Topics

Linear inequations and their regions

From equation to inequation

An equation such as $2x + 3y = 12$ is satisfied by the points on a line.
An inequation such as $2x + 3y \leq 12$ is satisfied by every point on one side of that line, and usually the line itself.
The solution to an inequation is a region, not a point. That is the whole difference, and it is what makes inequations the right model for "at most", "no more than" and "at least".

The words and the symbols

Words in the context	Symbol
at most, no more than, up to, maximum of, cannot exceed	$\leq$
at least, no fewer than, minimum of, must be	$\geq$
fewer than, under, less than	$<$
more than, over, exceeds	$>$

"At most" and "less than" are different. "At most 20" includes 20; "fewer than 20" does not. The distinction decides whether your boundary line is solid or dashed.

Drawing the region

Step 1 — Draw the boundary. Replace the inequality sign with $=$ $=$ and draw that line.
- Find two points, usually the intercepts: set $x = 0$ to get the $y$ -intercept, set $y = 0$ to get the $x$ -intercept.
Step 2 — Choose the line style.
- $\leq$ or $\geq$ → solid line: the boundary is included.
- $<$ or $>$ → dashed line: the boundary is not included.
Step 3 — Test a point that is not on the line.
- Use $(0, 0)$ whenever the line does not pass through it — the arithmetic is trivial.
- If the test point satisfies the inequation, shade its side. If not, shade the other side.
Step 4 — Shade and label. State clearly whether the shading shows the region that satisfies the inequation or the region that is excluded, and keep that convention for every inequation on the diagram.

Testing beats remembering. Rules like " $\leq$ means shade below" fail as soon as the coefficient of $y$ is negative. Substituting a point never fails.

Why the sign can flip

Multiplying or dividing an inequation by a negative number reverses the inequality sign.
- From $-2y > 6$ , dividing by $-2$ gives $y < -3$ .
This is the single most common algebraic error with inequations.
Avoid it entirely by not rearranging: leave the inequation as it is, draw the boundary from the intercepts, and test a point.

The non-negativity conditions

In almost every real context the variables cannot be negative:

$x \geq 0, \qquad y \geq 0$

These restrict everything to the first quadrant.
They are part of the system and must be written down. You cannot make $-3$ chairs, and a model that allows it is wrong even if the arithmetic that follows is right.

Reading an inequation off a graph

The reverse task — given a shaded half-plane, state its inequation:
- Find the equation of the boundary line from its intercepts or gradient.
- Decide the direction by testing a point that is clearly inside the shaded area.
- Decide the strictness from the line style: solid → $\leq$ or $\geq$ , dashed → $<$ or $>$ .
This is exactly the "connecting representations" EN4 asks for, and it appears in internals as often as the forward direction.

Tips and Tricks

Test $(0,0)$ whenever you can. Substituting zeros takes seconds and removes all doubt about which side to shade. Only pick another point when the boundary passes through the origin.

Solid or dashed is worth stating. $\leq$ gives a solid line and $<$ a dashed one, and an internal task that says "fewer than 20 hours" is testing whether you noticed.

The sign flips when you divide by a negative. If you must rearrange, write the flip down as its own step so it does not get lost.

Write $x \geq 0$ and $y \geq 0$ explicitly whenever the variables count real things. Assessors look for them, and they change the shape of the region.

Say which side your shading means. Shading the satisfying region and shading the excluded region are both used in practice; only one of them matches your answer, so label it.

Worked ExampleDrawing a single inequation

A community hall can be hired for events. Each hour of hire costs $40 and each extra staff member costs $60 for the event. The budget is at most $720. Write an inequation, draw its region, and interpret two specific points.

Step 1 — Define the variables

Step 2 — Form the inequation

Total cost is $40h + 60s$ dollars, and the budget cannot be exceeded, so:

$40h + 60s \leq 720$

"At most" gives $\leq$ , which means the boundary line will be solid.

Step 3 — Simplify before drawing

Every term divides by 20:

$2h + 3s \leq 36$

Step 4 — Draw the boundary line $2h + 3s = 36$

Find the intercepts:

Set $h = 0$ : $3s = 36$ , so $s = 12$ . Point $(0, 12)$ .
Set $s = 0$ : $2h = 36$ , so $h = 18$ . Point $(18, 0)$ .

Draw a solid line through $(0, 12)$ and $(18, 0)$ , because the boundary is included.

Step 5 — Test the origin

Substitute $(0, 0)$ into the inequation:

$2(0) + 3(0) = 0 \leq 36 \quad \textbf{TRUE}$

Step 6 — Add the non-negativity conditions

Hours and staff cannot be negative:

$h \geq 0, \qquad s \geq 0$

The region is therefore the triangle with corners $(0,0)$ , $(18, 0)$ and $(0, 12)$ .

Step 7 — Interpret two specific points

Point $(12, 4)$ — 12 hours with 4 extra staff:

$2(12) + 3(4) = 24 + 12 = 36 \leq 36 \quad \textbf{TRUE}$

Point $(15, 3)$ — 15 hours with 3 extra staff:

$2(15) + 3(3) = 30 + 9 = 39 > 36 \quad \textbf{FALSE}$

Step 8 — State what the region means

One practical restriction the graph does not show: only points with whole-number coordinates are usable, since the hall is hired by the hour and staff come as whole people. The region contains infinitely many points, but only the lattice points inside it are real options.

Linear inequations and their regions

Covers AS91269

From equation to inequation

An equation such as $2x + 3y = 12$ is satisfied by the points on a line.
An inequation such as $2x + 3y \leq 12$ is satisfied by every point on one side of that line, and usually the line itself.
The solution to an inequation is a region, not a point. That is the whole difference, and it is what makes inequations the right model for "at most", "no more than" and "at least".

