Sheet № 142 · Higher only · AQA · Edexcel · OCR
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 sol
§Key definitions
Question:
On a graph, show the region where y < 3.
Answer:
Dashed horizontal line at y = 3, shade below.
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.
§Formulas to memorise
≤ or ≥ means draw a solid line (the boundary is included)
< or > means draw a dashed line (the boundary is not included)
Test a point (usually (0, 0)) to determine which side to shade
Replace the inequality sign with = — to get the equation of the boundary line.
Draw the boundary line — on the graph. Use a solid line for ≤ or ≥ and a dashed line for < or >.
Choose a test point — that is not on the line. The origin (0, 0) is easiest if the line does not pass through it.
Substitute the test point — into the inequality. If it satisfies the inequality, shade the side containing that point. If not, shade the other side.
For multiple inequalities — , shade each region and identify the overlap — this is the feasible region.
Label the required region — clearly, as the question may ask you to mark it with R or leave it unshaded.
Worked example
On a graph, show the region where y < 3.
Working:
⚠ 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.