One-variable inequalities and number-line solutions | Year 8

01 / LEARN

Understand the idea.

Inequalities give sets of values

An inequality usually describes a set of values. The statement $x < 2$ includes every real number below $2$ , including fractions and decimals. The signs $\leq$ and $\geq$ also include the boundary value.

On a number line, use an open circle for an excluded boundary and a filled circle for an included one. Shade left for smaller values and right for larger values; the shading represents all solutions.

Know when the sign reverses

Adding or subtracting the same quantity preserves the inequality direction. Multiplying or dividing by a positive number does too. Multiplying or dividing by a negative number reverses the direction because it reverses number order.

For example, $1 < 3$ becomes $- 1 > - 3$ after multiplication by $- 1$ . Once you solve, test a value in the proposed region and check whether the boundary itself is included.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

An included boundary

Solve $x + 3 \leq 5$ and show the result.

Worked solution

1
Subtract 3 from both sides.
$x \leq 5 - 3 = 2$
Subtraction preserves the direction of the comparison.
2
Check and include the boundary.
$2 + 3 = 5$
The sign $\leq$ allows equality, so draw a filled endpoint at $2$ .
3
Show the full solution region.
$x \leq 2$
Shade left from $2$ : every smaller value also keeps the sum at most $5$ .

Answer

$x \leq 2$ .

EXAMPLE 02

A negative divisor

Solve $- 3 x > 6$ .

Worked solution

1
Divide both sides by negative 3.
$x < \frac{6}{- 3} = - 2$
Division by a negative number reverses the inequality sign.
2
Exclude the boundary.
$- 3 (- 2) = 6$
The original sign is strict, so equality at $- 2$ is not allowed; use an open endpoint.
3
Check a value to the left.
$- 3 (- 3) = 9 > 6$
The test supports the direction; shade left to represent every $x < - 2$ .

Answer

$x < - 2$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Solve $2 x - 1 > 5$ .

Show answer & reasoning

$x > 3$ .

Add $1$ to obtain $2 x > 6$ , then divide by positive $2$ to get $x > 3$ . Equality at $3$ is excluded, so the number line has an open endpoint and shading to the right.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Write an inequality for $x$ is at least 4.
2 marks
Solve $x - 5 < 2$ .
2 marks
Solve $3 x + 1 \leq 10$ .
2 marks
Solve $- 2 x \geq 8$ .
2 marks
Describe the supplied number line.
2 marks
Entry costs AUD 5 and each activity costs AUD 2. With at most AUD 17, what is the greatest whole number of activities?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write an inequality for $x$ is at least 4.
2 marks
At least includes the boundary.
Use greater than or equal to.
$x \geq 4$ .
Solve $x - 5 < 2$ .
2 marks
Add 5 to both sides.
The sign does not change.
$x < 7$ .
Solve $3 x + 1 \leq 10$ .
2 marks
Subtract 1, then divide by positive 3.
The value 3 is included.
$x \leq 3$ .
Solve $- 2 x \geq 8$ .
2 marks
Divide by -2 and reverse the direction.
Check the boundary gives 8.
$x \leq - 4$ .
Describe the supplied number line.
2 marks
The open endpoint excludes -1.
The ray points to larger values.
$x > - 1$ .
Entry costs AUD 5 and each activity costs AUD 2. With at most AUD 17, what is the greatest whole number of activities?
3 marks
Subtract 5 and divide by 2.
Activity counts are whole numbers.
$5 + 2 a \leq 17$ gives at most 6 activities.

Checking saved progress…

Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics (2026)

Multi-step equations; brackets and variables on both sides; equations with rational solutions; substitution checks; one-variable inequalities and number-line solutions; tables and straight-line graphs; horizontal/vertical lines; how coefficients change a graph; graphical solutions and intersection checks; real-world linear models with digital exploration.

Official curriculum source ↗

In this lesson

Understand the idea.

Inequalities give sets of values

Know when the sign reverses

See the method in action.

An included boundary

Subtract 3 from both sides.

Check and include the boundary.

Show the full solution region.

A negative divisor

Divide both sides by negative 3.

Exclude the boundary.

Check a value to the left.

Try it yourself.

Check your understanding

Keep practising