Hello! In our last lesson, we successfully tackled systems of equations where a straight line meets a curve. You learned how to use substitution to combine a linear and a quadratic equation, a process that relies heavily on the algebraic manipulation skills you've been building.
Today, we'll shift our focus from equations, which typically have one or two distinct solutions, to linear inequalities. This lesson directly addresses the final outcome in our "Algebraic Foundations" module: to solve linear inequalities and represent their solutions on a number line. Instead of finding a specific value for x, we'll be finding a whole range of values that satisfy a certain condition. This concept is fundamental and has direct parallels to your interests in programming and AI, where conditions like if health < 0 or while loop_counter <= 10 determine the flow of logic.
1. From Equations to Inequalities
You are already very comfortable solving linear equations like . The process for solving a linear inequality like is almost identical. The main difference is that instead of a single answer, you get a range of answers.
The language of inequalities uses four key symbols. This table provides a great visual summary of what they mean and how they look on a number line.
Notice the circles on the number line graphs.
- An open circle (o) is used for "greater than" (>) and "less than" (<). It means the number itself is not included in the solution.
- A closed circle (•) is used for "greater than or equal to" (≥) and "less than or equal to" (≤). It means the number itself is included in the solution.
2. The Rules of Solving
Solving a linear inequality involves rearranging it to isolate the variable, just as you would with an equation. You can add, subtract, multiply, or divide both sides by the same number to maintain the balance.
However, there is one critical rule that is unique to inequalities.
How To Solve Linear Inequalities, Basic Introduction, Algebra
This short clip from The Organic Chemistry Tutor explains the single most important new rule you need to learn for solving inequalities.
Watch the section from this explanation of what happens when you multiply or divide by a negative number.
To summarize that key point: Whenever you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign.
For example:
- We know that .
- If we divide both sides by -2, we get and .
- Clearly, is less than , so we must flip the sign: .
Let's now watch a full walkthrough that puts everything together.
Solving Inequalities & Drawing Solutions on a Number Line | Higher & Foundation | GCSE Maths Tutor
This video from The GCSE Maths Tutor, a channel we've used before, provides a great introduction to solving inequalities and representing them on a number line.
First, watch the initial example from the beginning. The presenter gives a helpful mnemonic: if the inequality symbol has "more ink" (like ≥ or ≤), then the circle on the number line also has "more ink" (it's filled in). Next, watch the example with variables on both sides from this section. You'll see that the process is exactly the same as for equations: you start by eliminating the smaller of the variable terms.
3. Your Turn: Practice Solving
Now it's time to try some questions yourself, which I know you prefer. The following video segment presents four problems and then walks through the solutions.
Solving Inequalities & Drawing Solutions on a Number Line | Higher & Foundation | GCSE Maths Tutor
Pause the video as the questions appear and try to solve them. Then, watch the worked solutions to check your answers and method.
Attempt the four practice questions shown from this section. Pay close attention to your signs, especially with the negative numbers in the third question.
Solving inequalities can sometimes involve brackets or more complex arrangements. The following resource provides a wealth of examples. You don't need to read all of them, but looking at a few specific cases will be very helpful.
Linear Inequalities - GCSE Maths - Steps, Examples & Worksheet
This guide from Third Space Learning breaks down the process and shows many types of problems you might encounter.
First, quickly review the three steps for solving under the heading How to solve. Then, work through these two examples to see how to handle brackets and more complex setups: Example 6, which involves expanding brackets first. Example 7, which reinforces how to handle variables on both sides. Finally, read the list of common errors. Being aware of these will help you avoid simple mistakes.
4. Compound Inequalities
Sometimes, a variable is bounded between two values. This is called a compound inequality, and it looks something like . This is read as "x is greater than or equal to -4 AND less than 2."
Solving these is very similar; you just perform the same operation on all three parts of the inequality to isolate x in the middle.
Solving Inequalities & Drawing Solutions on a Number Line | Higher & Foundation | GCSE Maths Tutor
This part of the same video explains how to solve compound inequalities and tackle a common exam question: finding the integer values that satisfy the inequality.
Watch the segment from this timestamp. Notice how subtracting 3 and dividing by 2 is applied to the left, middle, and right sides of the inequality.
5. From Graph to Algebra
Finally, you also need to be able to work in reverse: given a number line representation, write down the algebraic inequality it represents.
Solving Inequalities & Drawing Solutions on a Number Line | Higher & Foundation | GCSE Maths Tutor
This last segment shows you how to interpret a number line diagram and write it in algebraic form.
Watch from this section to see how to construct the inequality statement from a given graph. Then, try the two practice questions presented at the end of the clip.
To fully consolidate your skills in both directions, here is a worksheet. It is excellent practice for your IGCSE exam. I highly recommend you try all the questions in the first two sections.
Conclusion
In this lesson, you've rounded out your foundational algebra skills by moving from equations to inequalities. You've learned that the solving process is very similar, but with the crucial rule of flipping the sign when multiplying or dividing by a negative. You are now also able to translate between the algebraic form of an inequality and its visual representation on a number line.
Key Takeaways:
- Solving linear inequalities follows the same principles as solving linear equations.
- The Golden Rule: When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign (e.g.,
<becomes>). - Solutions are represented on a number line using an open circle (o) for
<and>(value not included) and a closed circle (•) for≤and≥(value included). - Compound inequalities, like , are solved by applying inverse operations to all three parts simultaneously.
This lesson completes our first module, "Algebraic Foundations." You've built a solid toolkit for manipulating and solving various expressions. In our next lesson, we will begin the "Understanding Functions and Their Graphs" module. We'll start by exploring function notation, f(x), a concept that is the very cornerstone of the advanced mathematics used in game development, AI, and beyond.