Solve linear inequalities in one variable, represent solutions on a number line and as integer lists, and solve double inequalities, including reversing the sign when dividing by a negative.

What this dot point is asking

WJEC requires you to solve linear inequalities in one variable, to represent the solution set on a number line, to list the integers that satisfy an inequality, and to handle double inequalities. The methods mirror solving equations almost exactly, with one crucial extra rule about dividing by a negative, and the representation skills (open versus closed circles, integer lists) are where marks are commonly won or lost. Inequalities appear on both components.

Solving linear inequalities

An inequality compares two expressions that are not equal, using $<$, $>$, $\le$ or $\ge$. Solve it exactly as you would an equation, applying inverse operations to both sides, but carry the inequality sign through each line.

For $2x + 3 < 11$: subtract $3$ to get $2x < 8$, then divide by $2$ to get $x < 4$. The solution is a range of values, not a single number.

The negative rule

There is one way inequalities differ from equations.

This rule exists because multiplying by a negative reverses order on the number line: $2 < 3$ but $-2 > -3$.

Number line representation

A solution set is shown on a number line with a circle at the boundary and an arrow for the range.

A closed (filled) circle marks an included boundary; an open (hollow) circle marks an excluded one.

Double inequalities and integer lists

A double inequality such as $-2 < 3x + 1 \le 10$ traps the variable between two bounds. Solve it by doing the same operation to all three parts at once: subtract $1$ to get $-3 < 3x \le 9$, then divide by $3$ to get $-1 < x \le 3$.

To list integers satisfying an inequality, find the range first, then write the whole numbers inside it, watching the boundaries: $-1 < x \le 3$ gives $0, 1, 2, 3$, because $-1$ is excluded (strict) but $3$ is included (inclusive). Care with which boundary is in or out is the most marked detail in these questions.

Inequalities on a graph (Higher)

At Higher tier, inequalities in two variables describe regions of the coordinate plane rather than ranges on a line. The line $y = 2x + 1$ divides the plane in two: $y > 2x + 1$ is the region above it and $y < 2x + 1$ the region below. A strict inequality uses a dashed boundary line (the line itself is excluded) and an inclusive one uses a solid line. To show the region satisfying several inequalities at once, shade where all the conditions overlap, testing a point such as the origin to decide which side of each line to keep.

Forming inequalities from words

WJEC sets worded inequality problems where you translate a constraint into algebra. "A taxi charges GBP 3 plus GBP 2 per mile, and I have at most GBP 15" becomes $3 + 2m \le 15$, solving to $m \le 6$, so at most $6$ miles. The phrases "at least" ($\ge$), "at most" ($\le$), "more than" ($>$) and "fewer than" ($<$) signal which symbol to use. Reading the wording carefully to pick the right symbol, then solving and interpreting the answer in context, is the full skill these questions test.

Why this matters

Inequalities extend equation-solving to ranges of values, which model real constraints such as budgets, capacities and tolerances. They connect to graphs through shaded regions and to the discriminant work in quadratics, and the careful boundary reasoning (open versus closed, included versus excluded) is exactly the precise communication WJEC rewards under AO2. Mastering the negative rule prevents a whole class of avoidable errors.