Two-variable inequalities and shaded regions | Year 10

01 / LEARN

The idea, step by step

Draw the boundary before shading

Replace the inequality by equality to obtain its boundary line. Use a solid line for $\leq$ or $\geq$ , since boundary points are included. Use a dashed line for $<$ or $>$ . Pick a point not on the boundary and substitute it into the original inequality to decide which side to shade.

More than one condition

A feasible region is the overlap of all allowed half-planes. Conditions such as $x \geq 0$ and $y \geq 0$ restrict the region to the first quadrant. Find corner points by intersecting boundaries, then check them against every condition. In contexts involving counts, only whole-number points in the region may be usable.

A common trap

Shading the side suggested by the direction of the inequality without substituting a point.

Instead: Rearranging may change how the boundary is written. A test point reliably identifies the correct half-plane.

02 / WORKED EXAMPLES

See the thinking

Example 1

A triangular feasible region

Describe the region $x \geq 0$ , $y \geq 0$ , $x + y \leq 6$ .

The first two conditions select the first quadrant.
The boundary $x + y = 6$ joins $(6, 0)$ and $(0, 6)$ and is included.
The origin satisfies $0 + 0 \leq 6$ , so select the side containing the origin.

The closed triangle with vertices

(0, 0), (6, 0), (0, 6)

Example 2

A strict boundary

Which side of $y = 2 x + 1$ satisfies $y < 2 x + 1$ ?

The boundary is dashed because equality is excluded.
Test $(0, 0)$ : $0 < 1$ is true.
Select the half-plane below the line.

Below the dashed line, excluding the boundary.

Check your understanding

Does $(2, 3)$ satisfy $x + y \leq 6$ and $y \geq x$ ?

Show answer

Yes.

$2 + 3 = 5 \leq 6$ and $3 \geq 2$ .

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Decide whether $(3, 2)$ satisfies $2 x + y \leq 9$ .
2 marks
Is the boundary of $y > 3 x - 2$ solid or dashed? Which side is allowed?
2 marks
Find the intercepts of the boundary $2 x + y = 8$ .
2 marks
Give the vertices of the feasible region $x \geq 0$ , $y \geq 0$ , $2 x + y \leq 8$ .
3 marks
On the supplied grid, shade the region $x \geq 0$ , $y \geq 0$ , $x + y < 5$ .
3 marks
A stall sells necklaces for 5 dollars and bracelets for 3 dollars. Write inequalities for buying at least 4 items using no more than 24 dollars. Define the variables.
3 marks

Read the worked solutions

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

Decide whether $(3, 2)$ satisfies $2 x + y \leq 9$ .
2 marks
$2 (3) + 2 = 8 \leq 9$ .
Yes.
Is the boundary of $y > 3 x - 2$ solid or dashed? Which side is allowed?
2 marks
Equality is excluded and larger $y$ values are required.
Dashed; above the line.
Find the intercepts of the boundary $2 x + y = 8$ .
2 marks
Set each variable to zero in turn.
$(4, 0)$ and $(0, 8)$
Give the vertices of the feasible region $x \geq 0$ , $y \geq 0$ , $2 x + y \leq 8$ .
3 marks
The boundary intercepts and origin form the triangle.
$(0, 0), (4, 0), (0, 8)$
On the supplied grid, shade the region $x \geq 0$ , $y \geq 0$ , $x + y < 5$ .
3 marks
The origin passes the inequality test.
The first-quadrant triangle below the dashed line from $(0, 5)$ to $(5, 0)$ ; the sloping boundary is excluded.
A stall sells necklaces for 5 dollars and bracelets for 3 dollars. Write inequalities for buying at least 4 items using no more than 24 dollars. Define the variables.
3 marks
Let $x$ count necklaces and $y$ count bracelets; total items and total cost give the two conditions.
$x + y \geq 4$ , $5 x + 3 y \leq 24$ , with $x, y$ nonnegative integers.

Checking saved progress…

Continue learning

Practical simultaneous/inequality models and checks

Next lesson

Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian Mathematics Curriculum

Year 10 simultaneous equations and inequalities with graphical, algebraic and practical interpretations.

Official curriculum source ↗

Back to Linear equations and inequalities