# Identify Regions in Inequalities

In this worksheet, students will learn how to illustrate inequalities graphically, accurately shading and identifying regions defined by inequalities.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Eduqas, OCR, AQA, Pearson Edexcel

Curriculum topic:   Algebra

Curriculum subtopic:   Solving Equations and Inequalities, Algebraic Inequalities

Difficulty level:

### QUESTION 1 of 10

When we are working with inequalities, we can illustrate these by either using a number line (if it has a single variable) or on a set of axes (if it has double variables).

We can also illustrate the region on a graph that satisfies numerous inequalities.

A reminder about which lines to use:

If the inequality you are drawing contains either a > or < symbol, you should draw a dashed line.

If the inequality contains either a or symbol, you should draw a solid line.

e.g. Illustrate the region that satisfies the inequalities:

x > 2, y ≥ x and x + y < 8

Step 1: Draw the limit of each inequality as a straight line.

If we draw these on individual graphs (you won’t need to do this), we can see the lines more clearly.

Notice that the middle graph has a solid line, this is because the inequality has a greater than or equal to symbol.

Usually, we would actually draw all of these onto one graph, like this:

Step 2: Illustrate the region that satisfies all three inequalities.

There’s a short cut we can use here.

There is only one section that works for all the inequalities - the bit in the middle of all three lines (shaded grey here):

Let's extend from this starting question now.

e.g. Are the points below in the region defined by the inequalities x > 2, y ≥ x and x + y < 8?

a) (3,4)

b) (2,6)

c) (3,3)

If we plot these on the graph we previously created, they will appear like this:

a) (3,4) – This is in the middle of the region so it is definitely within the region.

b) (2,6) – This is on a dashed line. As a dashed line means > or <, points on the line don’t count as being inside the region.

c) (3,3) – This point is on a solid line. A solid line means ≥ or ≤ so the line itself is included in the region.

Are you ready to investigate some regions independently now?

You may want to have some squared paper and a pencil handy in case you want to draw any of these graphs to compare with our maths teacher's versions.

Read the statement below about how to find regions, then type in the word that correctly fills the gap.

Read the statement below, then type the answers that correctly fill each gap.

Which of these graphs accurately shows the correct lines for the equations y = - 4, x = 2 and y = 2x + 1?

Left-hand graph

Right-hand graph

Which region in the graph below satisfies the inequalities: y > - 4, x < 2 and y < 2x + 1?

A

B

C

Your friend is trying to illustrate these inequalities:

-2 < x ≤ 1

y > -2

y < x + 1

In which of the two diagrams below have they drawn the inequalities correctly?

A

B

Consider this group of inequalities:

-2 < x ≤ 1

y > -2

y < x + 1

How many integer points are there that satisfy all three of these inequalities?

Your friend is trying to illustrate these inequalities:

3y + 2x > 12

y < x – 1

x < 6

In which of the two diagrams below have they drawn the inequalities correctly?

A

B

Consider this group of inequalities:

3y + 2x > 12

y < x – 1

x < 6

Do the pairs of values below satisfy all three inequalities?

Your friend is trying to illustrate these inequalities:

x ≥ 2

y > x

x + y ≤ 6

In which of the two diagrams below have they drawn the inequalities correctly?

A

B

Consider this group of inequalities:

x ≥ 2

y > x

x + y ≤ 6

Do the pairs of values below satisfy all three inequalities?

• Question 1

Read the statement below about how to find regions, then type in the word that correctly fills the gap.

EDDIE SAYS
Did you recall this fact from the Introduction? Occasionally you may see the alternative applied, but only when each inequality is added to a graph individually. This means that the unshaded area will become the region which satisfies. If the graph is presented to you with the inequalities already present (as in this activity), the shaded area will be the region which satisfies the inequalities in question. Review the Introduction now to commit this process to memory before you tackle the rest of these activities.
• Question 2

Read the statement below, then type the answers that correctly fill each gap.

