Here we will learn about solving inequalities including how to solve linear inequalities, identify integers in the solution set and represent solutions on a number line.
There are also solving inequalities worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
Solving inequalities is where we calculate the values that an unknown variable can be in an inequality.
Solving inequalities is similar to solving equations, but where an equation has one unique solution, an inequality has a range of solutions.
To do this we need to balance the inequality in the same way as we would when solving an equation. Solutions can be integers or decimals, positive or negative numbers.
4 is the only solution to this equation.
x can be any value that is less than 4
This changes the direction of the inequality sign.
\[\beginx can be any value that is greater than -4
In order to solve inequalities:
Get your free solving inequalities worksheet of 20+ questions and answers. Includes reasoning and applied questions.
Get your free solving inequalities worksheet of 20+ questions and answers. Includes reasoning and applied questions.
Solving inequalities 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:
In this case you are subtracting ‘6’ from both sides.
2 Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.
In this case you need to divide both sides by 4 .
3 Write your solution with the inequality symbol.
Any value less than 5 satisfies the inequality
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
In this case you need to add ‘4’ to both sides.
\[\beginRearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.
In this case you need to divide both sides by 5 .
\[\beginWrite your solution with the inequality symbol.
Any value greater than or equal to 6 satisfies the inequality
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
Let’s start by expanding the bracket
Then we need to add ‘12’ to both sides.
\[\beginRearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.
In this case you need to divide both sides by 3 .
\[\beginWrite your solution with the inequality symbol.
Any value less than or equal to 8 satisfies the inequality
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
In this case you need to subtract ‘2x’ from both sides.
\[\beginRearrange the inequality so that ‘x’ s are on one side of the inequality sign and numbers on the other.
In this case you need to add ‘6’ to both sides.
\[\beginRearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.
In this case you need to divide both sides by 3 .
\[\beginWrite your solution with the inequality symbol.
Any value greater than 7 satisfies the inequality
Rearrange the inequality to eliminate the denominator.
In this case you need to multiply both sides by 5 .
\[\beginRearrange the inequality so that ‘x’ s are on one side of the inequality sign and numbers on the other.
In this case you need to subtract ‘3’ from both sides
\[\beginWrite your solution with the inequality symbol.
Any value less than 7 satisfies the inequality
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
In this case you need to subtract ‘1’ from both sides.
\[\beginRearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.
In this case you need to divide both sides by 6 .
\[\beginThis can be simplified to \frac or the decimal equivalent.
Write your solution with the inequality symbol.
Any value less than \frac satisfies the inequality
Represent the solution on a number line
Rearrange the inequality so that ‘x’ s are on one side of the inequality sign and numbers on the other.
In this case you need to add ‘7’ to both sides.
Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.
In this case you need to divide both sides by 2 .
Represent your solution on a number line.
Any value less than 6 satisfies the inequality. An open circle is required at 6 and the value lower than 6 indicated with an arrow.
Rearrange the inequality so that ‘x’ s are on one side of the inequality sign and numbers on the other.
In this case you need to subtract ‘1’ from both sides.
\[\beginRearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.
In this case you need to divide both sides by negative 2 .
\[\beginChange the direction of the inequality sign.
Because you divided by a negative number, you also need to change the direction of the inequality sign.
List the integer values that satisfy
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
In this case you need to subtract ‘1’ from each part.
List the integer values satisfied by the inequality.
2 is not included in the solution set. 7 is included in the solution set. The integers that satisfy this inequality are:
List the integer values that satisfy
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
In this case you need to divide each part by ‘4’ .
\[\beginList the integer values satisfied by the inequality.
\frac<7> \leq x \leq5\frac is included in the solution set but it is not an integer. The first integer higher is ‘2’ . 5 is also included in the solution set. The integers that satisfy this inequality are:
List the integer values that satisfy
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
In this case you need to subtract ‘5’ from each part.
Rearrange the inequality so that ‘x’ is isolated. In this case you need to divide each part by 2.
Represent the solution set on a the number line
-4 is not included in the solution set so requires an open circle. 1 is included in the solution set so requires a closed circle. Put a solid line between the circles to indicate all the values that satisfy the solution set.
Not including the inequality symbol in the solution is a common mistake. An inequality has a range of values that satisfy it rather than a unique solution so the inequality symbol is essential
Errors can be made with solving equations and inequalities by not applying inverse operations or not balancing the inequalities. Working should be shown step-by-step with the inverse operations applied to both sides of the inequality.
2. Solve 4x-3\geq25
3. Solve 2(x-5)\leq8
4. Solve 6x – 5 > 4x + 1
6. Solve 8x+1\geq3
7. Represent the solution on a number line 5x – 2 < 28
An open circle is required and all values less than 6 indicated.
8. Solve 2 – 3x > 14
Change the direction of the inequality sign as you have divided by a negative number
9. List the integer values that satisfy 2 -1 is not included in the solution set as is greater than -1 . 10. List the integer values that satisfy 4\leq3x\leq21 The first integer greater than \frac is 2 . 11. List the integer values that satisfy -4<3x+2\leq5 -2 is not included in the solution set as x is greater than -2 . 1. John’s solution to 2x + 5 > 17 is shown on the number line Is John’s solution correct? (2 marks) Correctly solves the inequality (1) No, correct solution is x > 6 Indicates ‘no’ with correct reason or represents correct inequality on the number line (1) 2. (a) Solve 4x+1\leq3x-2 (4 marks) (a) (1) Correct solution x\leq-3 (1) (b) (1) Correct inequality or their inequality shown on the number line with aa closed circle and values on the left side of the circle indicated with an arrow. (1) 3. (a) Solve 5x – 1 > 9 (3 marks) (a) (1) Correct solution: x > 2 (1) (b) (1) You have now learned how to: 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 tuition programme. © 2024 Third Space Learning. All rights reserved. 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 This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience. Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information. Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.
2 is included in the solution set as x is less than or equal to 2 .
7 is included in the solution set as x is less than or equal to 7 .
1 is included in the solution set as x is less than or equal to 1 .Solving inequalities GCSE questions
Explain your reasoning.
(b) Represent your solution to (a) on the number line
Correct attempt at solving, for example eliminating ‘x’ . x+1\leq-2
‘-3’ or their value indicated on the number line with a closed circle
(b) Write down the smallest integer that satisfies 5x – 1 > 9
Correct attempt at solving: 5x > 10
Correct solution: 3Learning checklist
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