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Algebraic expressions Solving linear equations Quadratic equations by factorising Quadratic formulaThis topic is relevant for:

Here we will learn about inequalities including how to represent inequalities on a number line, list integer values in solution sets, solve linear inequalities and solve quadratic inequalities.

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

**Inequalities **compare numbers or expressions in order of size.

There are four ways we can compare terms using **inequality notation**.

**Less than**

E.g.

x < 2‘ x is **less than ** 2 ’

**Greater than**

E.g.

x > 2‘ x is **greater than ** 2 ’

**Less than or equal to**

E.g.

x \leq 2‘ x is **less than or equal **to 2 ’

**Greater than or equal to**

E.g.

x \geq 2‘ x is **greater than or equal to ** 2 ’

Inequalities are similar to linear equations but rather than there being one unique solution, there are many.

E.g.

Let’s solve this linear equation

\[4 x+1=21 \\
4 x=20 \\
x=5\]

The only solution is ‘ 5 ’.

However when the equals sign becomes an inequality sign there are many solutions.

\[4 x+1<21 \\
4 x<20 \\
x<5\]

Any value less than ‘ 5 ’ satisfies this inequality.

Solutions are real numbers that can be indicated on a number line and could be integers, fractions or decimals. They can also include positive numbers and negative numbers.

We can use inequalities in a variety of different ways including:

- Inequalities on a number line
- List integer values in a solution set
- Solving inequalities
- Quadratic inequalities

Let’s look at these in more detail below.

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

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

E.g.

Represent x< 5 on a number line

x< 5 means ‘ x is less than 5 ’ so an **open** circle is placed at 5 .

x< 5 means that x can have any value less than 5 so an arrow is placed on the left hand side of the circle.

**Step by step guide: **Inequalities on a number line

We can list all of the integer values represented by an inequality

E.g.

List the integer values represented by 2 < x < 8

As ‘<’ means less than, the values of x do **not** include the values given.

‘ 2 ’ and ‘ 8 ’ are **not** included in the solution set.

So the integer values satisfied by the inequality are 3, 4, 5, 6, 7

E.g.

List the integer values represented by 2 \leq x \leq 8

As ‘ \leq ’ means less than **or equal**, the values of x **do **include the values given.

‘ 2 ’ and ‘ 8 ’ are included in the solution set.

So the integer values that form the solution set for the inequality are 2, 3, 4, 5, 6, 7, 8

**Solving inequalities** is where we calculate the values that an unknown variable can be in an inequality.

E.g.

Solve 4x + 2 < 14

Start by subtracting ‘ 2 ’ from both sides of the inequality

4x < 12Then divide both sides by 4

x < 3This means that x can take any value that is **less than ** 3 .

**Step by step guide:** Solving inequalities

Linear inequalities are inequalities which only involve x with no higher powers. They are similar in style to linear equations.

For example, 3x+7>16

**Quadratic inequalities** are similar to quadratic equations and when plotted they display a parabola. We can solve quadratic inequalities to give a range of solutions rather than up to two unique solutions.

E.g.

Solve x^{2} + 6x + 8 \leq 0

Let’s start by finding the roots of the quadratic expression by either factorising or using the quadratic formula.

\[x^{2} + 4x+8\leq0\\
(x+4)(x+2)\leq0\]

Therefore the roots of the equation are -4 and -2 as these are the values of x that will make each bracket equal to zero.

We know that the roots are where the graph is equal to 0 (where it crosses the x axis) and the general shape is a ‘u’ shape (because the coefficient of the x^{2} term is positive).

So we can sketch the graph:

As the graph needs to be \leq 0 , we need values of x that will produce the graph that is below the x axis

-4\leq x \leq-2**Step-by-step guide:** Q__uadratic inequalities__

Representing inequalities on a number line can be extended into 2 dimensions. Graphical inequalities can also involve two variables.

E.g.

x>3The inequality x>3 can be represented on a x,y axes grid. There is a dashed line drawn at x=3 and the shaded region R is x>3.

You can find out more about graphical inequalities at:

**Step-by-step guide**: Inequalities on a graph

Get your free inequalities worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free inequalities worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREERepresent x > 3 on the number line

**Indicate the values on the number line with an open circle or closed circle**.

x > 3 means ‘ x is greater than 3 ’ so it is represented with an open circle.

2**Indicate the values that are satisfied by the inequality with an arrow.**

‘ x is greater than 3 ’ so an arrow is placed on the right hand side of the circle.

Represent -3<x \leq 6 on the number line

**Indicate the values on the number line with an open circle or closed circle**.

‘ x is greater than -3 ’ so it is represented with an open circle.

x \leq 6‘ x is less than** or equal to** 6 ’ so it is represented with a closed circle.

2**Indicate the values that are satisfied by the inequality with a line between the circles.**

List the integers that satisfy the inequality 1<x \leq 7

**Identify what the inequality signs indicate**.

‘ 1 ’ is not included in the solution set as x is greater than 1 .

x \leq 7‘ 7 ’ is included in the solution set as x is less than **or equal** to 7 .

