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Weekly online one to one GCSE maths revision lessons now available Learn more In order to access this I need to be confident with: 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. What are inequalities on a number line?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. The solution set can represent all the real numbers shown within the range and these values can also be negative numbers. What are inequalities on a number line?How to represent inequalities on a number lineIn order to represent inequalities on a number line:
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. How to represent inequalities on a number lineInequalities on a number line worksheetGet 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 worksheetGet your free Inequalities on a number line worksheet of 20+ questions and answers. Includes reasoning and applied questions. DOWNLOAD FREE Related lessons on inequalitiesInequalities 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:
Inequalities on a number line examplesExample 1: single valuesRepresent x > 3 on a 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 . Example 2: single valuesRepresent −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 . Example 3: values within a rangeRepresent 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. Example 4: values within a rangeRepresent −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. Example 5: writing an inequality from a number lineWrite 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} Example 6: writing an inequality from a number lineWrite 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. Example 7: listing integer values in a solution setList 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. Example 8: listing integer values in a solution set from a number lineList 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. Common misconceptions
A common error is to confuse open circles and closed circles: Open circles do not include the value so require a ‘<’ sign.
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. But −5 is less than −1 as they are ordered −5 , −4, −3, −2, −1 , 0, 1, 2, 3 on a number line.
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’ .
Usually integer values are requested to be listed in a solution set. ‘0’ can sometimes be forgotten.
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. Practice inequalities on a number line questions5 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. 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.
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. -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. 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 . -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. -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. 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.
-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. 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. Inequalities on a number line GCSE questions1. John buys x bananas and y pears. He buys
One of the inequalities for this information is x\geq5 Write down two more inequalities for this information (2 marks) Show answer 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
(i) Show the inequality -3\leq{x}<2 on this number line. (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) Learning checklistYou have now learned how to:
Still stuck?Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors. Find out more about our GCSE maths revision programme. We use essential and non-essential cookies to improve the experience on our website. Please read our Cookies Policy for information on how we use cookies and how to manage or change your cookie settings.Accept What is an inequality on a number line?In math, an inequality shows the relationship between two values in an algebraic expression that are not equal. Inequality signs can indicate that one variable of the two sides of the inequality is greater than, greater than or equal to, less than, or less than or equal to another value.
How do you graph an inequality?To graph an inequality, treat the <, ≤, >, or ≥ sign as an = sign, and graph the equation. If the inequality is < or >, graph the equation as a dotted line. If the inequality is ≤ or ≥, graph the equation as a solid line.
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