EDDIE SAYS
We need to differentiate between lines illustrating greater than or less than (> or <) regions, as opposed to greater / less than or equal to regions (≥ or ≤). A point that is on the line with a < or > inequality is not included in the region, whereas with ≤ or ≥ it would be. We use the two different types of lines (dashed and solid) to make this distinction clear.
• Question 3

Which of these graphs accurately shows the correct lines for the equations y = - 4, x = 2 and y = 2x + 1?

Left-hand graph
EDDIE SAYS
It’s really important to get our y = a and x = a directions the right way round. The line x = 2 will be vertical and pass through the x axis at the point (2, 0). The line y = -4 will be horizontal as pass through the y axis at the point (0, -4). Does that help you identify the correct graph?
• Question 4

Which region in the graph below satisfies the inequalities: y > - 4, x < 2 and y < 2x + 1?

B
EDDIE SAYS
We need to find only the points that are defined by all the inequalities. Region A shows the values satisfying the equation y > 2x + 1 (i.e. the opposite area on the graph to y < 2x + 1). Region C shows the values satisfying the equation x > 2 (i.e. the opposite area on the graph to x < 2). Only Region B shows the values satisfying all three equations.
• Question 5

Your friend is trying to illustrate these inequalities:

-2 < x ≤ 1

y > -2

y < x + 1

In which of the two diagrams below have they drawn the inequalities correctly?

B
EDDIE SAYS
Spotting the correct types of lines in each graph, is the quickest and easiest way on this occasion to reach the correct answer. All these inequalities use either a < or > sign, so all three lines need to be dashed. Remember that ≥ or ≤ symbols require us to use solid lines, as values on the line are also viable solutions.
• Question 6

Consider this group of inequalities:

-2 < x ≤ 1

y > -2

y < x + 1

How many integer points are there that satisfy all three of these inequalities?

6
EDDIE SAYS
'Integer points' means that the values of x and y must be whole numbers only (integers). In graphed inequalities, points on dashed lines don’t fit the criteria and points on solid lines do. Look at our graph below which illustrates these three inequalities:

Can you see that there are 6 points which satisfy all three inequalities, PLUS use whole numbers?
• Question 7

Your friend is trying to illustrate these inequalities:

3y + 2x > 12

y < x – 1

x < 6

In which of the two diagrams below have they drawn the inequalities correctly?

A
EDDIE SAYS
Once again, its all about the lines here. All of the inequalities use either a > or < sign, so all the lines present need to be dashed. Did you spot that?
• Question 8

Consider this group of inequalities:

3y + 2x > 12

y < x – 1

x < 6

Do the pairs of values below satisfy all three inequalities?

EDDIE SAYS
Let's draw our graph here then plot these points:

Any points that are inside the region do satisfy all three equations. Any that are outside the region don’t. The point will only satisfy the inequalities if it is on a solid line or within the region itself, not on a dashed line.
• Question 9

Your friend is trying to illustrate these inequalities:

x ≥ 2

y > x

x + y ≤ 6

In which of the two diagrams below have they drawn the inequalities correctly?

A
EDDIE SAYS
Were you looking to see if the lines needed to be dashed or solid? This was less helpful on this occasion, as we had a combination of both. The easiest spot here is to look for the line x = 2. In graph A, this is running vertically through the x axis at the point (0, 2) which is correct. In graph B, this line is incorrect as it is horizontal so has, in fact, been changed to y = 2. Always remember the golden rule that x = ... makes a vertical line, whereas, y = ... makes a horizontal line.
• Question 10

Consider this group of inequalities:

x ≥ 2

y > x

x + y ≤ 6

Do the pairs of values below satisfy all three inequalities?

EDDIE SAYS
Here is our graph with these three points plotted:

A point will only satisfy all three inequalities if it is on a solid line or within the region itself. Which points does this apply to? Don't let (3, 3) trip you up - it may be on one solid line but it is also on one dashed line. This means that it fails to satisfy all three equations. Great work completing this activity! Hopefully you are feeling much more confident now to draw and interpret graphed inequalities.
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