# Graphical representation of linear inequalities

> Edexcel International GCSE Further Pure Mathematics · 4PM1
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s3-graphical-representation-of-linear-inequalities/

This guide teaches you to represent two-variable linear inequalities graphically, identify feasible regions, and solve basic linear programming problems aligned with the Edexcel IGCSE Further Pure Math (4PM1) syllabus.

**Prerequisites:** [Solving linear inequalities in one variable](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s3-linear-inequalities-one-variable/); [Plotting straight line graphs](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s2-straight-line-graphs/)

## Learning objectives

- Draw and label boundary lines for linear inequalities in two variables
- Correctly shade the region that satisfies one or more linear inequalities
- Identify feasible regions for simple linear programming problems
- Find the optimal point of a basic linear objective function within a feasible region

## Drawing Boundary Lines for Linear Inequalities

**Boundary Line** — The straight line representing the equality version of a linear inequality, which separates the coordinate plane into two half-planes. Solid lines are used for ≤/≥ (points on the line satisfy the inequality), dashed lines for </> (points on the line do not satisfy).

*Example:* The inequality $2x + y \leq 4$ has a solid boundary line $2x + y = 4$.

To draw a boundary line, first rearrange the inequality into the form $y = mx + c$ or use the intercept method: find x and y intercepts by setting y=0 and x=0 respectively. Plot two points to draw the line, then check if it should be solid or dashed based on the inequality symbol.

**Worked example:** Draw the boundary line for the inequality $3x - 2y > 6$

1. 1. Write the equality version of the inequality: $3x - 2y = 6$
2. 2. Find the x-intercept: set $y=0$, so $3x = 6$ → $x=2$, giving point (2, 0)
3. 3. Find the y-intercept: set $x=0$, so $-2y = 6$ → $y=-3$, giving point (0, -3)
4. 4. The inequality uses > (strict), so draw a dashed line connecting (2,0) and (0,-3), and label the line with its equation $3x - 2y = 6$

> **Exam tip:** Always label boundary lines with their full equation: you will lose marks if you omit labels in your exam.

*Calculator:* allowed

## Identifying and Shading Feasible Regions

**Feasible Region** — The set of all points on the coordinate plane that satisfy all given linear inequalities simultaneously.

*Example:* For constraints $x \geq 0$, $y \geq 0$, $x + y \leq 3$, the feasible region is the triangle with vertices at (0,0), (3,0) and (0,3).

To find which side of a boundary line to shade, pick a test point not on the line (the origin (0,0) is easiest if it is not on the boundary line). Substitute the x and y values of the test point into the inequality: if the resulting statement is true, shade the side containing the test point; if false, shade the opposite side. If you have multiple inequalities, the feasible region is the area shaded by all constraints.

> **tip**
>
> If the origin lies on a boundary line, use another simple point like (0,1) or (1,0) as your test point to avoid errors.

**Worked example:** Shade the feasible region satisfying $x + y \geq 2$, $x \leq 3$, $y \leq 4$

1. 1. Draw all boundary lines: $x+y=2$ (solid, ≥), $x=3$ (solid, ≤), $y=4$ (solid, ≤)
2. 2. Test (0,0) for $x+y \geq 2$: $0 + 0 = 0 \geq 2$ is false, so shade the side of $x+y=2$ that does NOT contain (0,0)
3. 3. Test (0,0) for $x \leq 3$: $0 \leq 3$ is true, so shade the side of $x=3$ containing (0,0)
4. 4. Test (0,0) for $y \leq 4$: $0 \leq 4$ is true, so shade the side of $y=4$ containing (0,0)
5. 5. The overlapping shaded area is the feasible region: mark it clearly with an R or cross-hatching as directed by the question.

*Calculator:* allowed

## Solving Simple Linear Programming Problems

**Linear Programming** — A method used to find the maximum or minimum value of a linear objective function, subject to a set of linear inequality constraints.

*Notation:* Objective function is usually written as $P = ax + by$, where a and b are constants.

For Edexcel IGCSE Further Pure Math problems, the optimal (maximum or minimum) value of the objective function will always lie at one of the vertices (corner points) of the feasible region. To solve, first identify all vertices of the feasible region, substitute each vertex into the objective function, then select the vertex that gives the required maximum or minimum value.

**Worked example:** A factory makes chairs (x) and tables (y). Constraints are $x + 2y \leq 10$, $x \geq 0$, $y \geq 0$. Profit per chair is \$10, per table is \$15. Find the maximum possible profit.

