# Inequalities: Understanding and Using Algebraic Techniques

1. A Level Maths Topics
2. Algebra Topics
3. Inequalities

Inequalities are a concept encountered frequently in this syllabus, and this article will provide an overview of them, including their definitions and the techniques that can be used to solve them. Through a combination of clear explanations and examples, readers will gain the confidence to identify and conquer inequalities in their own studies. The first step in understanding inequalities is to define them. An inequality is a mathematical statement that two expressions are not equal. For example, the equation ‘x + 5 > 10’ can be read as ‘x plus five is greater than ten’.

In this case, x could take any value that is greater than 5, such as 6 or 7, and the inequality would still hold true. When approaching inequalities, it is important to understand the different types of symbols used to represent them. The most commonly used symbols are ‘greater than’ (>) and ‘less than’ (<). These symbols are used to indicate which values make the inequality true or false. In this case, x could take any value that is greater than 5, such as 6 or 7, and the inequality would still hold true. It is also important to understand the different types of operations that can be used to solve inequalities.

These include addition, subtraction, multiplication and division. Each operation has a different effect on the value of an inequality, so it is important to understand how each one works in order to solve a given problem. For example, if we wanted to solve the equation ‘x + 5 > 10’, we would first need to subtract 5 from both sides of the equation. This would give us the following: x > 5 We can then use this result to determine which values make the inequality true or false.

In this case, any value greater than 5 will make the inequality true, while any value less than 5 will make it false. In addition to these operations, there are also several other techniques that can be used to solve inequalities. These include using graphs to visualise solutions, using linear equations and using substitution. Each technique has its own strengths and weaknesses, so it is important to understand when it is appropriate to use each one. Finally, it is important to understand how inequalities can be used in real-world scenarios. For example, they can be used to analyse data sets or predict outcomes in business scenarios.

They can also be used in engineering and science projects to test hypotheses or model behaviour. By understanding how inequalities work in these contexts, students will gain a greater appreciation of their importance and relevance in the real world.

## Using Linear Equations

Linear equations are a powerful tool for solving inequalities, as they can provide more complex solutions than those available through graphing. To use linear equations to solve an inequality, it is necessary to set up a system of equations with one variable. This variable can then be solved for using the techniques outlined above.

For example, if we have the inequality 3x + 4 < 7, we can set up a system of equations by subtracting 4 from both sides of the inequality. This gives us 3x < 3, which can be written as 3x - 3 = 0. We can now solve for x by dividing both sides of the equation by 3, giving us x = 1.We can then substitute this value back into the original inequality to check that it is correct. Solving inequalities with linear equations can be beneficial because it allows for more complex solutions than those available through graphing.

For instance, if the inequality involves higher order terms or irrational numbers, it can be difficult to graph accurately. However, linear equations can provide an exact solution that is easy to interpret.

## Using Substitution

When using the substitution technique to solve inequalities, we can replace one expression with another in order to simplify the problem. This can be especially useful when dealing with multiple variables.

For example, in the equation 3x + 4y = 12, we can substitute x with 2y. This gives us the new equation: 6y + 4y = 12, which can then be simplified to 10y = 12, and then further simplified to y = 1.2. This substitution technique can also be used when solving inequalities. For example, in the inequality 3x + 4y < 12, we can substitute x with 2y.

This gives us the new inequality: 6y + 4y < 12, which can then be simplified to 10y < 12, and then further simplified to y < 1.2. Substitution can be a great way to simplify inequalities and make them easier to solve. However, it is important to remember that when substituting one expression for another, the inequality sign must remain unchanged.

## Using Graphs To Visualise Solutions

Graphs can be used to visualise solutions for inequalities by plotting points on a graph and then drawing a line that represents the boundary between the two solutions.

This allows for a quick and easy way of determining which values make the inequality true or false. When using a graph to visualise a solution for an inequality, it is important to remember that the points to be plotted must be chosen in such a way that they represent all possible solutions. For example, if solving an inequality such as x+2 > 5, then the points that would need to be plotted would be x=3, x=4, x=5, and x=6.This is because these points represent all possible solutions for this inequality. Once the points have been plotted, the next step is to draw a line that represents the boundary between the two solutions. This can be done by connecting the points in order of increasing x-value.

The resulting line will represent all points that make the inequality true or false. For example, if the line drawn from the points x=3, x=4, x=5, and x=6 is y=x+2, then all values of x that are greater than 3 will make the inequality true, while all values of x that are less than 3 will make the inequality false. By understanding how graphs can be used to visualise solutions for inequalities, students can become more confident in their ability to identify and conquer inequalities in their own studies. Inequalities are an essential concept within A Level Maths Topics and Algebra Topics syllabus. By understanding how they work, and the techniques such as using graphs to visualise solutions, using linear equations, and using substitution that can be used to solve them, students will gain a better appreciation of their importance in mathematics and their relevance in real-world scenarios. With the right level of knowledge and confidence, inequalities can be easily identified and conquered in any study.