GCSE — mathsHigher2 min read

Graphing Inequalities on a Coordinate Plane

Shade regions for linear inequalities on a graph for GCSE Maths. Identify feasible regions from multiple constraints.

Tutor AI Team2026-02-25T01:44:39.640Z

Graphing inequalities extends solving inequalities into two dimensions. You draw a line and shade the region that satisfies the inequality. This is a Higher-tier GCSE topic.

Core Concepts

Drawing the Boundary Line

$y < mx + c$ or $y > mx + c$ : dashed line (boundary not included).
$y \leq mx + c$ or $y \geq mx + c$ : solid line (boundary included).

Shading the Correct Region

For $y < mx + c$ : shade below the line.
For $y > mx + c$ : shade above the line.

Test point method: Substitute a point (often $(0,0)$ ) into the inequality. If true, that side is the solution.

Multiple Inequalities

The feasible region (solution region) satisfies ALL inequalities simultaneously. It's the overlap of all shaded regions.

Horizontal and Vertical Lines

$x > a$ : shade right of the vertical line $x = a$ .
$y < b$ : shade below the horizontal line $y = b$ .

Worked Example: Example 1

Problem

Shade the region for $y \leq 2x + 1$ :

Draw $y = 2x + 1$ as a solid line.
Shade below the line.

Solution

Worked Example: Example 2

Problem

Find the region satisfying $x \geq 1$ , $y > 0$ , and $x + y < 5$ .

$x = 1$ : solid vertical line, shade right.
$y = 0$ : dashed line (x-axis), shade above.
$x + y = 5$ : dashed line, shade below.
The feasible region is the triangle where all three overlap.

Solution

Practice Problems

1. Shade $y > x - 2$ .
1. Find the feasible region for $x \geq 0$ , $y \geq 0$ , $x + y \leq 6$ .
1. Which integer points lie in the region $1 \leq x \leq 4$ , $y < x + 1$ , $y > 0$ ?

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Key Takeaways

✓
Dashed line for $<$ or $>$ . Solid line for $\leq$ or $\geq$ .
✓
Use a test point to determine which side to shade.
✓
The feasible region satisfies all constraints simultaneously.

Tags:

#gcse#maths#algebra#inequalities#graphs

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