1. 1. Define the objective function: $P = 10x + 15y$
2. 2. Draw the feasible region, identify its vertices: (0,0), (10, 0), (0,5)
3. 3. Substitute each vertex into the profit function:
4. $$P(0,0) = 10(0) + 15(0) = 0$$
5. $$P(10,0) = 10(10) + 15(0) = 100$$
6. $$P(0,5) = 10(0) + 15(5) = 75$$
7. 4. The maximum profit is \$100, achieved when 10 chairs and 0 tables are made.

> **Exam tip:** Even if you can estimate the optimal point by eye, always substitute all vertices to confirm: you will get method marks even if your graph is slightly inaccurate.

*Calculator:* allowed

## Exam Presentation Rules for Graphical Inequalities

Following presentation rules is critical to score full marks in this topic, as examiners penalize missing labels or unclear shading.

1. Use a ruler to draw all straight boundary lines: freehand lines will lose marks.
2. Label every boundary line with its full equation.
3. Use solid lines for ≤/≥ and dashed lines for </>.
4. Clearly indicate the feasible region, either by shading it or shading the unwanted regions and marking the feasible region with an R.
5. Show all test point working when justifying your shaded region.

**Check your understanding**

1. What type of line should you use for the inequality $4x - 3y < 12$?

   - Solid line
   - Dashed line
   - Either is acceptable

   *Why:* Strict inequalities < or > use dashed lines, as points on the line do not satisfy the inequality.

2. Where is the optimal point for a linear objective function located?

   - At the centre of the feasible region
   - At a vertex of the feasible region
   - Along any edge of the feasible region

   *Why:* For all linear programming problems in this syllabus, the optimal value is always at a corner vertex of the feasible region.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using a solid line for strict inequalities (< or >)
  - Why it fails: Strict inequalities do not include points on the boundary line, so solid lines incorrectly imply points on the line are valid solutions.
  - Correct: Use a dashed line for < and >, solid line only for ≤ and ≥.
- **Wrong:** Shading the wrong side of the boundary line because you didn't test a point
  - Why it fails: Rearranging inequalities incorrectly can lead to reversing the inequality sign by mistake, leading to shading the wrong region.
  - Correct: Always test a point not on the boundary line (preferably (0,0) if possible) to confirm which side to shade.
- **Wrong:** Forgetting to label boundary lines with their equations
  - Why it fails: Exam mark schemes explicitly award marks for correctly labelled boundary lines; missing labels lead to lost marks even if the line is drawn correctly.
  - Correct: Write the full equation of each boundary line next to the line as soon as you draw it.
- **Wrong:** Calculating the optimal point by eye instead of substituting all vertices
  - Why it fails: Slight inaccuracies in graph drawing can make the optimal vertex look closer to the maximum/minimum than it is, leading to incorrect answers.
  - Correct: List all vertices of the feasible region, substitute each into the objective function, and select the value that meets the requirement (max or min).
- **Wrong:** Forgetting non-negativity constraints (x ≥ 0, y ≥ 0) in linear programming problems
  - Why it fails: Most real-world linear programming problems involve quantities that cannot be negative, so omitting these constraints expands the feasible region incorrectly.
  - Correct: Always check if the problem context (number of items, time, money) requires x and y to be non-negative, and add these constraints explicitly.

## Cheatsheet

| Task | Step-by-Step Action |
| --- | --- |
| Draw boundary line | 1. Write equality form; 2. Find intercepts; 3. Draw solid (≤/≥) or dashed (</>); 4. Label equation |
| Shade correct region | 1. Pick test point not on line; 2. Substitute into inequality; 3. Shade side where test point works; 4. Overlap for multiple constraints |
| Solve linear programming | 1. Define objective function; 2. Identify all feasible region vertices; 3. Substitute each vertex into objective; 4. Select max/min value |

## What's next

Now that you have mastered graphical representation of linear inequalities and basic linear programming, you are ready to move on to more advanced inequality topics in the Edexcel IGCSE Further Pure Math syllabus. This topic forms the foundation for applied problem-solving questions that often combine with coordinate geometry and algebraic manipulation in exam papers. Make sure you practice past paper questions to perfect your graph drawing speed and accuracy, as presentation is key to scoring full marks in this topic. You should also review algebraic methods for solving systems of inequalities to cross-verify your graphical solutions and avoid errors in high-pressure exam settings.

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