Regions from inequalities
Objective. Shade and interpret regions satisfying several inequalities.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Regions from inequalities are areas on a coordinate grid that satisfy two or more inequalities simultaneously, found by drawing each boundary line and shading the required side. A solid line is used for ≤ or ≥, meaning the boundary is included, while a dashed line is used for strict inequalities < or > where the boundary is excluded. To decide which side to shade, test a point not on the line, such as the origin, in each inequality. The final region satisfying all inequalities is where all the individual shaded regions overlap.
Worked example
Describe the region satisfying both inequalities y ≤ 2 and x > 1, identifying the boundary line types.
- 1The line y = 2 is solid since the inequality is ≤, including the boundary.
- 2The line x = 1 is dashed since the inequality is >, excluding the boundary.
- 3The region required is below or on y = 2 and strictly to the right of x = 1.
- 4This overlapping region is the solution set for the two regions' inequalities.
Answer: The region where y ≤ 2 (solid line) and x > 1 (dashed line) overlap
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Pupils shade the wrong side of a line by misreading the inequality direction, such as shading above y = x + 2 for y < x + 2. Always test a convenient point like (0,0) to check which side satisfies the inequality.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.