## What you need to know

### Graphical Inequalities

The way to express a graphical inequality is as follows:

– Treat the inequality as if it were an equation and plot the straight line. You should plot a solid line if the inequality is inclusive (i.e. ) and a dashed line if it is a strict inequality (i.e. ).

– Identify which side of the line the area that satisfies the inequality is on:

– If it is a greater than (or equal to) – – then you want the area above the line,

– If it is a less than (or equal to) – – then you want the area below the line.

– Shade the area that you identified in step 2.

**Note:** If you’re ever unsure about which way round your signs go, you can pick a coordinate from the region you think it is and see whether or not it satisfies your inequality. This way you can always check if you have completed the question correctly.

In order to really understand graphical inequalities you will need to have background knowledge of the following topics:

### Example 1: Graphical Inequalities

Shade the area that satisfies the inequality y+1 \geq 2x and mark it with the letter A.

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 below

### Example 2: Graphical Inequalities

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.

The first inequality will produce 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.

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 below.

### Example 3: Graphical Inequalities

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.

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 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

### Example Questions

1) Shade the region of a graph that satisfies the inequality x + 2y \leq + 8 and mark it with an R.

Firstly, rearrange this equation to make y the subject by subtracting x from both sides to get

2y \leq 8 - x

Then, divide both sides by 2 to get

y \leq -\dfrac{x}{2} + 4

Drawing this is an equation, the graph would be a solid line with gradient -\frac{1}{2} and y-intercept 4. Once drawn, we should shade and mark the region below the line. The result looks like:

2) Shade the region of a graph that is satisfied by the inequalities y \geq 1, y \leq 3, x>0, and y > x, and mark it with an R.

We’re going to treat the inequalities as equations and plot them as straight lines. The first one will be the solid plot of the line y=1, the second will be a solid plot of the line y=3, the third will be a dashed plot of the line x=0, and the fourth will be a dashed plot of the line y=x.

Now, we want to shade the area that is **below **the line y=1, **above** the line y=3, **to the right** of the line x=0, and **above **the line y=x.

The resulting graph looks like:

3) Shade the region of a graph that is satisfied by the inequalities y \leq x+1, y >-3, and x\leq 4, and label the area with an R.

We’re going to treat the inequalities as equations and plot them as straight lines. The first one will be the solid plot of the line y=x+1, the second will be a solid plot of the line x=4, and the third will be a dashed plot of the line y=3

Now, we want to shade the area that is **below **the line y=x+1, **above** the line y=-3, and **to the left** of the line x=4

The resulting graph looks like:

4) Determine that 3 inequalities that describe the shaded area on the graph below.

Firstly, determine the equations of the 3 lines and then from them, find the inequalities.

The horizontal line is clearly y = -2.

The dashed line has its y-intercept at 2 and a gradient of 2, so it is y=2x + 2

The final line has its y-intercept at 5 and a gradient of -3, so it is y = -3x + 5

Now, the shaded area is **above** y=-2 and the line is **solid**, so the inequality is

y\geq -2

The shaded area is **below **y=2x+2 and the line is **dashed**, so the inequality is

y <2x+2

The shaded area is **below** y=-3x+5 and the line is **solid**, so the inequality is

y \leq -3x+5

Therefore, the shaded area is described by the 3 inequalities

y\geq -2,\hspace{2mm} y<2x+2,\text{ and }y\leq-3x+5

5) Determine that 3 inequalities that describe the shaded area on the graph below.

Firstly, determine the equations of the 3 lines and then from them, find the inequalities.

The vertical line is clearly x = -2

The dashed line has its y-intercept at 2 and a gradient of 1/4, so it is y=\dfrac{1}{4}x + 2

The final line has its y-intercept at -6 and a gradient of 2, so it is y = 2x -6

Therefore, the shaded area is described by the 3 inequalities:

x> 2,\hspace{2mm} y<\dfrac{1}{4}x + 2,\text{ and }y\geq-2x -6

### Worksheets and Exam Questions

#### (NEW) Graphical Inequalities Exam Style Questions - MME

Level 6-7#### Inequalities on a graph - Drill Questions

Level 4-5#### Inequalities on a graph - Drill Questions

Level 6-7#### Defining inequalities regions - Drill Questions

Level 6-7### Videos

#### Graphical Inequalities Q1

GCSE MATHS#### Graphical Inequalities Q2

GCSE MATHS#### Graphical Inequalities Q3

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