The words and the symbols

Words in the context	Symbol
at most, no more than, up to, maximum of, cannot exceed	$\leq$
at least, no fewer than, minimum of, must be	$\geq$
fewer than, under, less than	$<$
more than, over, exceeds	$>$

"At most" and "less than" are different. "At most 20" includes 20; "fewer than 20" does not. The distinction decides whether your boundary line is solid or dashed.

Drawing the region

Step 1 — Draw the boundary. Replace the inequality sign with $=$ $=$ and draw that line.
- Find two points, usually the intercepts: set $x = 0$ to get the $y$ -intercept, set $y = 0$ to get the $x$ -intercept.
Step 2 — Choose the line style.
- $\leq$ or $\geq$ → solid line: the boundary is included.
- $<$ or $>$ → dashed line: the boundary is not included.
Step 3 — Test a point that is not on the line.
- Use $(0, 0)$ whenever the line does not pass through it — the arithmetic is trivial.
- If the test point satisfies the inequation, shade its side. If not, shade the other side.
Step 4 — Shade and label. State clearly whether the shading shows the region that satisfies the inequation or the region that is excluded, and keep that convention for every inequation on the diagram.

Testing beats remembering. Rules like " $\leq$ means shade below" fail as soon as the coefficient of $y$ is negative. Substituting a point never fails.

Why the sign can flip

Multiplying or dividing an inequation by a negative number reverses the inequality sign.
- From $-2y > 6$ , dividing by $-2$ gives $y < -3$ .
This is the single most common algebraic error with inequations.
Avoid it entirely by not rearranging: leave the inequation as it is, draw the boundary from the intercepts, and test a point.

The non-negativity conditions

In almost every real context the variables cannot be negative:

$x \geq 0, \qquad y \geq 0$

These restrict everything to the first quadrant.
They are part of the system and must be written down. You cannot make $-3$ chairs, and a model that allows it is wrong even if the arithmetic that follows is right.

Reading an inequation off a graph

The reverse task — given a shaded half-plane, state its inequation:
- Find the equation of the boundary line from its intercepts or gradient.
- Decide the direction by testing a point that is clearly inside the shaded area.
- Decide the strictness from the line style: solid → $\leq$ or $\geq$ , dashed → $<$ or $>$ .
This is exactly the "connecting representations" EN4 asks for, and it appears in internals as often as the forward direction.

Tips and Tricks

Test $(0,0)$ whenever you can. Substituting zeros takes seconds and removes all doubt about which side to shade. Only pick another point when the boundary passes through the origin.

Solid or dashed is worth stating. $\leq$ gives a solid line and $<$ a dashed one, and an internal task that says "fewer than 20 hours" is testing whether you noticed.

The sign flips when you divide by a negative. If you must rearrange, write the flip down as its own step so it does not get lost.

Write $x \geq 0$ and $y \geq 0$ explicitly whenever the variables count real things. Assessors look for them, and they change the shape of the region.

Say which side your shading means. Shading the satisfying region and shading the excluded region are both used in practice; only one of them matches your answer, so label it.

Worked ExampleDrawing a single inequation

Step 1 — Define the variables

Step 2 — Form the inequation

Total cost is $40h + 60s$ dollars, and the budget cannot be exceeded, so:

$40h + 60s \leq 720$

"At most" gives $\leq$ , which means the boundary line will be solid.

Step 3 — Simplify before drawing

Every term divides by 20:

$2h + 3s \leq 36$

Step 4 — Draw the boundary line $2h + 3s = 36$

Find the intercepts:

Set $h = 0$ : $3s = 36$ , so $s = 12$ . Point $(0, 12)$ .
Set $s = 0$ : $2h = 36$ , so $h = 18$ . Point $(18, 0)$ .

Draw a solid line through $(0, 12)$ and $(18, 0)$ , because the boundary is included.

Step 5 — Test the origin

Substitute $(0, 0)$ into the inequation:

$2(0) + 3(0) = 0 \leq 36 \quad \textbf{TRUE}$

Step 6 — Add the non-negativity conditions

Hours and staff cannot be negative:

$h \geq 0, \qquad s \geq 0$

The region is therefore the triangle with corners $(0,0)$ , $(18, 0)$ and $(0, 12)$ .

Step 7 — Interpret two specific points

Point $(12, 4)$ — 12 hours with 4 extra staff:

$2(12) + 3(4) = 24 + 12 = 36 \leq 36 \quad \textbf{TRUE}$

Point $(15, 3)$ — 15 hours with 3 extra staff:

$2(15) + 3(3) = 30 + 9 = 39 > 36 \quad \textbf{FALSE}$

Linear inequations and their regions

From equation to inequation

The words and the symbols

Drawing the region

Why the sign can flip

The non-negativity conditions

Reading an inequation off a graph

Step 1 — Define the variables

Step 2 — Form the inequation

Step 3 — Simplify before drawing

Step 4 — Draw the boundary line 2h+3s=362h + 3s = 362h+3s=36

Step 5 — Test the origin

Step 6 — Add the non-negativity conditions

Step 7 — Interpret two specific points

Step 8 — State what the region means

Linear inequations and their regions

From equation to inequation

The words and the symbols

Drawing the region

Why the sign can flip

The non-negativity conditions

Reading an inequation off a graph

Step 1 — Define the variables

Step 2 — Form the inequation

Step 3 — Simplify before drawing

Step 4 — Draw the boundary line 2h+3s=362h + 3s = 362h+3s=36

Step 5 — Test the origin

Step 6 — Add the non-negativity conditions

Step 7 — Interpret two specific points

Step 8 — State what the region means

Step 4 — Draw the boundary line $2h + 3s = 36$

Step 4 — Draw the boundary line $2h + 3s = 36$