GradeGen.AI

Mapping the Boundaries

Imagine putting up a fence around your garden. The physical fence marks the edge, but does the boundary itself belong to you or your neighbour? In mathematics, when we sketch inequalities on a graph, the line style tells us whether the boundary is included in our solution.

A strict inequality uses $<$ or $>$ symbols. Because the boundary values are not included in the solution set, we represent this on a graph using a dashed (or dotted) line.
An inclusive inequality uses $\leq$ or $\geq$ symbols. Because the boundary values are part of the solution, we represent this on a graph using a solid (unbroken) line.

Drawing the Lines

To sketch an inequality, first treat it as an equation to find the boundary line. This is the line formed by replacing the inequality sign with an equals sign.

Vertical lines take the form $x = k$ and pass through $k$ on the $x$ -axis.
Horizontal lines take the form $y = k$ and pass through $k$ on the $y$ -axis.
Diagonal lines take the form $y = mx + c$ and may require rearranging before plotting.

If you need to rearrange an equation and multiply or divide the inequality by a negative number, you must flip the inequality sign.

Finding the Feasible Region

Once your lines are drawn, you must determine which side of each line satisfies the inequality. The area where all given inequalities are satisfied simultaneously is called the feasible region.

AQA's official guidance is to create a shaded region over the unwanted (non-required) areas.
This leaves the wanted feasible region clear and unshaded, making it clearly legible.
In an exam, you must almost always label this clear area with a capital 'R'.

To check which side to shade, use the Test Point method. Choose a coordinate not on the line, such as $(0, 0)$ . Substitute these values into the inequality; if the mathematical statement is true, that side is the "wanted" region, so you should shade the opposite side. If the statement is false, that side is the "unwanted" region, so you should shade it.

Worked Example: Sketching Multiple Inequalities

Represent the region R satisfied by $x \leq 4$ , $y > 2$ , and $y \leq x + 1$ .

Step 1: Plot the boundary line $x = 4$ .

Because the inequality is $\leq$ (inclusive), use a solid line.
The wanted area is $x \leq 4$ (left), so shade the unwanted area to the right ( $x > 4$ ).

Step 2: Plot the boundary line $y = 2$ .

Because the inequality is $>$ (strict), use a dashed line.
The wanted area is $y > 2$ (above), so shade the unwanted area below ( $y < 2$ ).

Step 3: Plot the boundary line $y = x + 1$ .

Because the inequality is $\leq$ (inclusive), use a solid line (gradient 1, $y$ -intercept 1).
Test the point $(0, 0)$ : $0 \leq 0 + 1$ is true.
Since is below the line and is "wanted", shade the unwanted area above the line.

Step 4: Label the final region.

Mark the remaining clear triangular area with a capital 'R'.

[Insert diagram: A coordinate grid showing three boundary lines. A solid vertical line at $x = 4$ , a dashed horizontal line at $y = 2$ , and a solid diagonal line $y = x + 1$ . The area to the right of $x = 4$ , the area below , and the area above are all shaded out to represent the unwanted regions. This leaves a clear, unshaded triangular feasible region bounded by vertices at , , and . This clear region is labelled with a large capital 'R'.]

Integer Solutions

Higher Tier questions may ask you to "mark with a cross all points with integer coordinates in region R."

Points that sit exactly on solid lines are included in the region and should be marked.
Points that sit exactly on dashed lines are excluded from the region and must NOT be marked.

Students frequently lose a mark by using a solid line for a strict inequality (< or >); always double-check the symbol before putting pen to paper.

For AQA exams, you should shade the UNWANTED regions so that the region 'R' remains clear and easy for the examiner to see, unless the question explicitly tells you otherwise.

If a question tells you to 'Label the region R', simply shading or leaving an area clear is not enough—you must physically write a capital 'R' inside the correct space.

In Higher Tier questions asking for integer coordinates, remember that points sitting exactly on a dashed line are excluded and should not be marked.

An inequality using < or > symbols where the boundary value is excluded from the solution, represented by a dashed line.

An inequality using ≤ or ≥ symbols where the boundary value is part of the solution, represented by a solid line.

The line formed on a graph by replacing an inequality sign with an equals sign.

The area on a graph that is blocked out; in AQA exams, this convention is used to hide the unwanted (non-required) areas.

The set of all points on a coordinate grid that satisfy a system of multiple inequalities simultaneously, typically labelled with an 'R'.