Graphical Linear Inequalities

Finding the Solution Region

How do you know which side of a boundary line represents the correct mathematical zone if you only have a single equation? Once your line is drawn, you must identify the correct region by using a test point.

• First, select a coordinate that is definitely not sitting on your boundary line. The origin $(0, 0)$ is usually the fastest and easiest choice.
• Then, substitute the $x$ and $y$ values of this test point into your original inequality.
• Finally, evaluate the mathematical statement. If the statement is true, the side of the line containing your test point is the solution. If the statement is false, the opposite side of the line is the solution.
• If your boundary line happens to pass exactly through the origin, you cannot use $(0, 0)$; pick an alternative test point like $(1, 0)$ or $(1, 1)$ instead.

Shading Conventions and Region Labels

It might seem counterintuitive, but sometimes the most effective way to highlight a specific area is to scribble over everything else. OCR exams have specific conventions for how you should present your final graphical answers.

• It is highly recommended to shade the unwanted region. By crossing out the areas that do not satisfy the inequalities, the correct solution set remains entirely blank and clearly visible.
• Only shade the wanted region if the exam question explicitly instructs you to "Shade the region that satisfies...".
• When a question involves multiple inequalities at once, the final solution is the overlapping area where all statements are true simultaneously. You will frequently be instructed to label this specific overlapping area as Region R.

Worked Example

On a coordinate grid, identify and label the region R that satisfies both $y < 2x + 2$ and $y \geq 1$. Shade the unwanted regions.

Step 1: Draw the boundary lines.

• For $y \geq 1$: The symbol is inclusive, so draw a solid horizontal line passing through $y = 1$.
• For $y < 2x + 2$: The symbol is strict, so use a ruler to draw a dashed line passing through the coordinates $$ and .

Step 2: Use test points to identify the correct regions.

• For $y \geq 1$: Substitute $(0, 0)$. The statement $0 \geq 1$ is False. Therefore, the wanted region is the side not containing $(0, 0)$ (above the line).

Step 3: Shade the unwanted regions and label the solution.

• Shade the area below $y = 1$.
• Shade the area above $y = 2x + 2$.
• The clear, unshaded area that remains is the solution set. Because only two inequalities are used, this forms an unbounded, infinite wedge-shape. Write a large "R" inside this space.

Integer Coordinates and Set Notation

Finding a general area is useful, but sometimes examiners will ask you to pinpoint exact locations within that restricted zone. Some Higher tier OCR questions require you to mark all points with integer (whole number) coordinates inside the solution region with a cross.

• When listing or marking these integer coordinates, points that sit directly on a solid line are valid and must be included.
• However, points that sit directly on a dashed line are invalid and must be excluded.
• You may also encounter formal set notation in Higher tier papers, written in the format $\{ (x, y) : \text{inequality} \}$. For example, $\{ (x, y) : y \leq x + 3 \}$ translates simply to "the set of all coordinates where is less than or equal to ".