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

01 / LEARN

The idea, step by step

Inequalities give sets of values

The signs $<$ and $>$ exclude equality; $\leq$ and $\geq$ include it. Thus $x < 2$ means every real number less than 2, not just the integers.

Use an open endpoint when equality is excluded and a filled one when it is included. Shade left for smaller values and right for larger ones.

Know when the sign reverses

Adding or subtracting the same number preserves the direction. Multiplying or dividing by a positive number also preserves it. A negative multiplier or divisor reverses the ordering.

For instance, multiplying 1 < 3 by -1 gives -1 > -3. Use a sample solution and the boundary to check your result.

A common trap

Using a filled endpoint for a strict inequality.

Instead: A strict comparison excludes its boundary, so use an open endpoint.

02 / WORKED EXAMPLES

See the thinking

Example 1

An included boundary

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

Subtract 3 from both sides.
Equality is permitted, so fill the endpoint at 2.
Extend to the left.

x \leq 2

Example 2

A negative divisor

Solve $- 3 x > 6$ .

Divide both sides by -3 and reverse the sign.
The boundary -2 produces equality and is excluded.
Testing x = -3 gives 9 > 6, confirming the direction.

x < - 2

Check your understanding

Solve $2 x - 1 > 5$ .

Show answer

$x > 3$ .

Add 1 and divide by positive 2.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF 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…

Continue learning

Tables and straight-line graphs

Next lesson

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 ↗

Back to Linear equations and graphs