Linear inequalities | Year 12

01 / LEARN

Understand the idea.

Read the restriction carefully

Translate the wording before calculating: at most includes equality and gives $\leq$ ; at least includes equality and gives $\geq$ . Words such as less than or more than give strict inequalities, which exclude the boundary.

Adding or subtracting the same amount preserves order. Multiplying or dividing by a negative reverses it: for example, $2 < 5$ becomes $- 2 > - 5$ . This reversal matters when isolating a variable with a negative coefficient.

Choose the correct half-plane

For two variables, draw the equality boundary first. A solid line includes the boundary; a dashed line excludes it. Test a point away from that line to decide which half-plane satisfies the inequality.

Add restrictions from the situation separately. A resource bound alone does not prevent negative output or fractional chairs. Counts generally require non-negative integers, and a proposed plan must satisfy those conditions as well as the resource limit.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A negative coefficient

Solve $7 - 2 x \geq 15$ .

Worked solution

1
Remove the constant term.
$\begin{aligned} 7 - 2 x & \geq 15 \\ - 2 x & \geq 8 \end{aligned}$
Subtracting 7 from both sides preserves the direction of the inequality.
2
Divide by the negative coefficient.
$x \leq - 4$
Division by $- 2$ reverses the order, so the allowed values lie at or below $- 4$ .
3
Check the boundary and allowed side.
$\begin{aligned} 7 - 2 (- 4) & = 15 \\ 7 - 2 (- 5) & = 17 \geq 15 \end{aligned}$
The boundary is included, and a smaller $x$ produces an output above 15.

Answer

$x \leq - 4$ .

EXAMPLE 02

A resource limit

A chair takes 3 hours and a stool 2 hours. At most 30 hours are available. Write the restriction and test 6 chairs and 5 stools.

Worked solution

1
Define the production counts.
$x, y \in {0, 1, 2, \dots}$
Let $x$ count chairs and $y$ count stools; negative or fractional furniture is not permitted.
2
Express the hours used.
$3 x + 2 y \leq 30$
Chairs use 3 hours each and stools 2 hours each; at most 30 hours includes equality.
3
Test the proposed plan.
$3 (6) + 2 (5) = 18 + 10 = 28 h$
Substitute the chair and stool counts into the total-hours expression.
4
Compare with the available time.
$28 \leq 30, 30 - 28 = 2 h$
The counts are allowed integers and leave 2 hours unused, so this plan meets the stated constraints.

Answer

$3 x + 2 y \leq 30$ ; the proposed plan is feasible.

03 / YOUR TURN

Try it yourself.

Check your understanding

Solve $5 x + 4 < 24$ .

Show answer & reasoning

$x < 4$ .

Subtracting 4 gives $5 x < 20$ . Dividing by positive 5 preserves the direction, giving $x < 4$ ; the boundary $x = 4$ is excluded because it gives equality.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Solve $3 x - 5 \leq 16$ .
2 marks
Solve $10 - 4 x > 22$ .
2 marks
A venue holds no more than 90 people. Adult and child bookings are $a$ and $c$ . Write a restriction.
2 marks
A minimum revenue of AUD 180 is needed. Standard items earn AUD 12 and deluxe items AUD 20. Write the restriction.
2 marks
Does $(4, 3)$ satisfy both $2 x + y \leq 12$ and $x + 2 y \geq 11$ ?
2 marks
For $x + y < 8$ , is $(3, 5)$ allowed?
2 marks

Read the worked solutions

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

Solve $3 x - 5 \leq 16$ .
2 marks
Add 5, then divide by 3.
$x \leq 7$ .
Solve $10 - 4 x > 22$ .
2 marks
$- 4 x > 12$ ; dividing by negative 4 reverses the symbol.
$x < - 3$ .
A venue holds no more than 90 people. Adult and child bookings are $a$ and $c$ . Write a restriction.
2 marks
No more than includes the maximum.
$a + c \leq 90$ .
A minimum revenue of AUD 180 is needed. Standard items earn AUD 12 and deluxe items AUD 20. Write the restriction.
2 marks
Minimum means at least.
$12 x + 20 y \geq 180$ .
Does $(4, 3)$ satisfy both $2 x + y \leq 12$ and $x + 2 y \geq 11$ ?
2 marks
The first sum is 11, but the second is 10, below the required 11.
No.
For $x + y < 8$ , is $(3, 5)$ allowed?
2 marks
Its sum equals 8 and the strict inequality excludes equality.
No.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics: Modelling with Linear Relationships, a compulsory school-assessed topic. The external examination scope differs from the full course.

Official curriculum source ↗

In this lesson

Understand the idea.

Read the restriction carefully

Choose the correct half-plane

See the method in action.

A negative coefficient

Remove the constant term.

Divide by the negative coefficient.

Check the boundary and allowed side.

A resource limit

Define the production counts.

Express the hours used.

Test the proposed plan.

Compare with the available time.

Try it yourself.

Check your understanding

Keep practising