# Inequalities

## Inequalities Revision

**Inequalities**

When we deal with equations, the equals sign tells us that each side of the equation are equal to each other. Another type of equation is an inequality, where the sides of the equation do not equal each other. The **inequality** sign may tell us one side of the equation is** smaller** or** larger** than the other, or **smaller than or equal to**, or** larger than or equal to**, each other. You will need to understand how to **derive** and **solve** **inequalities**.

We can also display **inequalities** on a number line. You will need to understand how to **use** and** interpret** **inequalities** on a number line.

**Inequality Symbols**

There are 4 symbols used in** inequalities**, in place of an equals sign in an equation:

- >
**greater than**

**Example:** 4 > 3 means 4 is greater than 3

- <
**less than**

**Example:** 3 < 4 means 3 is less than 4

- \geq
**greater than or equal to**

**Example:** 4\geq4 means 4 is greater than or equal to 4 (6\geq4 is also an example that works)

- \leq
**less than or equal to**

**Example:** 2\leq3 means 2 is greater than or equal to 2 (3\leq3 is also an example that works)

We often see these sign used in **algebraic inequalities**:

3x < 4 means 3x is less than 4

**Solving Inequalities – Basics**

For the most part, we can treat and solve **inequalities** just as we do with equations (we will talk about the exception later).

**Example**: solve 3x+3\leq12,

We would treat this as an equation and start by minusing 3 from both sides:

3x\leq9

And now, divide both sides by 3:

x\leq3

Meaning x is **less than or equal to** 3. This **inequality** is now solved.

**Solving Inequalities with 2 Signs**

You may also come across** inequalities** with 2 signs.

**Example:**

12<2x+6<24

This means, 2x+6 is between 12 and 24. Or 2x+6 is **greater than** 12 but** less than** 24.

We can just as easily solve** inequalities** like this, in the same way as solving equations, but whatever is done to one side, must be done to all 3 sides:

12<2x+6<24

Start by subtracting 6 from each side to get 2x on its own:

6<2x<18

And then dividing by 2 to get x on its own:

3<x<9

Hence, x is **greater than** 3 but **less than** 9

**Note**: the two signs may not always be the same (for example 12>x\geq2, meaning x is **less than** 12 but **greater than or equal to** 2) but these are solved using the same method.

**Solving Inequalities: the Exception**

There is one case where solving** inequalities** differs from solving equations, and this occurs when **multiplying or dividing** by a **negative number**.

**RULE**: if you **multiply or divide** by a **negative number**, the **inequality** sign changes direction

When we flip a sign (this means the same as the sign changing direction):

< \rightarrow >\\

> \rightarrow <\\

\leq \rightarrow \geq\\

\geq \rightarrow \leq\\

**Example**: solve 19<-8x-5<27

Start as usual by adding 5 to each side:

24<-8x<32

Now, we want to isolate the x, so we need to divide by -8 but as we are diving by a **negative number**, we need to flip the signs to >,

-3 > x > -4

Meaning x is **less than** -3 but **greater than** -4

**Inequalities on a Number Line**

We can display **inequalities** on a number line.

Note: a filled circle, \bullet is used for the inequalities less than **or equal to**, and greater than **or equal to**. Whereas an open circle, \circ, is used for less than or greater than (without the equal to).

**Examples**:

**1.**

x>-6

You can see the arrow is pointed towards values **greater than** -6, because the inequality sign is a **greater than**, and the circle is open because it is not **greater than** or equal to.

**2.**

x\geq-6

**3.**

x<6

**4.**

x\leq6

Number lines can also be used for **inequalities** with 2 signs, for example:

-7<x\leq4

**Example 1: Rearranging and Solving Inequalities**

Solve the following **inequality**,

42+x\leq\dfrac{1}{4}x\leq51+x

**[3 marks]**

Let’s subtract x from each side to get rid of the x‘s on the outside of the **inequality**:

42\leq-\dfrac{3}{4}x\leq51

Now, let’s divide by -\dfrac{3}{4} to get x on its own. Remember – when dividing by a **negative number** we need to use the opposite signs:

-56\geq x\geq-68

**Example 2: Number Lines**

Express the following **inequality** on a number line,

-3\leq x<5

**[2 marks]**

So, as the first sign means **less than or equal to**, the circle on the number line will be filled, but the second sign, **less than**, will be an empty circle:

## Inequalities Example Questions

**Question 1**: List the integers that satisfy the inequality

-12<x\leq-3

**[2 marks]**

The values that satisfy this inequality are -11, -10, -9, -8, -7, -6, -5, -4, -3

**Question 2**: Express the following inequality on a number line

6\geq 2x\geq -4

**[3 marks]**

Firstly, we need x on its own, so let’s divide by 2,

3\geq x\geq -2

Which can also be written as,

-2\leq x\leq3

And we can plot this on a number line,

**Question 3**: Solve the following inequality,

-4x+11\leq47

**[3 marks]**

Start by minusing 11 from each side,

-4x\leq36

Divide through by -4, remembering to flip the sign:

x\geq-9

**Question 4**: Solve the following inequality,

26-6x\geq2\geq14-6x

**[4 marks]**

Start by adding 6x to get x in the middle and remove it from the sides,

26\geq2+6x\geq14

Minus 2 from each side,

24\geq6x\geq12

Divide through by 6,

4\geq x\geq2