Inequalities on a Graph

Graphical inequalities combine your algebra skills with coordinate geometry. Instead of solving for a single value, you shade a region on a graph that represents all the points satisfying one or more inequalities. This is a Higher-only topic that regularly appears on AQA, Edexcel, and OCR papers for 3–4 marks. Learning the rules about solid and dashed lines will help you pick up every mark.

What Are Inequalities on a Graph?

When you draw an inequality on a graph, you start by drawing the boundary line (the equation you get by replacing the inequality sign with =). Then you decide which side of the line satisfies the inequality and shade that region.

Key Formulas

Step-by-Step Method

1. Replace the inequality sign with = to get the equation of the boundary line.
2. Draw the boundary line on the graph. Use a solid line for ≤ or ≥ and a dashed line for < or >.
3. Choose a test point that is not on the line. The origin (0, 0) is easiest if the line does not pass through it.
4. Substitute the test point into the inequality. If it satisfies the inequality, shade the side containing that point. If not, shade the other side.
5. For multiple inequalities, shade each region and identify the overlap — this is the feasible region.
6. Label the required region clearly, as the question may ask you to mark it with R or leave it unshaded.

Worked Example 1 — Foundation Level

Question: On a graph, show the region where y < 3.

Working:

• Boundary line: y = 3 (a horizontal line through 3 on the y-axis).
• The inequality is strict (<), so draw a dashed line.
• Test (0, 0): is 0 < 3? Yes ✓. So shade the side containing (0, 0), which is below the line.

Answer: Dashed horizontal line at y = 3, shade below.

Worked Example 2 — Higher Level

Question: Show the region satisfying y ≤ 2x + 1 on a graph.

Working:

• Boundary line: y = 2x + 1. This has gradient 2 and y-intercept 1.
• The inequality is ≤, so draw a solid line.
• Test (0, 0): is 0 ≤ 2(0) + 1 = 1? Yes, 0 ≤ 1 ✓. Shade the side containing (0, 0), which is below the line.

Answer: Solid line y = 2x + 1, shade below and on the line.

Worked Example 3 — Exam Style

Question: Show the region R satisfying all three inequalities: x ≥ 1, y ≥ 0, and x + y ≤ 5.

Working:

• x = 1: vertical solid line through x = 1. Test (2, 0): 2 ≥ 1 ✓. Region is to the right of the line.
• y = 0: this is the x-axis. Solid line. Region is above the x-axis.
• x + y = 5: solid line from (0, 5) to (5, 0). Test (0, 0): 0 + 0 = 0 ≤ 5 ✓. Region is below this line.

The feasible region R is the triangle with vertices (1, 0), (1, 4), and (5, 0).

Answer: The triangular region bounded by x = 1, y = 0, and x + y = 5, labelled R.

Common Mistakes

• Using the wrong line type. Solid lines include the boundary (≤, ≥). Dashed lines exclude it (<, >). Using the wrong type loses a mark.
• Shading the wrong side. Always use a test point — do not guess which side to shade. The origin (0, 0) is the simplest choice unless the line passes through it.
• Not labelling the region. If the question says "label the region R," you must write R in the correct area.

Exam Tips

• Some exam boards ask you to shade the unwanted region and leave the required region clear. Read the instructions carefully.
• When drawing boundary lines, plot at least three points to ensure accuracy.
• For questions with multiple inequalities, draw all boundary lines first, then identify the overlap.
• Integer coordinate questions may ask you to list all integer points in the feasible region — check each one satisfies every inequality.

Practice Questions

Q1 (Foundation): Draw the region where x > −2 on a graph.

Q2 (Higher): Show the region satisfying y ≥ x − 1 and y < 4.

Q3 (Higher): Find the integer coordinates in the region where x ≥ 0, y ≥ 0, and 2x + y ≤ 4.

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Related Topics

• Solving Inequalities
• Inequalities on a Number Line
• Linear Graphs and Equation of a Line

Summary

• Draw the boundary line by replacing the inequality with =.
• Use a solid line for ≤ or ≥ and a dashed line for < or >.
• Test a point (usually the origin) to decide which side to shade.
• For multiple inequalities, the feasible region is where all shaded areas overlap.
• Read the question carefully — some exams ask you to shade the unwanted region instead.