Graphical Representation of Inequalities

Linear inequalities often need to be rearranged into the standard straight-line format ($y = mx + c$) so you can easily identify the gradient and y-intercept. Remember that $x = k$ creates a vertical line passing through $k$ on the x-axis, while $y = k$ creates a horizontal line passing through $k$ on the y-axis.

Line Styles: Solid vs. Dashed

The style of the line you sketch is crucial because it tells the examiner whether the values exactly on the line are included in the solution set.

• An inclusive inequality ($\le$ or $\ge$) is represented by a solid line. Points sitting exactly on this line are part of the solution.
• A strict inequality ($<$ or $>$) is represented by a dashed line (or dotted line). Crucially, points exactly on a dashed line are not part of the solution set.

Finding the Feasible Region

The feasible region (or Region R) is the specific area on a graph where all given inequalities in a system are satisfied simultaneously. To find which side of a boundary line forms this region, you use a test point.

A test point is a coordinate $(x, y)$ that does not sit on the boundary line itself. The origin $(0, 0)$ is usually the easiest coordinate to test. You substitute the $x$ and $y$ values of your test point into the original inequality. If the mathematical statement is true, the side containing your test point is the "wanted" region. If it is false, the opposite side is the wanted region.

As a general shortcut for vertical lines, $x > k$ means the region is to the right, and $x < k$ means the region is to the left. For $y > mx + c$, the region is above the line, and for $y < mx + c$, it is below.

Shading Conventions

Edexcel accepts two primary methods for indicating the final region on your sketch:

1. Shading the "wanted" region: Colouring the area where all inequalities are true.
2. Shading out the unwanted region: Colouring the areas that do not satisfy the inequalities, leaving the feasible region blank and clear.

Whichever method you choose, you must always label the final overlapping area satisfying all inequalities with a clear capital R.

Worked Example

Sketch the region R satisfying $3x + 2y \ge 12$, $y < 2x$, and $x < 3$.

Step 1: Find and test the first boundary line ($3x + 2y = 12$).

• Rearrange into $y = mx + c$: $2y = -3x + 12$ becomes $y = -1.5x + 6$.
• Sketch as a solid line because the inequality is $\ge$.
• Test point $(0,0)$: $3(0) + 2(0) \ge 12 \rightarrow 0 \ge 12$ is False.
• The wanted side is above the line.

Step 2: Find and test the second boundary line ($y = 2x$).

• Sketch as a dashed line because the inequality is $<$.
• The line passes through $(0,0)$, so you cannot use it as a test point. Test $(1,0)$ instead: $0 < 2(1) \rightarrow 0 < 2$ is True.
• The wanted side contains $(1,0)$ (below the line).

Step 3: Find and test the third boundary line ($x = 3$).

• Sketch a vertical dashed line at $x = 3$ because the inequality is $<$.
• The wanted side is to the left ($x < 3$).

Step 4: Finalise the sketch.

• Identify the central triangular area where all "wanted" sides overlap.
• Shade out the unwanted areas to keep the drawing clean.
• Label the clear triangular space with a capital R.