Two-variable inequalities and shaded regions | Year 10

01 / LEARN

Understand the idea.

Draw the boundary before shading

A linear inequality in two variables selects a half-plane. Replace the inequality by equality to draw its boundary. Use a solid boundary when equality is permitted and a dashed boundary for a strict inequality; boundary points are then excluded.

Choose a point away from the boundary and substitute it into the original inequality. A true result selects the side containing that point; a false result selects the other side. A point on the boundary cannot distinguish the sides.

More than one condition

A feasible region contains points satisfying every condition together. Shade the overlap, including or excluding each boundary as required. Axis restrictions such as $x \geq 0$ and $y \geq 0$ include the axes. For object counts, only nonnegative whole-number points in the overlap are meaningful.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A triangular feasible region

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

Equal-scale first-quadrant axes run from zero to seven in unit steps. A pale sage triangle has vertices (0,0), (6,0) and (0,6), with its sloping solid boundary labelled x plus y equals six.

Worked solution

1
Apply the nonnegative coordinate conditions.
$x \geq 0, y \geq 0$
These select the first quadrant together with its axes.
2
Find the intercepts of the remaining boundary.
$\begin{aligned} y = 0 & ⟹ x = 6 \\ x = 0 & ⟹ y = 6 \end{aligned}$
The line $x + y = 6$ passes through $(6, 0)$ and $(0, 6)$ ; equality means it is solid.
3
Test the origin to choose the allowed side.
$0 + 0 \leq 6$
The true statement selects the half-plane containing the origin.
4
Describe the intersection.
$(0, 0), (6, 0), (0, 6)$
The three included boundaries enclose the closed triangle with these vertices, including its interior.

Answer

The closed triangle with vertices $(0, 0), (6, 0), (0, 6)$ .

EXAMPLE 02

A strict boundary

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

Worked solution

1
Draw the boundary with the correct style.
$y = 2 x + 1$
The original inequality is strict, so the line is dashed and its points are excluded.
2
Test a point away from the line.
$(0, 0) : 0 < 2 (0) + 1$
The origin is not on the boundary and makes the original inequality true.
3
Identify the selected half-plane.
$y < 2 x + 1$
For each fixed $x$ , allowed $y$ values are less than the line height, so shade below the dashed line.

Answer

Below the dashed line, excluding the boundary.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Yes.

At $(2, 3)$ , the first condition gives $2 + 3 = 5 \leq 6$ and the second gives $3 \geq 2$ . Both are true, so the point belongs to their overlap.

Keep practising

6 worksheet questions · 15 marks

Open worksheet 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.

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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 ↗

In this lesson

Understand the idea.

Draw the boundary before shading

More than one condition

See the method in action.

A triangular feasible region

Apply the nonnegative coordinate conditions.

Find the intercepts of the remaining boundary.

Test the origin to choose the allowed side.

Describe the intersection.

A strict boundary

Draw the boundary with the correct style.

Test a point away from the line.

Identify the selected half-plane.

Try it yourself.

Check your understanding

Keep practising