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Ordering numbers including negatives and decimals

This topic is relevant for:

Here we will learn about inequalities on a number line including how to represent inequalities on a number line, interpret inequalities from a number line and list integer values from an inequality.

There are also inequalities on a number line worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**Inequalities on a number line** allow us to visualise the values that are represented by an inequality.

To represent inequalities on a number line we show the range of numbers by drawing a straight line and indicating the end points with either an open circle or a closed circle.

An open circle shows it **does not include** the value.

A closed circle shows it **does include** the value.

E.g.

The solution set of these numbers are all the real numbers between 1 and 5 .

As 1 has an open circle, it **does not include ‘ 1 ’** but does include anything higher, up to and **including 5 **as this end point is indicated with a closed circle.

We can represent this using the inequality 1 < x \leq5

We can also state the **integer **values (whole numbers) represented by an inequality.

In this example, the integers 2, 3, 4 and 5 are all greater than 1 but less than or equal to 5 .

The solution set can represent all the real numbers shown within the range and these values can also be negative numbers.

In order to represent inequalities on a number line:

**Identify the value(s) that needs to be indicated on the number line.****Decide if it needs an open circle or a closed circle;**

< or > would need an open circle

\leq or \geq would need a closed circle.**Indicate the solution set with a straight line to the left hand side or right hand side of the number or with a straight line between the circles.**

E.g.

Represent x < 3 on a number line

An open circle needs to be indicated at ‘ 3 ’ on the number line.

As x < 3 is ‘ x is less than 3 ’, the values to the left hand side of the circle need to be indicated with a line.

E.g.

Represent 2<{x}\leq{6} on a number line.

An open circle needs to be indicated above ‘ 2 ’ and a closed circle needs to be indicated above ‘ 6 ’.

Then draw a line between the circles to indicate any value between these circles.

Get your free Inequalities on a number line worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free Inequalities on a number line worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE**Inequalities on a number line** is part of our series of lessons to support revision on **inequalities**. You may find it helpful to start with the main inequalities lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Represent x > 3 on a number line.

**Identify the value that needs to be on the number line.**

In this example it is 3 .

**2Decide if this needs to be indicated with an open circle or a closed circle.**

As the symbol is > then it will be an open circle.

**3Decide if the straight line needs to be drawn to the right or the left of the circle.**

As x is greater than 3 the straight line needs to be drawn to the right hand side of the circle to show the solution set of values greater than 3 .

Represent −2\geq{x} on a number line.

**Identify the value that needs to be on the number line.**

In this example it is −2 .

**Decide if this needs to be indicated with an open circle or a closed circle.**

As the symbol is \geq then it will be a closed circle.

**Decide if the straight line needs to be drawn to the right or the left of the circle.**

As x is less than or equal to −2 the straight line needs to be drawn to the left hand side of the circle to show the solution set of values less than −2 .

Represent 2\leq{x}\leq{7} on a number line.

**Identify the values that need to be indicated on the number line.**

In this example they are 2 and 7 .

**Decide if they need to be indicated with open circles or closed circles.**

As the symbols are both there will be two closed circles.

**Draw a straight line between the circles to represent the solution set.**

Represent −2<{x}\leq{3} on a number line.

**Identify the values that need to be indicated on the number line.**

In this example they are −2 and 3 .

**Decide if they need to be indicated with open circles or closed circles.**

As the symbols are < and \leq there will be an open circle and a closed circle.

**Draw a straight line between the circles to represent the solution set.**

Write the inequality that is shown on this number line.

**Identify the value indicated.**

In this example it is ‘ 4 ’.

**Decide which inequality symbol to use.**

As the circle is closed and the values indicated are greater than 4 we use the inequality is x\geq{4}

Write the inequality that is shown on this number line:

**Identify the values indicated on the number line.**

In this example they are −2 and 4 .

**Decide which inequality symbol to use.**

As the circle above the −2 closed we include the −2 and use -2\leq{x}.

As the circle above the 4 is open we do not include the 4 and use x < 4 .

**Put the inequalities together.**

-2\leq{x}<4

List the integer values satisfied by the inequality -4\leq{x}<2

**Identify the values indicated on the number line.**

In this example they are −4 and 2 .

−4 is included as it is followed by \leq

2 is not included as < is before it.

**List the integer values.**

−4, −3, −2, −1, 0, 1

List the integer values satisfied by the inequality shown on the number line below.

**Identify the values indicated on the number line.**

In this example they are -2 and 4 .

−2 is not included as it is represented by an open circle.

4 is included as it is represented by a closed circle.

** List the integer values.**

−1, 0, 1, 2, 3, 4

**Incorrect identification inequality symbols**

A common error is to confuse open circles and closed circles:

Open circles do not include the value so require a ‘<’ sign.

Closed circles do include the value so require a ‘\leq'

**Incorrect ordering of negative numbers**

A common error is to not recognise the symmetry about ‘0’ on the number line, and therefore not comparing the size of negative numbers correctly.

E.g.

5 is greater than one as they are ordered ** 1 **, 2, 3, 4, ** 5 ** on a number line.

But −5 is less than −1 as they are ordered ** −5 **, −4, −3, −2, ** −1 **, 0, 1, 2, 3 on a number line.

