Solve linear inequalities in one variable, represent solutions on a number line, and find integer values that satisfy an inequality

What this dot point is asking

SEAB wants you to solve linear inequalities in one variable, show the solution on a number line, and list integer values that satisfy an inequality or a pair of bounds. The methods mirror solving equations, with one crucial extra rule about negatives.

The answer

Inequality signs

The four signs are less than $<$, greater than $>$, less than or equal to $\le$, and greater than or equal to $\ge$. A strict sign ($<$ or $>$) excludes the boundary value; an inclusive sign ($\le$ or $\ge$) includes it.

Solving like an equation, with one exception

Solve an inequality almost exactly as you solve an equation: add, subtract, expand and collect terms freely. The one exception is that multiplying or dividing both sides by a negative number reverses the direction of the inequality. So from $-3x < 12$ you divide by $-3$ to get $x > -4$, with the sign flipped.

Representing the solution on a number line

Draw the solution as a shaded region on a number line. Use an open (hollow) circle at a boundary that is excluded by a strict inequality, and a closed (filled) circle at a boundary included by an inclusive inequality, then shade in the direction of the solution.

Double inequalities and integer solutions

A double inequality such as $-3 \le x < 5$ bounds the variable on both sides; operate on all three parts at once. To list integer solutions, identify the smallest and largest integers inside the range, taking care over which boundary is included.

Examples in context

Example 1. A budget constraint. If items cost $4$ dollars each and you have at most $30$ dollars, the number bought $n$ satisfies $4n \le 30$, so $n \le 7.5$, meaning at most $7$ items since $n$ must be a whole number. Inequalities model limits and budgets naturally.

Example 2. Acceptable measurements. A component is acceptable if its length $L$ lies within tolerance, say $49.5 \le L \le 50.5$. This double inequality describes the allowed range, and any reading outside it fails the check.

Try this

Q1. Solve $2x - 5 < 9$. [2 marks]

• Cue. Add $5$: $2x < 14$, then divide by $2$: $x < 7$.

Q2. Solve $-4x \ge 20$. [2 marks]

• Cue. Divide by $-4$ and flip the sign: $x \le -5$.

Q3. List the integers satisfying $-2 \le x < 3$. [2 marks]

• Cue. Include $-2$, exclude $3$: the integers are $-2, -1, 0, 1, 2$.