Every time you set a budget filter while shopping online, you are defining a zone of acceptable options rather than a single specific item. Similarly, a linear inequality in two variables defines an entire half-plane of valid coordinates on a graph, rather than just a single line. The edge of this region is called the boundary line, which separates the coordinates that satisfy the inequality from those that do not.
When graphing an inequality, the style of the boundary line visually communicates whether the numbers exactly on the line are included in the solution:
To easily plot the boundary line, you must usually rearrange the inequality into the format.
Once the boundary line is drawn, you must determine which side of the line represents the correct solution. To do this, pick a test point — a specific coordinate pair not located on the line — and substitute it into the original inequality.
Calculate and represent the solution set for the inequality graphically. Shade the region NOT required.
Step 1: Rearrange into form to find the boundary line.
Step 2: Determine line style.
Step 3: Identify the solution region using a test point.
Step 4: Final annotated graph with the correct region shaded.
When multiple inequalities are plotted together, they enclose an overlapping area known as Region R. The corners of this region are called vertices.
While some vertices align perfectly with the grid, OCR Higher Tier questions often use the "Calculate" command word to demand algebraic proof of exactly where two boundary lines intersect. To find the exact coordinates of a vertex, you must solve the equations of the two intersecting boundary lines using simultaneous equations.
Calculate the exact coordinates of the vertex formed by the intersection of the boundary lines and .
Step 1: Set up the simultaneous equations.
Step 2: Substitute the isolated variable into the other equation.
Step 3: Solve for .
Step 4: Substitute back to find .
Often, exams will ask you to identify the integer solutions within Region R. These are specific grid coordinates where both values are whole numbers. When listing these, check the interior of the clear region first, then evaluate the boundary lines: you must include integer coordinates that fall on solid boundary lines, but strictly ignore any that fall on a dashed line.
Forgetting to flip the inequality sign when dividing by a negative number during rearrangement, which guarantees you will shade the wrong side of the boundary line.
In OCR exams, always read the shading instructions carefully; you will frequently be asked to 'shade the area NOT required', which leaves the target solution area (Region R) blank and clearly visible.
Do not replace the inequality sign with an equals sign in your intermediate algebraic working, as OCR examiners may penalise this; only use an equals sign when explicitly writing the final equation of the boundary line.
When asked to list integer solutions, deliberately check points that land exactly on the boundaries: include them if the line is solid, but exclude them if the line is dashed.
If a boundary line passes exactly through the origin (0,0), you cannot use it as a test point; choose a simple alternative like (1,0) or (0,1) instead.
Linear inequality in two variables
An mathematical statement involving x and y that defines a region or half-plane on a Cartesian graph rather than a single line.
Boundary line
The straight line, drawn solid or dashed, that separates the coordinate plane into a region that satisfies an inequality and one that does not.
Strict inequality
An inequality using < or > symbols that explicitly excludes the boundary value, represented graphically by a dashed line.
Inclusive inequality
An inequality using ≤ or ≥ symbols that includes the boundary value, represented graphically by a solid line.
Test point
A specific coordinate pair, frequently (0,0), substituted into an inequality to verify which side of a boundary line satisfies the condition.
Feasible region
The area on a graph where all given inequalities in a system are satisfied simultaneously.
Region R
The specific area on a graph representing the intersection of multiple linear inequalities, typically left unshaded in OCR exams.
Vertex
An intersection point (corner) where two boundary lines meet, calculated accurately using simultaneous equations.
Integer solutions
Specific coordinates within a valid region where both the x and y values are whole numbers.
Put your knowledge into practice — try past paper questions for Mathematics
Linear inequality in two variables
An mathematical statement involving x and y that defines a region or half-plane on a Cartesian graph rather than a single line.
Boundary line
The straight line, drawn solid or dashed, that separates the coordinate plane into a region that satisfies an inequality and one that does not.
Strict inequality
An inequality using < or > symbols that explicitly excludes the boundary value, represented graphically by a dashed line.
Inclusive inequality
An inequality using ≤ or ≥ symbols that includes the boundary value, represented graphically by a solid line.
Test point
A specific coordinate pair, frequently (0,0), substituted into an inequality to verify which side of a boundary line satisfies the condition.
Feasible region
The area on a graph where all given inequalities in a system are satisfied simultaneously.
Region R
The specific area on a graph representing the intersection of multiple linear inequalities, typically left unshaded in OCR exams.
Vertex
An intersection point (corner) where two boundary lines meet, calculated accurately using simultaneous equations.
Integer solutions
Specific coordinates within a valid region where both the x and y values are whole numbers.