ (here $-3$), and the curve is U-shaped. For a reciprocal $y = \dfrac{4}{x}$, the curve approaches but never touches the x-axis and y-axis, so both axes are asymptotes. A sketch does not need to be to scale, but the shape, intercepts and symmetry must be right.

Try this

Q1. Solve $5 - 2x \ge 1$. [2 marks]

• Cue. Subtract $5$: $-2x \ge -4$. Divide by $-2$ and flip: $x \le 2$.

Q2. Write down the integer values of $x$ that satisfy $-3 < x \le 2$. [2 marks]

• Cue. $-2, -1, 0, 1, 2$.