Inequalities On a Number Line

Inequalities On a Number Line

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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.

Level 4-5 AQA Edexcel OCR WJEC

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.

Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

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,

Inequality On A Number Line Greater Than Or Equal To

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 strict inequality, e.g. x\lt2.

Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

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.

Inequalities On A Number Line Greater Than Or Equal To Less Than

Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC
Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

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,

Inequalities On A Number Line Greater Than Less Than

Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

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

Inequalities On A Number Line Less Than Or Equal To And Greater Than Or Equal To

Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

Example Questions

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

 

Inequalities On A Number Line Greater Than Or Equal To Example

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

 

Inequalities On A Number Line Less Than Or Equal To Example

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.

Inequalities On A Number Line Less Than Greater Than Example

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

Inequalities On A Number Line Greater Than Or Equal To Less Than Or Equal To Example

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).

 

Inequalities On A Number Line Less Than Or Equal To Greater Than Example

Worksheet and Example Questions

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

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Drill Questions

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Inequalities - Drill Questions

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Algebra Inequalities - Drill Questions

Level 4-5 GCSE

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