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Solving Inequalities
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.
What is solving inequalities?
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.
What is solving inequalities?
Solving an equation
Solve
4 is the only solution to this equation.
2 × 4 + 1 = 9Solving an inequality
Solve
x can be any value that is less than 4
Multiplying and dividing by a negative number
This changes the direction of the inequality sign.
E.g.
x can be any value that is greater than -4
How to solve inequalities
In order to solve inequalities:
- Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
- Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.
- Write your solution with the inequality symbol.
Explain how to solve inequalities
Solving inequalities worksheet
Get your free solving inequalities worksheet of 20+ questions and answers. Includes reasoning and applied questions.
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Solving inequalities worksheet
Get your free solving inequalities worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREERelated lessons on inequalities
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:
Solving linear inequalities examples
Example 1: solving linear inequalities
Solve
4x+6 < 26- Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
In this case you are subtracting ‘6’ from both sides.
2Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.
In this case you need to divide both sides by 4 .
3Write your solution with the inequality symbol.
x < 5Any value less than 5 satisfies the inequality
Example 2: solving linear inequalities
Solve
5x-4\geq26Rearrange 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.
Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.
In this case you need to divide both sides by 5 .
Write your solution with the inequality symbol.
Any value greater than or equal to 6 satisfies the inequality
Example 3: solving linear inequalities with brackets
Solve
3(x-4)\leq12Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
Let’s start by expanding the bracket
3x-12\leq12Then we need to add ‘12’ 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 3 .
Write your solution with the inequality symbol.
Any value less than or equal to 8 satisfies the inequality
Example 4: solving linear inequalities with unknowns on both sides
Solve
5x - 6 > 2x + 15Rearrange 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.
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 ‘6’ 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 3 .
Write your solution with the inequality symbol.
Any value greater than 7 satisfies the inequality
Example 5: solving linear inequalities with fractions
Solve
\frac{x+3}{5}<2Rearrange the inequality to eliminate the denominator.
In this case you need to multiply both sides by 5 .
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 ‘3’ from both sides
Write your solution with the inequality symbol.
Any value less than 7 satisfies the inequality
Example 6: solving linear inequalities – non integer solutions
Solve
6x+1\geq4Rearrange 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.
Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.
In this case you need to divide both sides by 6 .
This can be simplified to \frac{1}{2} or the decimal equivalent.
Write your solution with the inequality symbol.
Any value less than \frac{1}{2} satisfies the inequality
Example 7: solving linear inequalities and representing solutions on a number line
Represent the solution on a number line
2x - 7 < 5Rearrange 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.
Example 8: solving linear inequalities with negative x coefficients
Solve
1 - 2x <7Rearrange 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.
Rearrange 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 .
Change the direction of the inequality sign.
Because you divided by a negative number, you also need to change the direction of the inequality sign.
x > -3Example 9: solving linear inequalities and listing integer values that satisfy the inequality
List the integer values that satisfy
3<x+1\leq8Rearrange 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:
3, 4, 5, 6, 7Example 10: solving linear inequalities and listing integer values that satisfy the inequality
List the integer values that satisfy
7\leq4x\leq20Rearrange 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’ .
List the integer values satisfied by the inequality.
\frac{7}{4} 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:
2, 3, 4, 5Example 11: solving linear inequalities and representing the solution on a number line
List the integer values that satisfy
-3<2x+5\leq7Rearrange 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.
Common misconceptions
- Solutions as inequalities
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
E.g.
When solving x + 3 < 7 giving a solution of ‘4’ or ‘x = 4’ is incorrect, the answer must be written as an inequality ‘x < 4’
- Balancing inequalities
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.
E.g.
When solving x + 3 < 7 , adding ‘3’ to both sides rather than subtracting ‘3’ from both sides
Practice solving inequalities questions
1. Solve 3x+7 < 31
2. Solve 4x-3\geq25
3. Solve 2(x-5)\leq8
4. Solve 6x – 5 > 4x + 1
5. Solve \frac{x-4}{2}>6
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<x+3\leq5
-1 is not included in the solution set as is greater than -1 .
2 is included in the solution set as x is less than or equal to 2 .
10. List the integer values that satisfy 4\leq3x\leq21
The first integer greater than \frac{4}{3} is 2 .
7 is included in the solution set as x is less than or equal to 7 .
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 is included in the solution set as x is less than or equal to 1 .
Solving inequalities GCSE questions
1. John’s solution to 2x + 5 > 17 is shown on the number line
Is John’s solution correct?
Explain your reasoning.
(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
(b) Represent your solution to (a) on the number line
(4 marks)
(a)
Correct attempt at solving, for example eliminating ‘x’ . x+1\leq-2
(1)
Correct solution x\leq-3
(1)
(b)
‘-3’ or their value indicated on the number line with a closed circle
(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
(b) Write down the smallest integer that satisfies 5x – 1 > 9
(3 marks)
(a)
Correct attempt at solving: 5x > 10
(1)
Correct solution: x > 2
(1)
(b)
Correct solution: 3
(1)
Learning checklist
You have now learned how to:
- Solve inequalities
- Represent your solutions on a number line
- List integer values that satisfy your inequality
The next lessons are
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