Solve simple linear inequalities in one unknown and represent the solution on a number line

What this dot point is asking

SEAB wants you to solve simple linear inequalities in one unknown and show the solution on a number line. An inequality describes a range of values rather than a single answer, and it is solved almost exactly like an equation - with one important extra rule about negatives.

The answer

The inequality symbols

Four symbols describe how two quantities compare:

• $<$ means "less than".
• $>$ means "greater than".
• $\leq$ means "less than or equal to".
• $\geq$ means "greater than or equal to".

So $x > 3$ means $x$ is any number bigger than $3$ (but not $3$ itself), while $x \geq 3$ includes $3$.

Solving an inequality

Solve an inequality using the same balance steps as an equation: add, subtract, multiply or divide both sides to get the unknown alone. The inequality symbol stays the same for these operations as long as you are not multiplying or dividing by a negative.

For $2x + 1 < 9$, subtract $1$ to get $2x < 8$, then divide by $2$ to get $x < 4$.

The one special rule

When you multiply or divide both sides by a negative number, you must reverse the inequality sign. For $-3x > 12$, divide both sides by $-3$ and flip the sign: $x < -4$.

A safe way to avoid this is to keep the unknown term positive. For $-3x > 12$, you could instead add $3x$ to both sides and subtract $12$, leading to the same answer without flipping.

Representing the solution on a number line

A number line shows the range of solutions clearly:

• Use an open (unfilled) circle for $<$ or $>$, because the endpoint is not included.
• Use a closed (filled) circle for $\leq$ or $\geq$, because the endpoint is included.
• Draw an arrow in the direction of all the valid values.

Integer solutions

If a question asks for integer solutions, list the whole numbers in the range. For $x \leq 5$ with $x$ a positive integer, the solutions are $1, 2, 3, 4, 5$.

Examples in context

Example 1. A budget limit. A student has \30$and pays a$\$6$ entry fee plus \3$per ride at a fair. The number of rides$n$must satisfy$6 + 3n \leq 30$. Solving gives$3n \leq 24$, so$n \leq 8$. The student can take at most$8$ rides. The inequality captures "no more than", which is exactly what a budget means.

Example 2. A measurement range. A spring stretches between $5\ \text{cm}$ and $9\ \text{cm}$, so its length $L$ satisfies $5 \leq L \leq 9$. This double inequality reads as $L$ being at least $5$ and at most $9$, a single statement describing the whole safe range. Ranges like this appear in measurement and graph questions.

Try this

• Cue. Solve $4x + 3 > 19$. Subtract $3$ to get $4x > 16$, then divide by $4$ to get $x > 4$.
• Cue. Solve $7 - x \geq 2$. Subtract $7$ to get $-x \geq -5$, then divide by $-1$ and flip to get $x \leq 5$.
• Cue. List the positive integers satisfying $x < 4$. They are $1, 2$ and $3$.