Inequalities on Graphs
Inequalities on graphs describe a region of the coordinate grid where every point makes one or more inequalities true, for example y < 2x + 1 or x + y >= 4. Each inequality is drawn as a boundary line, solid for >= or <= and dashed for a strict < or >, with shading showing which side is included. This Higher GCSE topic often combines two or three inequalities to define one shaded region.
Before you start
Make sure you're comfortable with these topics first:
Method
- Rearrange each inequality if needed so you can identify its boundary line (treat the inequality sign as an equals sign to find the line).
- Decide whether each boundary line should be solid (for <= or >=) or dashed (for a strict < or >).
- Draw each boundary line accurately on the grid, using a table of values or the gradient and intercept.
- Test a point that is not on any of the lines, often the origin (0, 0), in each inequality to decide which side satisfies it.
- Shade out the region(s) that do NOT satisfy each inequality, or shade in the wanted region and label it R.
- Where several inequalities apply together, the final labelled region must satisfy all of them at the same time.
Worked example
On a grid, show the region R that satisfies all three inequalities: x >= 1, y < 4 and x + y <= 5.
- Draw the line x = 1 as a solid vertical line, since x >= 1 is not a strict inequality.
- Draw the line y = 4 as a dashed horizontal line, since y < 4 is strict.
- Draw the line x + y = 5 as a solid line through (0, 5) and (5, 0), since x + y <= 5 is not strict.
- Test the point (2, 2): 2 >= 1 is true, 2 < 4 is true, and 2 + 2 = 4 <= 5 is true, so (2, 2) lies inside region R.
- Shade and label the triangular region bounded by these three lines, containing the point (2, 2), as region R.
Practice questions
Try each question, then tap to reveal the answer.
Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
Solve the inequality 4x + 5 > 2x - 3, and describe the solution set on a number line.
(a) Draw the graphs of x = 2, y = 1 and x + y = 7 on a grid. (b) Shade the region R that satisfies all three inequalities x >= 2, y >= 1 and x + y <= 7.
A region R satisfies both y <= 8 - x^2 and y >= x + 2. Show that a point with x-coordinate x can only lie in R when -3 <= x <= 2.
Free printable worksheet
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