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

The Eduqas algebra content asks you to solve linear inequalities and show the solution on a number line, to solve quadratic inequalities at Higher tier, and to recognise and sketch the standard non-linear graphs: quadratic, cubic, reciprocal and exponential. Inequalities behave almost exactly like equations with one crucial exception, and graph recognition turns up in both components. The quadratic inequality is a Higher-tier task that combines factorising with reading a parabola's sign.

Solving linear inequalities

An inequality is solved by the same balancing as an equation, with one rule that does not apply to equations.

For $3x + 4 \ge 19$, subtract $4$ to get $3x \ge 15$, then divide by $3$ (positive, no flip) to get $x \ge 5$. A double inequality such as $-1 < 2x + 3 \le 7$ is solved by doing the same to all three parts: subtract $3$ to get $-4 < 2x \le 4$, then divide by $2$ to get $-2 < x \le 2$.

Representing on a number line

Mark the solution on a number line using circles and an arrow or line. An open circle means the endpoint is not included ($<$ or $>$); a closed (filled) circle means it is included ($\le$ or $\ge$). For $x \ge 5$, place a closed circle at $5$ with an arrow to the right. For $-2 < x \le 2$, place an open circle at $-2$ and a closed circle at $2$, joined by a line.

Quadratic inequalities (Higher)

A quadratic inequality is solved by combining factorising with the shape of the parabola.

The direction matters: a "less than zero" inequality gives the region between the roots, while a "greater than zero" inequality gives the two regions outside them.

Recognising non-linear graphs

Know the standard shapes. A quadratic $y = ax^2 + bx + c$ is a parabola (U-shaped for positive $a$, n-shaped for negative $a$). A cubic $y = x^3$ has a characteristic S-shape passing through the origin. A reciprocal $y = \dfrac{k}{x}$ has two separate branches with the axes as asymptotes (the curve approaches but never touches them). An exponential $y = k^x$ grows ever more steeply for $k > 1$ and passes through $(0, 1)$. Matching an equation to its graph, or sketching from the equation, is a recurring short question.

Integer solutions and combined inequalities

Eduqas often asks for the integer values that satisfy an inequality. For $-2 \le x < 3$ where $x$ is an integer, the values are $-2, -1, 0, 1, 2$ (note that $3$ is excluded by the strict $<$). Reading the endpoints carefully, deciding whether each is included, is exactly where marks are won or lost. When two conditions are combined, such as $x > 1$ and $x \le 5$, the solution is the overlap $1 < x \le 5$. A worded constraint problem ("a number is at least $4$ but less than $10$") translates to $4 \le n < 10$ in the same way, and listing the satisfying integers is a common follow-up.

Why inequalities and graphs matter

Inequalities express constraints, which is how real problems are framed: a budget that must not be exceeded, a minimum mark to pass, a safe range for a measurement. Being able to set up and solve them, and to read the answer back as a range rather than a single value, is a genuinely useful skill that Eduqas tests through context. Curve recognition matters because the shape of a graph tells a story at a glance: a parabola has a single turning point, a reciprocal models inverse proportion, and an exponential captures rapid growth or decay, all of which recur across the ratio and statistics areas.