2**List the integers that are satisfied by the inequality.**

Solve 3x + 2 < 14

**Rearrange the inequality so that all the unknowns are on one side of the inequality sign.**

\[3 x+2<14 \\
3 x<12\]

2**Rearrange the inequality by dividing by the ** x **coefficient so that a single unknown is shown as an inequality with a value.**

\[3 x<12 \\
x<4\]

3**Ensure you have the correct direction of the inequality sign.**

The answer is x < 4 .

(Not ‘ 4 ’ or x = 4 )

Solve x^{2}+14 x+40 \leq 0

**Factorise the quadratic**.

2**Find the roots of the quadratic**.

The values of x that make each bracket equal zero are x=-4 and x= -10

3**Write the solution using interval notation.**

Solve 3x^{2}+4 x-5 < 0

**Identify values of a, b and c to substitute into the quadratic formula**.

2**Substitute the values into the quadratic formula**.

3**Simplify to calculate the solutions of the inequality**.

x = -2.12 and x = 0.786

4**Use interval notation to represent the inequality**.

We need 3x^{2}+4 x-5 < 0 , therefore we need the solutions between the two roots.

-2.12 < x < 0.7.86**Closed and open circles**

A common error is to incorrectly identify if an open circle or closed circle is required by interpreting the inequality sign.

x < 4 would require an open circe

x\leq4 would require a closed circle

**Writing solutions as an inequality**

When solving an inequality, a common error is to not include the inequality sign in the solution.

E.g.

x + 5 < 8The solution is x < 8 .

x = 8 or ‘ 8 ’ are not acceptable as it does not identify the range of values that satisfy the inequality.

1. Represent x \geq -2 on the number line.

x \geq -2

x is greater than or equal to -2 so a closed circle is placed at ‘ -2 ’.

The values to the right hand side of the closed circle are indicated by an arrow as these values are greater than -2 .

2. Represent -4\leq x < 7 on the number line.

x \geq -4

x is greater than or equal to -4 so a closed circle is placed at ‘ -4 ’.

x < 7x is less than 7 so an open circle is placed at ‘ 7 ’.

The values that satisfy the inequality are between these values and indicated with a line drawn between them.

3. List the integers that satisfy the inequality 3\leq x < 8

3, 4, 5, 6, 7, 8

4, 5, 6, 7, 8

4, 5, 6, 7

3, 4, 5, 6, 7

3\leq x

‘ 3 ’ is included in the solution set as the inequality sign indicates that x is greater than or equal to 3 .

x < 8‘ 8 ’ is not included in the solution set as the inequality sign indicates that x is less than 8 .

4. List the integer values that satisfy the inequality -5<2 x+1 \leq 3

-2, -1, 0, 1, 2

-2, -1, 1

-2, -1, 0, 1

-1, 0, 1

-6 < 2x \leq 2

Subtract one from all parts of the inequality.

-3 < x 1Divide all parts by 2 .

‘ -3 ’ is not included in the solution set as x is greater than -3 .

‘ 1 ’ is included in the solution set as x is less than or equal to 1 .

So the integer values satisfied by the solution set as -2, -1, 0, 1

5. Solve x^{2}-12x+27<0

-9 < x <-3

3\leq x \leq9

-3 < x < 9

3 < x < 9

x^{2}-12x+27<0

Factorise the quadratic

(x – 9)(x – 3) < 0Find the roots by calculating which values of x make each bracket equal zero.

x = 9 and x = 3

We want the values of x that produce the curve below the x-axis

Write the solution using interval notation.

3 < x < 9

6. Solve 2x^{2}+6x-3<0

-3.44 < x <0.44

0.44 < x <3.44

-3.44 < x <-0.44

-3.44\leq x\leq0.44

Identify the values of a, b and c from ax^{2} + bx + c

a = 2, b = 6, c = -3Substitute the values into the quadratic formula

x=\frac{-6 \pm \sqrt{6^{2}-4 \times 2 \times-3}}{2 \times 2}Simplify to find the roots

x=\frac{-6 \pm \sqrt{36–4}}{4} x=\frac{-6 \pm \sqrt{60}}{4}Write the solution using interval notation

-3.44 < x <0.44

1. (a) Write down the inequality represented on this numbers line

(b) List the integer values satisfied by the inequality in part a).

**(3 marks)**

Show answer

(a)

Identifying -3 and 4

**(1)**

**(1)**

(b)

-2, -1, 0, 1, 2, 3, 4**(1)**

2. (a) Solve this inequality 5x – 1 < 19

(b) Represent your solution to part a) on the number line below

**(4 marks)**

Show answer

(a)

Rearranging to 5x < 20 or identifying a solution of 4

**(1)**

**(1)**

(b)

Indicates 4 , or their solution from part a) on the number line

**(1)**

**(1)**

3. (a) Solve the inequality x^{2}+x-12\leq 0

(b) List the integer values satisfied by the inequality in part a).

**(5 marks)**

Show answer

(a)

Factorises the inequality to (x + 4)(x – 3)

**(1)**

Identifies solutions of ‘ -4 and 3

**(1)**

**(1)**

(b)

Lists all but two correct integers

**(1)**

**(1)**

You have now learned how to:

- Represent inequalities on a number line
- List integers that satisfy an inequality
- Solve linear inequalities
- Solve quadratic inequalities

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