Graphical Inequalities GCSE Revision and Worksheets

We want to pretend this is an equation and plot the line, so we must firstly rearrange the inequality, so it looks like our familiar straight-line equation. Subtracting 1 from both sides, we get

y \geq 2x - 1

Now, we must plot y = 2x - 1 as a solid line – since it is an inclusive inequality – and then shade the area above it – since it is a “greater than or equal to” – and mark that area with an A. The result looks like the graph on the right.

More often than not, you are presented with multiple inequalities and asked to shade the area that satisfies all of them. In that case, you repeat step 1 and 2 above for each of the inequalities and once you’re done, you shade the portion of the graph that is in all of the satisfied areas. Let’s have a look.

Example: Shade the region of a graph that satisfies the inequalities y>2, x\geq-1, and y < -x + 5 and mark it with an A.

So, we’re pretending that all 3 inequalities are equations and plotting them as such. The first one will be a dashed-line plot of y=2 since is a strict inequality; the second will be a solid-line plot of x=-1 since it is an inclusive inequality; the third will be a dashed-line plot of y=-x+5, which is a line with gradient -1 and y-intercept 5.

We want to shade the area that satisfies all 3 inequalities, or in other words the area that is above the line y=2, to the right of the line x=-1, and below the line y=-x+5.

The completed drawing, with the shaded region marked A, looks like the graph on the right.

Note: you don’t have to write the equations on usually, they’re just there to show what’s going on.

The only other thing to consider now is: what happens when you’re given a pre-drawn graph like the one above and have to deduce the inequalities from it? Let’s see.

Example: Determine the 3 inequalities that describe the shaded area on the graph below.

Firstly, we need to find the equations of the 3 lines that have been drawn on this graph. Once this is done, we can consider them as inequalities.

The first two we can read off: the horizontal line is y=-2, and the vertical line is x=2. Then, we can see that the slanted line has a y-intercept of -1 and (either by doing the triangle method or otherwise) a gradient of 3, so its equation must be

y=3x-1

Now we must convert them to the appropriate inequalities.

The line y=-2 is dashed and the shaded area is above it, so the inequality must be y>-2. The line x=2 is solid and the shaded area is to the left of it, so the inequality must be x\leq 2. Finally, the line y=3x-1 is solid and the shaded area is below it, so the inequality must be y\leq 3x-1. So, the 3 inequalities are

y>-2,\hspace{2mm}x\leq2\text{, and }y\leq 3x-1

If you’re ever unsure about which way round your signs go, or whether you should be looking above/below/left/right of a line, you can pick a coordinate from the region you think it is and see whether or not it satisfies your inequality.

Inequalities is an expansive topic in the new GCSE Maths course with new types of graphical inequality questions appearing in new exams and the specimen papers. For GCSE Maths tutors in Leeds to York Maths teachers, the graphical inequality resources on this page are suitable for a number of different exercises including monitoring student progress as well as setting worthwhile homework.