Linear inequalities | Year 12

01 / LEARN

The idea, step by step

Read the restriction carefully

At most means less than or equal to; at least means greater than or equal to. Strict inequalities exclude their boundaries. Solving works like an equation except that multiplying or dividing by a negative reverses the order. Adding or subtracting a number does not reverse the symbol.

Choose the correct half-plane

Draw the equality boundary first. Use a solid boundary when equality is allowed and a dashed one when it is excluded. Test a point that is not on the line to identify the permitted side. Add non-negativity or whole-number restrictions if required by the context; they are not automatically included in a resource inequality.

A common trap

Every boundary belongs to its feasible region.

Instead: Strict inequalities exclude their boundary; only non-strict inequalities allow equality.

02 / WORKED EXAMPLES

See the thinking

Example 1

A negative coefficient

Solve $7 - 2 x \geq 15$ .

Subtract 7: $- 2 x \geq 8$ .
Divide by negative 2 and reverse the inequality.
Check $x = - 4$ gives equality and $x = - 5$ gives an allowed value.

x \leq - 4

Example 2

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.

Let x count chairs and y count stools. Then $3 x + 2 y \leq 30$ , with non-negative integer counts.
At $(6, 5)$ , hours used are $18 + 10 = 28$ .
Because $28 \leq 30$ , the plan meets the time limit.

3 x + 2 y \leq 30

; the proposed plan is feasible.

Check your understanding

Solve $5 x + 4 < 24$ .

Show answer

$x < 4$ .

Subtract 4, then divide by positive 5.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF 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 ↗

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