**Incorrect interpretation of the inequality symbol**

The direction of the inequality sign shows if the solution set is ‘greater than’ or ‘less than’. This can be confused when both sides of the inequality are switched. For example x > 8 is the same as 8 < x and ‘x’ is greater than 8 as the inequality sign is open towards the ‘x’ .

**Not listing all of the possible values in a solution set**

Usually integer values are requested to be listed in a solution set. ‘0’ can sometimes be forgotten.

**Not considering real numbers**

In the inequality -2\leq{x}<4 , the highest integer value that satisfies the inequality is ‘3’ . However, real numbers larger than 3 but less than 4 are also satisfied by this inequality.

1. Show x > 5 on the number line

5 is not included in the solution set as it is ‘>’ so an open circle is needed. The inequality sign is open towards the ‘x’ indicating it has values greater than 5 so the line is drawn to the right hand side of the circle.

2. Show x\leq{7} on the number line

7 is included in the solution set as it is ‘\leq’ so a closed circle is needed. The inequality sign is closed towards the ‘x’ indicating it has values less than 7 so the line is drawn to the left hand side of the circle.

3. Show 1<x\leq{8} on the number line

1 is not included in the solution set as it is ‘<’ so an open circle is needed. 8 is included in the solution set as it is ‘ \leq ‘ so a closed circle is needed. A line is drawn between the circles to indicate that all values in between are in the solution set.

4. Show -3 < x < 4 on the number line

-3 and 4 are not included in the solution set as both signs are ‘<’ so open circles are needed. A line is drawn between the circles to indicate that all values in between are in the solution set.

5. Write down the inequality for x that is shown on this number line

x \leq 6

x < 6

x > 6

x \geq 6

6 is indicated with a closed circle so this value is included in the solution set. The arrow is drawn to the left hand side to indicate values less than 6 .

6. Write down the inequality for x that is shown on this number line.

-4 < x < 2

-4<{x}\leq{2}

-4\leq{x}\leq{2}

-4\leq{x}<2

-4 is indicated by an open circle so this value is not included in the solution set therefore requires a ‘<’ symbol. 2 is indicated by a closed circle so this value is included in the solution set therefore requires a ‘\leq’ symbol. A line between the circles indicates all values in between are in the solution set.

7. List the integer values satisfied by the inequality -3<{x}\leq{4}

-3, -2, -1, 0, 1, 2, 3, 4

-2, -1, 0, 1, 2, 3

-3, -2, -1, 0, 1, 2, 3

-2, -1, 0, 1, 2, 3, 4

‘<’ follows -3 which means this value is not included in the solution set. ‘\leq’ is before 4 which means this value is included in the solution set. All the integers greater than -3 and up to and including 4 are in the solution set.

8. List the integer values satisfied by the inequality -5 < x < 1

-4, -3, -2, -1, 0, 1

-5, 4, -3, -2, -1, 0, 1

-4, -3, -2, -1, 0

-5, 4, -3, -2, -1, 0

Both inequality signs are ‘<’ which means these values are not included in the solution set. All the integers greater than -4 and less than -1 are in the solution set.

9. List the integer values satisfied by the inequality shown on the number line below.

-1, 0, 1, 2, 3, 4

0, 1, 2, 3

0, 1, 2, 3, 4

-1, 0, 1, 2, 3

-1 is indicated with a closed circle so this value is included in the solution set. 4 is indicated with an open circle so this value is not included in the solution set. All of the integers greater than and including -1 and up to 4 are included in the solution set.

10. List the integer values satisfied by the inequality shown on the number line below.

-1, 0, 1, 2

1, 0, 1

0, 1, 2

0, 1

Both -1 and 2 are indicated with closed circles so these values are included in the solution set. All of the integers greater than and including -1 and up to and including 2 are included in the solution set.

1. John buys x bananas and y pears.

He buys

- At least 5 bananas
- At most most 9 pears
- He buys more pears than bananas

One of the inequalities for this information is x\geq5

Write down two more inequalities for this information

**(2 marks)**

Show answer

y\leq9

**(1)**

y>x

**(1)**

2.

(a) Show the inequality x > 4 on this number line.

(b) Write down the inequality for x that is shown on this number line

**(3 marks)**

Show answer

(a)

Open circle at 4

**(1)**

Arrow indicating values greater than 4

**(1)**

(b)

x\leq7**(1)**

3.

(a) Write down an inequality for x that is shown on this number line

(b)

(i) Show the inequality -3\leq{x}<2 on this number line.

(ii) List the integers that are included in the solution set

**(6 marks)**

Show answer

(a)

2 < x or x\leq 7

**(1)**

2 < x \leq 7

**(1)**

(b)

Closed circle at -3 or for open circle at 2

**(1)**

**(1)**

-2, -1, 0, 1

**(1)**

-3, -2, -1, 0, 1

**(1)**

You have now learned how to:

- Identify inequalities from a number line
- Show inequalities on a number line
- List integer values in the solution set

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#### GCSE Maths Papers - November 2022 Topics

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Practice paper packs based on the November advanced information for Edexcel 2022 Foundation and Higher exams.

Designed to help your GCSE students revise some of the topics that are likely to come up in November exams.