Graphing Inequalities: Shading Regions
Some IGCSE questions ask you to show an inequality as a region on a graph rather than as a solution on a number line. This page explains how to draw the boundary line for an inequality, how to decide which side to shade, when to use a solid or dashed line, and how to find the region that satisfies several inequalities at once, with worked examples and a practice room of randomised, auto-marked questions.
On a number line, an inequality such as \(x > 2\) is shown as a section of the line. On a pair of axes, an inequality is shown as a region: a whole area of the grid where every point satisfies the inequality. This page shows how to draw the boundary line, decide which side to shade, choose a solid or dashed line, and combine several inequalities into one region.
Step 1: Draw the Boundary Line
Every inequality has a matching boundary line: you get it by replacing the inequality sign with an equals sign. So the region for \(y > 2x + 1\) has the boundary line \(y = 2x + 1\). Draw that line first, exactly as you would any straight line.
📌 Solid or dashed?
The type of line shows whether points on the line are included:
- Solid line for \(\le\) or \(\ge\): the boundary is part of the region.
- Dashed line for \(<\) or \(>\): the boundary is not part of the region.
| Sign | Meaning | Line |
|---|---|---|
| \(<\) | less than | Dashed |
| \(>\) | greater than | Dashed |
| \(\le\) | less than or equal to | Solid |
| \(\ge\) | greater than or equal to | Solid |
A Quick Reminder: Vertical and Horizontal Lines
The simplest boundaries are vertical and horizontal lines, and they appear constantly in region questions. A line like \(x = 8\) is vertical (every point on it has \(x\)-coordinate 8); a line like \(y = -2\) is horizontal (every point on it has \(y\)-coordinate \(-2\)). The shading works the same way: pick the side that satisfies the inequality.
💡 The region x ≤ 8
Solid vertical line at \(x = 8\); shade the side where x is 8 or less (to the left).
Everything on or left of the line satisfies x ≤ 8.
💡 The region y ≥ -2
Solid horizontal line at \(y = -2\); shade the side where y is \(-2\) or more (above).
Everything on or above the line satisfies y ≥ -2.
📌 Easy to mix up
"\(x = \,\)" lines are vertical; "\(y = \,\)" lines are horizontal. A quick check: \(x = 8\) only fixes the across position, so the line runs straight up and down through 8 on the x-axis.
Step 2: Decide Which Side to Shade
Once the line is drawn, you need the correct side. The reliable method is to pick a test point not on the line (the origin \((0,0)\) is easiest when the line does not pass through it), and check whether it satisfies the inequality. If it does, that point is in the region; if not, the region is on the other side.
📌 The test-point method
- Substitute the test point into the inequality.
- If the inequality is true, shade the side containing that point.
- If it is false, shade the other side.
Note: in some exam questions you shade the required region; in others you shade the unwanted region and label the answer R. Always read the instruction carefully.
Worked Examples
💡 Example 1: A single inequality
Show the region that satisfies \(y \ge 2x - 1\).
Boundary line: \(y = 2x - 1\) (solid, because of \(\ge\)).
Test the origin \((0,0)\):
\[ 0 \ge 2(0) - 1 \;\Rightarrow\; 0 \ge -1 \]True (✓), so shade the side containing the origin (above the line).
Solid line; region shaded above (the origin side).
What's happening?
The \(\ge\) sign means the line itself counts, so it is solid. The origin makes the inequality true, so the region is the side that contains it.
💡 Example 2: Three inequalities Higher
Show the region satisfying \(x \ge 1\), \(y \ge 1\) and \(x + y \le 5\).
Boundaries: \(x = 1\), \(y = 1\), \(x + y = 5\) (all solid).
Test the origin \((0,0)\):
\[\begin{aligned} x \ge 1: &\; 0 \ge 1 \text{ false} \\ y \ge 1: &\; 0 \ge 1 \text{ false} \\ x + y \le 5: &\; 0 \le 5 \text{ true} \end{aligned}\]Shade right of \(x=1\), above \(y=1\), origin side of \(x+y=5\).
The shaded triangle satisfies all three at once.
What's happening?
Each inequality removes part of the grid. The only area satisfying all three is the triangle where the three shaded sides overlap.
🔑 Key Points
- Replace the inequality sign with \(=\) to get the boundary line, then draw it.
- Solid line for \(\le\) and \(\ge\); dashed line for \(<\) and \(>\).
- "\(x=\,\)" lines are vertical; "\(y=\,\)" lines are horizontal.
- Use a test point (the origin if possible) to decide which side to shade.
- For several inequalities, the region is where all the shaded sides overlap.
- Read whether the question wants the required or the unwanted region shaded.
⚠ Common Mistakes
- Wrong line type: solid for \(<\)/\(>\) or dashed for \(\le\)/\(\ge\).
- Shading the wrong side by guessing instead of testing a point.
- Using the origin as a test point when the line passes through it.
- Confusing vertical and horizontal: \(x=\) is vertical, \(y=\) is horizontal.
- Stopping at one boundary when several inequalities are given.