**Inequalities**

The equations we are most familiar with are equalities, where the left hand side and the right hand side are identical or **equal**. It is also useful to have a way of expressing a **range of values** instead of a single specific value which can be achieved by using **inequalities**. A common way to display inequalities is using a number line.

**Inequality Symbols and their Meaning **

Before learning about how to **display inequalities** on a number line, you firstly need to know the following symbols and their definitions:

**> means “greater than”,****\geq means “greater than or equal to”,****< means “less than”,****\leq means “less than or equal to”.**

We call \leq and \geq **inclusive **inequalities, and we call < and > **strict** inequalities. For example x\leq 8 **includes** the value 8 as a possibility for x, whilst the inequality x<8 does not.

**Type 1: ****Inequalities on a Number Line**

The inequality, x\geq -5 means that x can take any value that is **bigger than** -5, including -5.

We can display this on a number line by drawing a filled in circle at -5 and an **arrow pointing** to the right hand side indicating the numbers that are greater than -5. This should look like,

When drawing inequalities on a number line it is important to remember that you should use an **closed circle**, \bullet, for an **inclusive inequality**, e.g. x\geq-5 and use a **open circle**, \circ, for a **st rict inequality**, e.g. x\lt2.

**Type 2: Inequalities on a Number Line**

Inequalities can describe a range of values between an upper and lower limit. For example the inequality -3 \leq x \lt 5 means that x can take any value greater than or equal to -3 but also has to be less than 5.

Here, the first part of the inequality is an **inclusive inequality** so is drawn with a filled in circle and the second part is a **strict inequality** so is drawn with a empty circle.

**Example:** Draw -3 \leq x \lt 5 on the number line below.

**Example 1**

Show the inequality -1 \lt x \lt 10 on a number line.

**[2 marks]**

**Key points when drawing the number line. **

- The circles indicate the limits of the inequality (-1 and 10).
- Both are strict inequalities they should be open circles.
- A connecting line between the two circles indicates the values x can take between the two limits.

Putting it all together the number line should look like,

**Example 2**

Display the inequalities x \leq 0 and x \geq 3 on a number line.

**[2 marks]**

This is similar to the previous example but this time,

- Both are inclusive inequalities so should be drawn with closed circles
- x can take any value less than or equal to 0 so an arrow should be drawn to the left hand side of 0
- x can take any value grater or equal to 3 so an arrow should be drawn to the right hand side of 3

### Take an Online Exam

#### Inequalities on a Number Line Online Exam

### Example Questions

**Question 1:** Display the inequality -1\geq x on a number line.

**[1 mark]**

The inequality, -1\geq x, will require a closed circle at 3 and an arrow pointing right.

**Question 2:** Display the inequality x \le 4 on a number line.

**[1 mark]**

The inequality, x \le 4, will require an closed circle at 4 and an arrow pointing left.

**Question 3:** Display the inequalities y>3 and y<-2 on the same number line.

**[2 marks]**

The first inequality, y>3, will require an open circle at 3 and an arrow pointing right.

The other inequality, y<-2, will require an open circle at -2 and an arrow pointing left.

**Question 4:** Display the inequality -1\leq x \leq8 on a number line.

**[2 marks]**

The lower bound, -1\leq x, will require a closed circle at x= -1

The upper bound x \leq8, will require an closed circle at x=8

**Question 5:** In order to be profitable, a bus company requires a certain number of passengers for each tour; however, the bus can only hold so many seated passengers.

Write an inequality for a bus, b, if the company requires more than 6 passengers to be profitable but can only hold 54 passengers and display this on a number line.

**[3 marks]**

Forming the correct inequality 6< b \leq 54 and displaying with an open circle for representing the strict inequality (6) and a closed circle representing the non-strict inequality (54).

### Worksheets and Exam Questions

#### (NEW) Inequalities On a Number Line Exam Style Questions - MME

Level 4-5 New Official MME### Drill Questions

#### Inequalities - Drill Questions

#### Algebra Inequalities - Drill Questions

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